Time-Frequency Analysis

TIMEFREQUENCY ANALYSIS Leon Cohen Hunter College and Graduate Center of The City University of New York Prentice Hall PTR Upper Saddle River New Jersey 07458 Library of Congress CataloginginPublication Data Cohen Leon Timefrequency analysis I Leon Cohen p cm Includes bibliographical references and index ISBN 013594532I 1 Signal processing 2 Timeseries analysis 3 Frequency spectra I Title TK5102 9 C557 1995 621 382 23dc2G 9439843 CIP Editorialproduction supervision booksorks Manufacturing buyer Alexis R Heydt 1995 by PrenticeHall PIR A Pearson Education Company Upper Saddle River NJ 07458 The publisher offers discounts on this book when ordered in hulk quantities For more information contact Corporate Sales Department Prentice Hall PTR I Lake Street Upper Saddle River New Jersey 07694 Phone 8003823419 Fax 2012367141 Email corporatesalesprenhallcom All rights reserved No part of this book may he reproduced in any form or by any means without permission in writing from the publisher Printed in the United States of America 10 9 8 7 6 5 4 ISBN 0135945321 PrenticeHall International UK Limited London PrenticeHall of Australia Pty Limited Sydney PrenticeHall Canada Inc Toronto PrenticeHall Hispanoamericana SA Mexico PrenticeHall of India Private Limited New Delhi PrenticeHall of Japan Inc Tokyo Pearson Education Asia Pte Ltd Singapore Editoria PrenticeHall do Brasil Ltda Rio De Janeiro To Carol Valerie Ken Livia and Douglas Contents Preface Notation in Brief 1 The Time and Frequency Description of Signals 11 Introduction 1 12 Time Description of Signals 2 13 Frequency Description of Signals 6 14 Simple Calculation Tricks 8 15 Bandwidth Equation 15 16 AM and FM Contributions to the Bandwidth 17 17 Duration and Mean Time in Terms of the Spectrum 19 18 The Covariance of a Signal 20 19 The Fourier Transform of the Time and Frequency Densities 22 110 Nonadditivity of Spectral Properties 23 111 Classification of Signals 25 2 Instantaneous Frequency and the Complex Signal 21 Introduction 27 22 Reasons for the Complex Signal 28 23 The Analytic Signal 30 24 Calculating the Analytic Signal 31 25 Physical Interpretation of the Analytic Signal 35 26 The Quadrature Approximation 36 27 Instantaneous Frequency 39 28 Density of Instantaneous Frequency 41 3 The Uncertainty Principle 31 Introduction 44 32 The Uncertainty Principle 46 33 Proof of the Uncertainty Principle 47 34 The Uncertainty Principle for the ShortTime Fourier Transform 50 vii Viii Contents 4 Densities and Characteristic Functions 41 Introduction 53 42 One Dimensional Densities 53 43 One Dimensional Characteristic Functions 56 44 Two Dimensional Densities 59 45 Local Quantities 63 46 Relation Between Local and Global Averages 64 47 Distribution of a New Variable 65 48 Negative Densities 69 5 The Need for TimeFrequency Analysis 51 Introduction 70 52 Simple Analytic Examples 71 53 Real Signals 75 54 Why Spectra Change 80 6 TimeFrequency Distributions Fundamental Ideas 61 Introduction 82 62 Global Averages 84 63 Local Average 84 64 Time and Frequency Shift Invariance 85 65 Linear Scaling 86 66 Weak and Strong Finite Support 86 67 Uncertainty Principle 87 68 The Uncertainty Principle and Joint Distributions 88 69 Uncertainty Principle and Conditional Standard Deviation 90 610 The Basic Problems and Brief Historical Perspective 91 7 The ShortTime Fourier Transform 71 Introduction 93 72 The ShortTime Fourier Transform and Spectrogram 94 73 General Properties 74 Global Quantities 99 75 Local Averages 100 76 Narrowing and Broadening the Window 101 77 Group Delay 102 78 Examples 103 79 Inversion 108 710 Expansion in Instantaneous Frequency 109 711 Optimal Window 110 8 The Wigner Distribution 81 Introduction 113 82 The Wigner Distribution 114 83 General Properties 117 84 Global Averages 118 Contents 85 Local Averages 119 86 Examples 120 87 The Wigner Distribution of the Sum of Two Signals 124 88 Additional Properties 127 89 Pseudo Wigner Distribution 130 810 Modified Wigner Distributions and Positivity 132 811 Comparison of the Wigner Distribution with the Spectrogram 133 9 General Approach and the Kernel Method 91 Introduction 136 92 General Class 136 93 The Kernel Method 140 94 Basic Properties Related to the Kerrkl 141 95 Global Averages 146 96 Local Averages 147 97 Transformation Between Distributions 149 10 Characteristic Function Operator Method 101 Introduction 152 102 Characteristic Function Method 152 103 Evaluation of the Characteristic Function 154 104 The General Class 156 105 Averages 157 106 The Moment Method 158 11 Kernel Design for Reduced Interference 111 Introduction 162 112 Reduced Interference Distributions 162 113 Kernel Design for Product Kernels 165 114 Projection Onto Convex Sets 166 115 BaraniukJones Optimal Kernel Design 166 12 Some Distributions 121 Introduction 168 122 ChoiWilliams Method 168 123 ZhaoAtlasMarks Distribution 172 124 BornJordan Distribution 174 125 Complex Energy Spectrum 174 126 Running Spectrum 175 13 Further Developments 131 Introduction 178 132 Instantaneous Bandwidth 178 133 Multicomponent Signals 182 ix x Contents 134 Spatial SpatialFrequency Distributions 184 135 Delta Function Distribution for FM Signals 185 136 Gabor Representation and TimeFrequency Distributions 186 137 Expansion in Spectrograms 188 138 Spectrogram in Terms of Other Distributions 189 139 Singular Value Decomposition of Distributions 190 1310 Synthesis 191 1311 Random Signals 192 1312 Numerical Computation 193 1313 Signal Analysis and Quantum Mechanics 195 14 Positive Distributions Satisfying the Marginals 141 Introduction 198 142 Positive Distributions 198 143 The Method of Loughlin Pitton and Atlas 201 15 The Representation of Signals 151 Introduction 204 152 Orthogonal Expansion of Signals 204 153 Operator Algebra 209 154 Averages 213 155 The Uncertainty Principle for Arbitrary Variables 216 16 Density of a Single Variable 161 Introduction 219 162 Density of a Single Variable 219 163 Mean Values 222 164 Bandwidth 223 165 Arbitrary Starting Representation 224 17 Joint Representations for Arbitrary Variables 171 Introduction 225 172 Marginals 225 173 Characteristic Function Operator Method 225 174 Methods of Evaluation 226 175 General Class for Arbitrary Variables 229 176 Transformation Between Distributions 229 177 Local Autocorrelation 230 178 Instantaneous Values 231 179 Local Values for Arbitrary Variable Pairs 232 1710 The Covariance 233 1711 Generalization of the ShortTime Fourier Transform 234 1712 Unitary Transformation 235 1713 Inverse Frequency 238 1714 Appendix 240 Contents 18 Scale 181 Introduction 242 182 The Scale and Compression Operator 242 183 The Scale Eigenfunctions 244 184 The Scale Transform 245 185 Signals with High Scale Content 248 186 Scale Characteristic Function 249 187 Mean Scale and Bandwidth 250 188 Instantaneous Scale 251 189 Uncertainty Principle for Scale 251 1810 Frequency and Other Scaling 252 1811 Appendix 253 19 Joint Scale Representations 191 Introduction 255 192 Joint TuneScale Representations 255 193 General Class of TuneScale Representations 256 194 Joint FrequencyScale Representations 258 195 Joint Representations of Time Frequency and Scale 258 196 Appendix 260 Bibliography 263 Index 291 Preface Changing frequencies is one of the most primitive sensations since we are sur rounded by light of changing color by sounds of varying pitch and by many other phenomena whose periodicities change in time A sunset is dramatic because of the colors and the change in colors The aim of timefrequency analysis is to describe how the frequency or spectral content of a signal evolves and to develop the phys ical and mathematical ideas needed to understand what a timevarying spectrum is The attempt to represent a signal simultaneously in time and frequency is full of challenges both physical and mathematical and I hope this book conveys the fascination and elegance the subject holds for me My aim is to give a simple exposition of the physical ideas and origins motiva tions methods underpinnings and scaffolding of the field I have attempted to be clear on questions of what is known what is not known what is speculation and the limitations of our current knowledge I never understood or learned unless I saw simple examples illustrating the ideas introduced Naturally I assume everyone else is like me So whenever possible I illustrate each point with examples The basic ideas and methods that have been developed are readily understood by the uninitiated the book is selfcontained The mathematicsis elementary with the possible exception of the last few chapters There is an attitude these days that one should use from the beginning the most sophisticated mathematics The rea son generally given is that the sophisticated mathematics has to be learned even tually I have attempted to do everything with the simplest of mathematics and only use sophisticated methods when absolutely needed or when there is an over whelming advantage either from a manipulative point of view or a simplification of the physical ideas Timefrequency analysis spans many fields including engineering physics as tronomy chemistry geophysics biology medicine and mathematics I have strived for a minimum of jargon so that the book may be understandable to a wide audi ence I wish to express my appreciation to Carol Frishberg Pat Loughlin Jim Pitton and Ted Posch for reading the manuscript and making many valuable suggestions Leon Cohen New York xiii Notation in Brief The main notational conventions we use are as follows 1 Integrals All integrals without limits imply integration from minus oo to oc f t 2 Fourier Transform Pairs We use st to denote a signal and Sw its Fourier transform and normalize symmetrically Sw 2r 1 st e2tdt st f Sw etdw vf2 7r For other quantities there will be exceptions in regard to the factor of 2n due to historically accepted conventions We consistently use angular frequency w 3 Magnitude and Phase It is often advantageous to express a signal and its Fourier transform in terms of their respective amplitudes and phases The notation we use is st Ateivt Sw Bwdow and we use phase and spectral phase to denote cp and Ow respectively and amplitude and spectral amplitude to denote At and Bw 4 Functions We often use the variable to denote a function That is f x and f y may not necessarily be the same function the individuality of the functions being denoted by the variable x or y Where confusion may arise we will use different notation to emphasize the distinction 5 Averages Global and conditional averages are denoted by the following con ventions w w h 02 w2 w2 Qwlhw2hwh eg average weight eg average weight for a given height eg standard deviation of weight eg standard deviation of weight for a given height 6 Operators Symbols in calligraphic letters are operators For example the fre quency operator W and time operator T are W3dt Tj d AJ xv Chapter 1 The Time and Frequency Description of Signals 11 INTRODUCTION In this chapter we develop the basic ideas of time and frequency analysis from the perspective that the standard treatments already contain the seeds and motivations for the necessity of a combined timefrequency description Signal analysis is the study and characterization of the basic properties of signals and was historically developed concurrently with the discovery of the fundamental signals in nature such as the electric field sound wave and electric currents A signal is generally a function of many variables For example the electric field varies in both space and time Our main emphasis will be the time variation although the ideas developed are easily extended to spatial and other variables and we do so in the latter part of the book The time variation of a signal is fundamental because time is fundamental How ever if we want to gain more understanding it is often advantageous to study the signal in a different representation This is done by expanding the signal in a complete set of functions and from a mathematical point of view there are an infinite number of ways this can be done What makes a particular representation important is that the characteristics of the signal are understood better in that rep resentation because the representation is characterized by a physical quantity that is important in nature or for the situation at hand Besides time the most important representation is frequency The mathematics of the frequency representation was invented by Fourier whose main motivation was to find the equation governing the behavior of heat The contributions of Fourier were milestones because indeed he did find the fundamental equation governing heat and in addition he invented the remarkable mathematics to handle discontinuities 1807 He had to be able to I 2 Chap 1 The Time and Frequency Description of Signals handle discontinuities because in one of the most basic problems regarding heat namely when hot and cold objects are put in contact a discontinuity in temper ature arises Founers idea that a discontinuous function can be expressed as the sum of continuous functions an absurd idea on the face of it which the great scientists of that time including Laplace and Lagrange did not hesitate to call ab surd in nondiplomatic language turned out to be one of the great innovations of mathematics and science However the reason spectral analysis is one of the most powerful scientific methods ever discovered is due to the contributions of Bunsen and Kirchhoff about sixty years after Fourier presented his ideas 1807 and about 35 years after his death in 1830 Spectral analysis turned out to be much more impor tant than anyone in Fouriers time could have envisioned This came about with the invention of the spectroscope2 and with the discovery that by spectrally ana lyzing light we can determine the nature of matter that atoms and molecules are fingerprinted by the frequency spectrum of the light they emit This is the modern usage of spectral analysis Its discoverers Bunsen and Kirchhoff observed around 1865 that light spectra can be used for recognition detection and classification of substances because they are unique to each substance This idea along with its extension to other waveforms and the invention of the tools needed to carry out spectral decomposition certainly ranks as one of the most important discoveries in the history of mankind It could certainly be argued that the spectroscope and its variations are the most important scientific tools ever de vised The analysis of spectra has led to the discovery of the basic laws of nature and has allowed us to understand the composition and nature of substances on earth and in stars millions of light years away It would be appropriate to refer to spectral analysis as BunsenKirchhoff analysis 12 TIME DESCRIPTION OF SIGNALS Fundamental physical quantities such as the electromagnetic field pressure and voltage change in time and are called time waveforms or signals We shall denote a signal by s t In principle a signal can have any functional form and it is possible to produce signals such as sound waves with extraordinary richness and complexity Fortunately simple signals exist hence the motivation to study and characterize the simple cases first in order to build up ones understanding before tackling the more complicated ones Laplace and Lagrange werent thrilled about Founers theory of heat either However his ideas were eventually widely accepted in his own lifetime and he succeeded toLagranges chair Fourier was heavily involved in politics and had his ups and downs in that realm also At one time he accompanied Napoleon to Egypt and had a major impact in establishing the field of Egyptology The spectroscope was invented by Fraunhofer around 1815 for the measurement of the index of refraction of glasses Fraunhofer was one of the great telescope makers and realized that the accurate determination of the index of refraction is essential for building optical instruments of high quality In using the spectroscope for that purpose Fraunhofer discovered and catalogued spectral lines which have come to be known as the Fraunhofer lines However the full significance of spectral analysis as a finger print of elements and molecules was first understood by Bunsen and Kirchhoff some fifty years after the invention of the spectroscope Sec 2 Time Description of Signals 3 The simplest timevarying signal is the sinusoid It is a solution to many of the fundamental equations such as Maxwell equations and is common in nature It is characterized by a constant amplitude a and constant frequency wo st acoswot 11 We say that such a signal is of constant amplitude This does not mean that the signal is of constant value but that the maxima and minima of the oscillations are constant The frequency wo has a clear physical interpretation namely the number of oscillations or ups and downs per unit time One attempts to generalize the simplicity of the sinusoid by hoping that a gen eral signal can be written in the form st at cost9t 12 where the amplitude at and phase i9t are now arbitrary functions of time To emphasize that they generally change in time the phrases amplitude modulation and phase modulation are often used since the word modulation means change Difficulties arise immediately Nature does not break up a signal for us in terms of amplitude and phase Nature only gives us the lefthand side st Even if the signal were generated by a human by way of Eq 12 with specific amplitude and phase functions that particular at and t9t would not be special since there are an infinite number of ways of choosing different pairs of amplitudes and phases that generate the same signal Is there one pair that is special Also it is often advantageous to write a signal in complex form st At ejwt s js 13 and we want to take the actual signal at hand to be the real part of the complex sig nal How do we choose A and cp or equivalently how do we choose the imaginary part s It is important to realize that the phase and amplitude of the real signal are not generally the same as the phase and amplitude of the complex signal We have emphasized this by using different symbols for the phases and amplitudes in Eqs 12 and 13 How to unambiguously define amplitude and phase and how to define a com plex signal corresponding to a real signal will be the subject of the next chapter From the ideas and mathematics developed in this chapter we will see why defin ing a complex signal is advantageous and we will lay the groundwork to see how to do it In this chapter we consider complex signals but make no assumptions re garding the amplitude and phase Energy Density or Instantaneous Power How much energy a signal has and specif ically how much energy it takes to produce it is a central idea In the case of elec tromagnetic theory the electric energy density is the absolute square of the elec tric field and similarly for the magnetic field This was derived by Poynting using Maxwells equations and is known as Poyntings theorem In circuits the energy 4 Chap 1 The Time and Frequency Description of Signals density is proportional to the voltage squared For a sound wave it is the pressure squared Therefore the energy or intensity of a signal is generally I st 12 That is in a small interval of time At it takes I st J2ot amount of energy to produce the signal at that time Since I st I2 is the energy per unit time it may be appropriately called the energy density or the instantaneous power since power is the amount of work per unit time Therefore 1 st 12 energy or intensity per unit time at time t energy density or instantaneous power I st I2 At the fractional energy in the time interval At at time t Signal analysis has been extended to many diverse types of data including eco nomical and sociological It is certainly not obvious that in those cases we can mean ingfully talk about the energy density per unit time and take I st I2 to be its value However that is what is done by analogy which is appropriate if the results are fruitful Total Energy If I st I 2 is the energy per unit time then the total energy is obtained by summing or integrating over all time E f I st I2 dt 14 For signals with finite energy we can take without loss of generality the total en ergy to be equal to one For many signals the total energy is infinite For example a pure sine wave has infinite total energy which is reasonable since to keep on pro ducing it work must be expended continually Such cases can usually be handled without much difficulty by a limiting process Characterization of Time Wave Forms Averages Mean Time and Duration If we consider I st 12 as a density in time the average time can be defined in the usual way any average is defined t f 15 The reasons for defining an average are that it may give a gross characterization of the density and it may give an indication of where the density is concentrated Many measures can be used to ascertain whether the density is concentrated around the average the most common being the standard deviation art given by T2 of f ft t 2 18t I2 dt 16 t2 t2 17 Sec 2 Time Description of Signals 5 where t2 is defined similarly to t The standard deviation is an indication of the duration of the signal In a time 20 most of the signal will have gone by If the standard deviation is small then most of the signal is concentrated around the mean time and it will go by quickly which is an indication that we have a signal of short duration similarly for long duration It should be pointed out that there are signals for which the standard deviation is infinite although they may be finite energy signals That usually indicates that the signal is very long lasting The average of any function of time gt is obtained by 9t f 9t I st I2 dt 18 Note that for a complex signal time averages depend only on the amplitude Example 11 Gaussian Envelope Consider the following signal where the phase is arbitrary st a7r 114 etto22iwt 19 The mean time and duration are calculated to be t V tt02 dt to 7r 2 a 2 2 110 dt 2a to 111 t V t e o Hence a2 t2t22a 112 Example 12 Rectangular Amplitude A signal with constant amplitude from time t1 to t2 and zero otherwise is st ejt tl t t2 113 V7t2 tl The mean time and duration are P2 t 1 tdt z t2 tl r I t2 tl Jtl which gives t dt 114 3 t2 t2t1 ti 115 at t2 t1 116 For this case the signal unambiguously lasts t2 tj However 2ot is pretty close to the true duration and has the advantage that it applies to any signal 2 6 Chap 1 The Time and Frequency Description of Signals 13 FREQUENCY DESCRIPTION OF SIGNALS There are four main reasons for frequency analysis or spectral analysis First by spectrally analyzing a waveform we learn something about the source That is how we have learned about the composition of the stars paper blood and almost ev erything else Second the propagation of waves through a medium generally depends on fre quenc That is why visible light goes through glass but not through aluminum while Xrays go through aluminum but not so readily through glass The propa gation of a wave through a medium is quite complicated but the basic effect is that waves of different frequencies propagate with different velocities This is called dispersion because the earliest discovered manifestation was that a prism can dis perse white light into different colors The other important effect in propagation is the attenuation the dying out or absorption of a wave The amount of attenuation depends on the medium and the frequency In the case of sound in normal condi tions there is almost no attenuation which is why we are able to hear from far away In contrast high frequency electromagnetic waves are damped within a short dis tance of entering the surface of a conductor To study the propagation through fre quency dependent mediums we decompose the signal into its different frequency components do the analysis for each frequency component and then reconstruct the signal to obtain the resulting wave form Hence the need to decompose a signal into individual frequencies which is what Fourier analysis does The third reason for spectral decomposition is that it often simplifies our un derstanding of the waveform Simple sinusoids are common in nature which is consistent with the fact that for some of the fundamental equations of motion si nusoids are possible solutions So are sums of sinusoids if the equation governing the physical quantity is linear In general a signal is messy but often the mess is really the simple superposition of sine waves which is simpler to understand and characterize Finally Fourier analysis is a powerful mathematical tool for the solution of or dinary and partial differential equations Fourier Expansion The signal is expanded in terms of sinusoids of different fre quencies st 2 J Sw etdw 117 The waveform is made up of the addition linear superposition of the simple wave forms ejt each characterized by the frequency w and contributing a relative amount indicated by the coefficient Sw Sw is obtained from the signal by Sw J st a2t dt 118 and is called the spectrum or the Fourier transform Since Sw and st are uniquely Sec 3 Frequency Description of Signals 7 related we may think of the spectrum as the signal in the frequency domain or fre quency space or frequency representation Spectral Amplitude and Phase As with the signal it is often advantageous to write the spectrum in terms of its amplitude and phase Sw Bw eiG 119 We call Bw the spectral amplitude and 7w the spectral phase to differentiate them from the phase and amplitude of the signal Energy Density Spectrum In analogy with the time waveform we can take I Sw 12 to be the energy density per unit frequency Sw 12 energy or intensity per unit frequency at frequency w energy density spectrum Sw 12 Ow the fractional energy in the frequency interval w at frequency w That I Sw 12 is the energy density can be seen by considering the simple case of one component st Swo e310t characterized by the frequency wo Since the signal energy is I st12 then for this case the energy density is I Swo12 Since all the energy is in one frequency I Swo 12 must then be the energy for that frequency In Chapter 15 we consider arbitrary representations and discuss this issuein greater detail Also the fact that the total energy of the signal is given by the integration of I Sw 12 over all frequencies as discussed below is another indication that it is the density in frequency The total energy of the signal should be independent of the method used to calculate it Hence if the energy density per unit frequency is I Sw 12 the total energy should be the integral of I Sw 12 over all frequencies and should equalthe total energy of the signal calculated directly from the time waveform E J I st2 dt J I Sw I2 dL 120 This identity is commonly called Parcevals or Rayleighs theorem3 To prove it con sider 3The concept of the expansion of a function in a set of orthogonal functions started around the time of Laplace Legendre and Fourier However the full importance and development of the theory of orthogonal functions is due to Rayleigh some one hundred years later around 1890 8 Chap 1 The Time and Frequency Description of Signals E J st 12 it 121 27r fff Sw Sw ej t dw dw dt ff S w Sw 6w w dw dw 122 J 123 where in going from Eq 121 to 122 we have used 1 ejt dt 6w w 124 27r Mean Frequency Bandwidth and Frequency Averages If I Sw 12 represents the density in frequency then we can use it to calculate averages the motivation be ing the same as in the time domain namely that it gives a rough idea of the main characteristics of the spectral density The average frequency w and its standard deviation v commonly called the root mean square bandwidth and signified by B are given by w J I 125 B2c22 Jww2 ISwI2dw 126 w2 w2 and the average of any frequency function gw is 127 gw JgwISw12dw 128 What Does the Energy Density Spectrum Tell Us The energy density spectrum tells us which frequencies existed during the total duration of the signal It gives us no indication as to when these frequencies existed The mathematical and physical ideas needed to understand and describe how the frequencies are changing in time is the subject of this book 14 SIMPLE CALCULATION TRICKS Suppose we want to calculate the average frequency From the definition Eq 125 it appears that we first have to obtain the spectrum But that is not so There is an important method or trick that avoids the calculation of the spectrum simplifies the algebra immensely and moreover will be central to our development in the later chapters for deriving timefrequency representations In this chapter we merely Sec 4 Simple Calculation Tricks 9 discuss the method in the context of its significant calculational merit To calculate averages of frequency functions we do not have to calculate the Fourier transform Sw It can be done directly from the signal and done simply We first state the result for average frequency and average square frequency give a few examples to display the power of the method and then discuss the general case The average frequency and average square frequency are given by w fw I Sw Iz dw fst d st dt 129 z w2 fw2 I Sw Iz dw Jst j dt st dt 130 J s t d Zt st dt 131 dt st Idt 132 That is to calculate the average frequency we differentiate the waveform and carry out the integration as per Eq 129 For the average square frequency we have a choice We can calculate the second derivative and use Eq 131 or calculate just the first derivative and use Eq 132 Either way we get the same answer The bandwidth is given by Qz w fw w2 I Sw 12dw 133 r lz J st w J st dt 1d2 j dt w st dt 134 135 These results are far from obvious although they are easy to prove and we will do so shortly First a few examples Example 13 Quadratic Phase with Gaussian Envelope Take Now 3t an14 ect22jOt223 0t 3 dt st jat at wo J 9t 137 and therefore 10 Chap 1 The Time and Frequency Description of Signals 2 s t jut 3t wo st dt J jat fit wo et dt wo 138 Also which gives a 2 a2 q2 2a 140 For this case the spectrum and energy density spectrum may be obtained without difficulty Sw a e022j 141 7 jQ Sw I2 Q29 7ra2 Q2 and the averages may be calculated directly as a check 142 Example 14 Sinusoidal Modulation This example is chosen to impress the reader with the simplicity and power of the method Suppose we have the signal st and we want to calculate the average frequency and bandwidth 143 The hard way The reader is welcome to try to calculate the Fourier transform of st and then use it to calculate the average frequency and bandwidth The easy way Taking the derivative of the signal we have j dtst LjatQtrnwmCOSwmtwost 144 and therefore using Eq 129 w J s t dt s t dt 145 fjatQt rnwmCOSwmtwoeQ22dt 146 mwm eW24 wo 147 Sec 4 Simple Calculation Tricks For the average square frequency we immediately get using Eq 132 11 z at j3t jMW COsWt jwo 12 a dt 148 w2 7 This integral is easy to evaluate since all the terms in the integrand that are linear in time drop out due to symmetry All complex terms drop out also since we know that the answer has to be real therefore the complex terms must add up to zero and do not have to be calculated The remaining terms are simple integrals Evaluation leads to 2 2 w2 a 2a3 wp 2 m 1 eWJ which gives 149 2 2 2 2 2 012 a 2 Q M 2 1 e2 150 The Frequency Operator and the General Case For convenience one defines the frequency operator by 1d 3 dt 151 and it is understood that repeated use denoted by Wn is to mean repeated differ entiation Wnst 17 n dtn 3t 152 We are now in a position to state and prove the general result that the average of a frequency function can be calculated directly from the signal by way of 9w fgwSwi2dw 153 f st gW st dt 154 fstg4 st dt 155 In words Take the function gw and replace the ordinaryvariable w by the operator dt operate on the signal multiply by the complex conjugate signal and integrate Before we prove this we must face a side issue and discuss the meaning of gW for an arbitrary function If g is wn then the procedure is dear as indicated by Eq 152 If g is the sum of powers then it is also dear For a general function we first expand the function in a Taylor series and then substitute the operator W for w That is if gw E 97lwn then gW E gnW 156 12 Chap 1 The Time and Frequency Description of Signals To prove the general result Eq 154 we first prove it for the case of the average frequency w J 2 JJ J ws t st ejdw dt dt 157 2 JJJ st stat ejtt dt dt 158 f f s t 6t t st dt dt 159 Jsltstdt 160 These steps can be repeated as often as necessary to prove it for g w Hence w s t fl st dt f tW st dt 161 Having proved the general result for functions of the form g w we now prove it for an arbitrary function gw by expanding the function in a Taylor series 9w JgwSJ2dw f I 162 E gnf s t W st dt 163 fstWstdt 164 Manipulation Rules The frequency operator is a Hermitian operator which means that for any two signals s1t and s2t f si t W s2t dt f s2t W slt dt 165 This is readily proved by integrating by parts Also a real function gw of a Her mitian operator gW is also Hermitian That is I sit9W s2t f s2t 9 W sit if gw is real 166 Sec 4 Simple Calculation Tricks 13 An important property of Hermitian operators is that their average value as de fined by Eq 164 must be real so in the manipulation of averages we can simply discard the imaginary terms since we are assured that they add up to zero We now derive the second simplification Eq 132 for the average square of frequency We have w2 fstW2stdt fstWWstdt 167 J W st Wst dt 168 J I W st 12 dt 169 This is an immense simplification since not only do we not have to find the spec trum we also avoid a double differentiation The Time Operator In the above discussion we emphasized that we can avoid the necessity of calculating the spectrum for the calculation of averages of frequency functions Similarly if we have a spectrum and want to calculate time averages we can avoid the calculation of the signal The time operator is defined by 170 T i du and the same arguments and proofs as above lead to 9t f gt st12 dt f S w gT Sw dw 171 In particular t f t2 J t2 I st 12 dt f S w 1 d Sw dw 172 I d 2 dw Sw dw 173 174 2 Sw dw f S w I A2 S S dw 175 14 Chap 1 The Time and Frequency Description of Signals Example 15 Mean Time and Duration from the Spectrum Consider the normalized spectrum a2n1 Sw 2n wn e12jtow w 0 176 We want to calculate the mean time and duration directly from the spectrum We have TSw Sw to ja2 jnw Sw 177 and therefore 1 to ja2 jnw I Sw 12 dw to 178 In evaluating this we dont have to do any algebra The complex terms must be zero and hence need no evaluation Also since the spectrum is normalized the real part integrates to to Similarly t2 J to ja2 jnw 121 Sw12 dw to 1 a2 179 0 42n1 which gives T2 t2 t2 1 a2 4 2n 1 The signal corresponding to this spectrum is a2n1 1 st n 27r2n a2 jt to n1 and the above average may be checked directly Also for this signal w 2n 1 a 2n22n1 2 2n1 2 w a2 B a2 180 181 182 The Translation Operator Many results in signal analysis are easily derived by the use of the translation operator eTN where r is a constant Its effect on a function of time is eTwft ftT 183 That is the translation operator translates functions by z Note that it is not Hermi tian To prove Eq 183 consider eTWft 00 E jT f t E n d f t 184 n0 n0 But this is precisely the Taylor expansion of f t T and hence Eq 183 follows Similarly the operator aj9T translates frequency functions ei9T Sw Sw 0 185 Sec 5 Bandwidth Equation 15 The Operator Method We have shown how the properties of the time and fre quency operators can be used to simplify calculations Indeed it is a powerful method for that purpose of which hopefully by now the reader is convinced How ever the operator method is not only a calculational tool but one of fundamen tal significance which will be developed and discussed in the later chapters of this book 15 BANDWIDTH EQUATION We now aim at expressing the mean frequency and bandwidth in terms of the time waveform1111313531 The results obtained will give a strong motivation for time frequency analysis and for the introduction of the complex signal Mean Frequency Consider first Wst WAt et At ewt 186 j At st 187 At w fw jSw j2 dw fst d st dt 188 r cpt j t A2t dt 189 The second term is zero This can be seen in two ways First since that term is purely imaginary it must be zero for w to be real Alternatively we note that the integrand of the second term is a perfect differential that integrates to zero Hence w fcdt st 2 dt I pt A2t dt 190 This is an interesting and important result because it says that the average fre quency may be obtained by integrating something with the density over all time This something must be the instantaneous value of the quantity for which we are calculating the average In this case the something is the derivative of the phase which may be appropriately called the frequency at each time or the instantaneous frequency wzt wtt P t 191 Instantaneous frequency as an empirical phenomenon is experienced daily as changing colors changing pitch etc Whether or not the derivative of the phase meets our intuitive concept of instantaneous frequency is a central issue and is ad dressed in subsequent chapters In addition this brings up the question that if 16 Chap 1 The Time and Frequency Description of Signals instantaneous frequency is the derivative of the phase what phase are we to use According to this definition the instantaneous frequency of a real signal is zero which is clearly an absurd result The means to get around these difficulties are developed in the next chapter with the introduction of the complex signal which corresponds to the real signal Bandwidth Equation 1114411 Now consider the bandwidth B 2 0z U f wwz ISwI2dw 192 or fst z dt w st dt 193 r ld 2 J j dt w 8t dt 194 I 1 At 0t w 2 A2t dt 195 r B2 J z A2t dt J cpt wz A2tdt 1 While A2t can be canceled in the first term the expression as written is prefer able because it explicitly shows that the bandwidth is the average of two terms one depending on the amplitude and the other depending only on the phase What is the meaning of this formula What is the significance of the two terms The expla nation will come with the ideas of describing a signal jointly in time and frequency which is the topic of this book The same steps lead to wz fw2 I Sw 12 dw 197 f At2 A2t dt f cp12t A2t dt 198 Calculation Techniques These equations besides being inherently interesting and offering a challenge for their interpretation are very useful for practical calcula tions as the following example shows Example 16 Cubic and Quadratic Phase Modulation Consider st Y7r14 et227t33j8t2jot 199 Sec 6 AM and FM Contributions to the Bandwidth 17 The denvative of the phase is given by gyp wo 3t yt2 and hence w f pt I st12 dt fwo pt yt2 I st12 dt 2a wo 1100 For the average square frequency we note that AA at and therefore ft2 I st12 fwo Qt yt22 I st12 1101 23 z w 2 3Y2 4a2 yao 2a from which we obtain 2 U2 a2o2 72 2a 2a2 1102 1103 16 AM AND FM CONTRIBUTIONS TO THE BANDWIDTH135 145 What contributes to the bandwidth The bandwidth is an indication of the spread in frequencies for the duration of the signal If a sound is produced at 1000 Hz and increased in frequencies to 1200 Hz at more or less constant amplitude we expect that the spread in frequencies will be about 200 Hz and that is indeed the case However if we have a signal of constant frequency at 1000 Hz then we can also achieve the same bandwidth by making it a short duration signal or by varying the amplitude rapidly Therefore the bandwidth does not give us a good indication of whether the spread of frequencies is due to deviations from the average frequency or to fast amplitude change or a to combination of both We now develop a measure of these two qualitatively different contributions But first we illustrate with an example Examine the set of signals shown in Fig 11 They all have the same bandwidth but they are qualitatively different In a the variation in frequency from the mean is zero while the variation in amplitude is large For the signal d the variation in frequency is high but has low amplitude variation The signals a d are progressions from one extreme to the other These two contributions to the bandwidth are apparent in the bandwidth equation Eq 196 Since the first term averages an amplitude term over all time and the second term averages a phase dependent term it is natural to define the AM and FM contributions by BAM JA12 t dt BFM f t dt 1104 with B2 BA2 2 M BFM 1105 We also define the fractional contributions and the ratio of the two contributions by rFM BB rAM BB 1106 To gain some insight we consider a few examples 18 Chap 1 The Time and Frequency Description of Signals a Or 0 0 5 0 c O D 10 a t 5 50 0 50 Fig 11 The real part of the signal st a7r1a eate2jat22 for various val ues of a and 0 All signals have the same bandwidth There is a large amplitude modulation in a and a small amplitude modulation in d but large frequency de viations from the mean frequency The AM and FM contributions as defined by Eq 1104 are a measure of the two qualitatively different effects that are contributing to the bandwidth The AM contributions are 71 18 13 and 04 respectively as we progress from a to d The values for a3 are 1616 1 556 0005 04 and 00005 0127 and each pair gives a bandwidth of 4 Example 17 Arbitrary Amplitude with Constant Frequency Consider the signal st At ejot 1107 where the amplitude At is arbitrary Since the instantaneous frequency is a constant we immediately see that BFM is zero and all the contributions to the bandwidth come from the amplitude modulation TAM 1 i rFM 0 1108 This is reasonable because a constant frequency does not contribute to the bandwidth since the bandwidth indicates the spread in frequency Example 18 Linear Frequency Modulation Take st a9r14 eate2jOt22tjwOt 1109 Direct calculation yields that 0 B B 1 110 FM AM 2 a p 1 111 TAM a2 N2 rFM 2 a2 F Sec 7 Duration and Mean Time in Terms of the Spectrum 19 A large bandwidth can be achieved in two qualitatively different ways Since the instantaneous frequency is w wo fit we can get a large spread by waiting long enough We achieve this by making the duration long which is accomplished by tak ing a small a However we can also get a large bandwidth by making the duration very small that is by taking a large a Example 19 Sinusoidal FM As a further example consider a sinusoidal modulated frequency with a Gaussian am plitude modulation 8t ait14et22jmlmwmt3wot The amplitude and frequency modulation contributions are BAM 2 R m 1 Cw12n 1113 The instantaneous frequency is mw coswt wo and we see that we can indeed achieve a large bandwidth if we have a large modulation index m or large w since by making either one large we get a large instantaneous frequency change ifwe wait long enough As with the previous example we can achieve the same bandwidth by making the duration small Qualitatively we have two very different effects and the AM and FM expressions measure these two effects 17 DURATION AND MEAN TIME IN TERMS OF THE SPECTRUM We have written the bandwidth and mean frequency in terms of the phase and am plitude of the signal The identical derivations can be used to write the mean time and duration in terms of the amplitude and phase ofthe spectrum In particular t f VYw I Sw I2 d 1114 and T2 Ut J Bp2 B2w Caw f Ow t2 B2w dw Bw 1115 Examine Eq 1114 It says that if we average 0w over all frequencies we will get the average time Therefore we may consider iw to be the average time for a particular frequency This is called the group delay and we shall use the following notation for it t9w w 1116 In Section 16 we showed how amplitude and frequency modulation contribute to the bandwidth Similarly the amplitude and phase variations of the spectrum 20 Chap 1 The Time and Frequency Description of Signals contribute to the duration We define the spectral amplitude modulation SAM and spectral phase modulation SPM contributions to the duration by TSAM fB2dw TSPM fhw t2 B2w dw 1117 with T 2 TSAM TSPM 1118 Example 110 Linear Frequency Modulation Consider the spectrum of the signal 3t a7r14 et2 27At227ot 1119 which we have given in Eq 141 and which we rewrite here in terms of the spectral phase and amplitude Sw a7r14 expp Wo2 i3 2 1 1 120 va 2a2 32 2a2 02 j Therefore t 1 121 a w WO g a2 p2 Also 2 a 2 2 1 122 TSAM T SPM 2aa2 2 2a2 02 18 THE COVARIANCE OF A SIGNAL If we want to determine in a crude way whether there is a relationship between height and weight of a population we do it by calculating the covariance or the cor relation coefficient In the same way we want a measure of how time and instan taneous frequency are related To see this we introduce the concept of covariance or correlation for signals Consider the quantity t Pt ftot 18t 12 dt 1123 which may be thought of as the average of time multiplied by the instantaneous frequency Now if time and frequency have nothing to do with each other then we would expect t cpt to equal t t t w Therefore the excess of t pt over t w is a good measure of how time is correlated with instanta neous frequency This is precisely what is called the covariance for variables such as height and weight and we similarly define the covariance of a signal by Sec 8 The Covariance of a Signal 21 Covtw tcpt tw 1124 The correlation coefficient is the normalized covariance r Covtw 1125 QtOw The reason for defining the correlation coefficient in the standard considerations such as for height and weight is that it ranges from minus one to one and hence gives an absolute measure That is not the case here but nonetheless it does give a good indication of the relationship between time and frequency Covariance in Terms of the Spectrum Suppose we place ourselves in the frequency domain so that time is t9 and frequency is w It is reasonable to define the covariance by Covtw t9w tw 1126 with t9w f wOwI SwI2dw 1127 Are these two definitions Eq 1124 and Eq 1127 identical For them to be identical we must have f t cpt I st l2 dt Jw w I Sw 12 dW 1128 In fact this equation is true but not obvious It can be proven by brute force but a very simple proof is given in Section 154 It is an interesting identity because it connects the phases and amplitudes of the signal and spectrum When Is the Covariance Equal to Zero If the covariance is to be an indication of how instantaneous frequency and time are related then when the instantaneous frequency does not change the covariance should be zero That is indeed the case Consider st At et 1129 where the amplitude modulation is arbitrary Now tWt But since w wo we have f twoIAtI2dt wot 1130 t wt w 1 131 o and therefore the covariance and correlation coefficient are equal to zero Similarly if we have a spectrum of the form Sw Bweiwto then there is no correlation 22 Chap 1 The Time and Frequency Description of Signals between time and frequency In general Covt 0 r 0 for st At of or 1132 Sw Bw e3to 1133 Covariance of a Real Signal or Spectrum Since the phase of a real signal is zero the derivative is zero and hence the covariance of a real signal is always zero This result misrepresents the physical situation and is another reason for defining a complex signal as we discuss in the next chapter Similarly signals that are symmetric in time have real spectra and their covariance is zero Example 111 Chirp For the signal st aa14 et2Jpt22jwot 1134 the average time is zero and therefore the covariance is cove t cpt ft 3t WO at 12 dt p 1 t21 st 12 dt 1135 This is readily evaluated to give Cove 2 r A 2 1136 a 02 When p 0 the correlation coefficient goes to zero in conformity with our discussion that for a constant frequency signal time and frequency are not correlated As a 0 the correlation coefficient goes to 1 depending on the sign of Q That is for constant amplitude we have total correlation If f3 is positive the instantaneous frequency in creases with increasing time and that is what the value of 1 for r tells us There is high positive correlation in the sense that when time is large the value of the instan taneous frequency is also large For Q negative we have negative correlation which is also reasonable since as time is increasing the instantaneous frequency is decreasing For a oo we have the correlation coefficient going to zero which is also reason able because for that case we have a short duration signal and the frequencies bear no relation to the chirping but are simply due to the short duration 19 THE FOURIER TRANSFORM OF THE TIME AND FREQUENCY DENSITIES Both I st I2 and Sw I 2 are densities The Fourier transform of a density is called the characteristic function It is a powerful method for studying densities as we will see in the later chapters Here we present a simple way to calculate the charac teristic function of I st I2 and I Sw 12 and show the relationship to the translation Sec 10 Nonadditivity of Spectral Properties 23 operator The characteristic function for the energy density spectrum is RT J I Sw 12 ejTw dw r sit ejTwst dt 1137 But we know from Section 14 that ejw is the translation operator and therefore Rr fststrdt 1138 Since this function compares or correlates the signal at two different times it is com monly called the deterministic autocorrelation function Inversely we have I Sw 12 1 f Rr ajwr dr 1139 2a The generalization of this result to random signals is the WienerKhinchin theorem Similarly the characteristic function in the frequency domain is R9 f I st12 ejet dt f Sw eoTSw dc f Sw Swe dw 1140 and hence f RO Cite d9 1141 I st 12 27r 110 NONADDITIVITY OF SPECTRAL PROPERTIES Many of the conceptual difficulties associated with timefrequency analysis are a reflection of the basic properties of signals and spectra If these properties are un derstood in their basic form then the curiosities encountered later will not be so paradoxical The fundamental idea to appreciate and always keep in mind is that the frequency content is not additive Suppose we have a signal composed of two parts the spectrum will be the sum of the corresponding spectrum of each part of the signal S S1 S2 1142 However the energy density is not the sum of the energy densities of each part 1 S 12 I S1 5212 15112 15212 2 Re Si S2 0 IS112IS212 Thus the frequency content is not the sum of the frequency content of each signal One cannot think of frequency content of signals as somethingwe have in a bucket that we add to the frequency content or bucket of another signal The physical reason is that when we add two signals the waveforms may add and interfere in all sort of ways to give different weights to the original frequencies Mathematically 24 Chap 1 The Time and Frequency Description of Signals stF ISUI 0 A 0 t J W r 13 0 1s 0 13 0 13 0 A 15 0 15 0 Fig 12 The energy density spectrum for the sum of two signals each localized in time The signal is given by Eq 1145 where a ctiz W12 5 10 ar 1 t1 5 The values for t2 are 5 7 9 and 11 in ad Even for large time separations as in c and d the energy density spectrum is affected The density of frequency is not just the sum of the frequencies of each part The implications for timefrequency analysis are explained in the next figure 15 this is reflected by the fact that the energy density spectrum is the absolute square of the sum of the spectra which results in nonlinear effects How the intensities change is taken into account by Eq 1143 Even if the signals are well localized and greatly separated in time we still cannot add frequency content For example consider two signals each lasting a second but one happening today and the other a million years from now Suppose we add the two signals Because of the large separation in time we have the intuitive sense that physical properties should somehow also be separable and simply add but that is not the case Even though the two signals have in some sense nothing to do with each other since they are greatly separated in time nonetheless the sum of the two signals produces an energy density spectrum that is not simply the sum of the two energy density spectra Let us take a particular example st Al e01ttl22jwittl A2 eQ2tt2 2 27W2tt2 1145 If we take eel and a2 to be large in relation to It2 tt I the two parts will be well separated in time A few typical situations are illustrated in Fig 12 where the en ergy density spectrum is also shown The energy density spectrum changes with signal separation even when the separation is large This is an important consider ation since it leads to seemingly paradoxical results in the distribution of intensity in the timefrequency plane Even though we have not begun our discussion of timefrequency distributions it is important to understand this nonadditivity prop erty Suppose we have two signals each concentrated around the timefrequency points w1 tl and w2i t2 and where these two points are well separated in both time and frequency If we add these signals we may think that since they are well sepa rated in time and frequency the resulting timefrequency density would be the sum of the same two dumps However that cannot be the case since it does not take into account the term Si S2 S1 S2 in the energy density spectrum This is illustrated in Fig 13 13 N Sec 11 Classification of Signals t2w2 tz w2 ti wi 0 c Fig 13 In a we have a signal that is localized at the timefrequency point t2 w2 and in b we have another signal localized at tlwl Since ti wi and t2 W2 are separated and the densities localized a seemingly plausible representation of the energy density in the timefrequency plane for the sum of the two signals of a and b is shown in c However this cannot be correct because that would imply that the energy density spectrum is the sum of the density spectra of each signal which is not the case as discussed in the previous figure 111 CLASSIFICATION OF SIGNALS 25 The types of signals found in nature vary greatly and a rich terminology has arisen to characterize them in broad terms If a signal does not change in some sense then one says it is stationary otherwise it is nonstationary If a signal lasts a short time it is generally called a transient burst or wave packet Short is a relative term that may mean a million years for astronomical signals or a billionth of a second in the case of atomic physics If the signal is explicitly known we say we have a deterministic signal Very often because of our ignorance or because the physical process producing the sig nal is governed by random events we have many possible signals in which case we say we have a collection or ensemble of signals a random signal or a stochastic signal A particular signal of the collection is said to be a realization For example if we produce sinusoids where the frequency is determined by some random event then we would have a random signal The spectral content is sometimes used to classify signals Signals whose spec tra are concentrated in a small band relative to the mean frequency are called nar row band otherwise they are called broadband However these classifications are crude For signals whose spectrum is changing they do not give a true sense of what is going on and can be misleading We have already seen for example that the bandwidth can be caused by two physically different mechanisms For exam 26 Chap 1 The Time and Frequency Description of Signals ple if a signal varies from 100 to 5000 Hz in 10 seconds in a pure and steady way then to classify it as broadband does not present a complete picture since at each time there may have been only one frequency On the other hand we can produce a signal that has the same bandwidth where indeed at each time there is a broad range of frequencies Timefrequency analysis enables us to classify signals with a considerably greater reflection of the physical situation than can be achieved by the spectrum alone Chapter 2 Instantaneous Frequency and the Complex Signal 21 INTRODUCTION Signals in nature are real Nevertheless it is often advantageous to define a complex signal that in some sense or other corresponds to the real signal In this chapter we describe the motivations for seeking a complex signal representation and its relation to the concept of instantaneous frequency We have seen in Chapter 1 that it is natural to define instantaneous frequency as the derivative of the phase because its average over time is the average frequency Thus far we have left open the question of how to get the phase One of the motives for defining the complex signal is that it will allow us to define the phase from which we can obtain the instantaneous frequency We seek a complex signal zt whose real part is the real signal srt and whose imaginary part slit is our choice chosen to achieve a sensible physical and mathematical description zt Sr jsi At eavt 21 If we can fix the imaginary part we can then unambiguously define the amplitude and phase by At sr s apt arctansrsi 22 which gives sisr srsi A2 23 27 28 Chap 2 Instantaneous Frequency and the Complex Signal for the instantaneous frequency The issue is then how to define the imaginary part Interest in the proper defini tion of instantaneous frequency first arose with the advent of frequency modulation for radio transmission in the 1920s Historically there have been two methods the quadrature method and the analytic signal method Before the introduction of the analytic signal by Gabor the main idea for forming a complex signal was the sense that for a signal of the form st At cos cot the complex counterpart should simply be At pt That begs the question because it requires first writing the signal in the form At cos cot and there are an infinite number of ways that can be done although in some situations it is intuitively obvious This idea is called the quadrature procedure and is discussed in Section 26 In 1946 the fundamental issues were crystallized by Gabor with the introduction of the analytic Signal 210 As we will see the analytic signal procedure devised by Gabor results in a complex signal that has a spectrum identical to that of the real signal for positive frequencies and zero for the negative frequencies Because of this fact there has been a tendency in the recent past to introduce the analytic signal by merely saying that the negative frequencies do not exist anyway so lets get rid of them However lets drop the negative frequencies is neither the historical nor the physical rea son for seeking a complex signal The reasons for doing so are that a complex signal offers a way to overcome difficulties that arise when considering only real signals Similarly there has been a tendency lately to define instantaneous frequency as the derivative of the phase of the analytic signal However instantaneous frequency is a primitive concept and not a question of mere mathematical definition The issue is whether any particular idea or definition does indeed match our intuitive sense and adequately represents that concept and whether it leads to further fruitful ideas As we will see the derivative of the phase of the analytic signal does meet our intuitive sense of instantaneous frequency for many cases but also produces many counter intuitive consequences That is all right because it is the counterintuitive situations that test the ideas Alternatively if an idea works well in some cases but apparently not in others then perhaps it is our interpretation of these apparently curious cases that is wanting In fact it will turn out that timefrequency analysis offers a frame work which explains many of the curiosities and difficulties We point out some of the difficulties with instantaneous frequency in this chapter One should keep an open mind regarding the proper definition of the complex signal that is the ap propriate way to define phase amplitude and instantaneous frequency Probably the last word on the subject has not yet been said 22 REASONS FOR THE COMPLEX SIGNAL First and most importantly for a real signal the spectrum satisfies Sw S w and therefore the energy density spectrum I Sw 12 is always symmetric about the origin Fig 21 symbolically draws such a density which is a perfectly good density Because of the symmetry the average frequency will always come out to be zero That is not what we want because it does not give us a sense of what is really going Sec 2 Reasons for the Complex Signal 29 2 vw 2 or b ISwI2 w4 wl 0 Fig 21 A real signal has a symmetrical density in frequency as illustrated in a Therefore the average frequency is zero and the spread in frequency is roughly the distance between the two bumps as illustrated in a Neither of these results is in dicative of the physical situation The analytic signal is defined to give the identical spectrum for the positive frequencies and zero for the negative frequencies as in b This results in an average frequency and bandwidth which better reflect the physical situation in that the average frequency falls somewhere in the middle of the bump and the bandwidth is the spread of the bump on We want the answer to come out to be somewhere in the middle of the right hand bump Also the spread in frequencies will be roughly the distance between the two bumps while what we want is the spread of one bump What can we do to obtain a value for average frequency that is roughly centered in the middle of the right hand bump We can achieve this by simply neglecting the left bump in the averaging w f00 wISw12dw 24 There are now two approaches we can take First we can continue to consider real signals and when taking spectral averages integrate from zero to infinity rather than oo to oo Or we can define a new signal that has the same spectrum for the pos itive frequencies and a zero spectrum for the negative frequencies The advantage of the second approach is that we can calculate frequency averages directly from the signal and therefore it is advantageous to have the signal once and for all In particular the new signal zt as yet unknown will assure that w f fzt4ztdt zt 25 The second reason for wanting to form a complex signal is that it will allow us to obtain the phase and amplitude of a signal unambiguously and that allows us to obtain an expression for instantaneous frequency 30 Chap 2 Instantaneous Frequency and the Complex Signal 23 THE ANALYTIC SIGNAL If the real signal st has the spectrum Sw then the complex signal zt whose spectrum is composed of the positive frequencies of Sw only is given by the in verse transform of Sw where the integration goes only over the positive frequen cies zt 2 1 f Sw ei t dt 27r o j zt 2 I st a et d dw The factor of 2 is inserted so that the real part of the analytic signal will be st otherwise it would be one half of that We now obtain the explicit form for zt in terms of the real signal st Since 00 Sw 2 J st ejt dt r 0 we have 1 r r st dt dw o f and using we obtain yielding 26 210 zt 1 18t1 17r 6t t t t dt 211 As zt st 2 J tt dt 212 We use the notation As to denote the analytic signal corresponding to the signal s The reason for the name analytic is that these types of complex functions satisfy the CauchyRiemann conditions for differentiability and have been traditionally called analytic functions The second part of Eq 212 is the Hilbert transform of the signal and there are two conventions to denote the Hilbert transform of a function st and Hst IIn a series of seminal papers Vakman 1W 549 5511 has addressed the concepts of instantaneous fre quency and the analytic signal and has brought forth the fundamental issues regarding these subjects The classical review article on the subject is by Vakman and Vainshtein15501 Sec 4 Calculating the Analytic Signal 31 Hst st fsctdt 213 The integration in the integral implies taking the principle part The Analytic Signal of a Complex Signal We have emphasized the physical mo tivation for defining a complex signal that corresponds to a real signal However there is no reason why st in Eq 212 cannot be any function real or complex For manipulative purposes it is important to allow for that possibility Energy of the Analytic Signal Because we have insisted that the real part of the complex signal be the original signal normalization is not preserved Recall that the spectrum of the original real signal satisfies I Sw I I Sw I and therefore the energy of the original signal is E f I Sw I2 dw 2f I Sw I2 dw 1 J 12Sw I2 dw 2Ez 214 0 2 0 That is the energy of the analytic signal is twice the energy of the original signal In addition the energy of the real part is equal to the energy of the imaginary part Ea EHfa 215 which can be seen by considering I zt 12 I st jHs 12 When this is expanded the middle term is s t st st 87 t dt dt 0 ff t t since the integrand is a two dimensional odd function 24 CALCULATING THE ANALYTIC SIGNAL 216 It might seem reasonable to now study the properties of the Hilbert transform so that we may better understand the properties of the analytic signal However that is not the case It is easier to study the analytic signal directly than to first develop theorems and results for the Hilbert transform The main point to keep in mind is that the analytic signal is formed from the positive part of the spectrum of the real signal and multiplied by two This will carry us fax Consider ejt whose spectrum is a delta function at w If w is negative then there is no positive frequency to retain and the answer is zero If it is positive then we just multiply by two Therefore A e3t 0 if w 0 2 eit if 0 217 32 Chap 2 Instantaneous Frequency and the Complex Signal This simple result is very important because if we can express the signal in terms of exponentials then all we have to do to form the analytic signal is drop the terms with negative frequency and multiply the positive frequency terms in the expansion by two A few examples will make this clear Example 21 The Analytic Signal of cos I w I t Write cos Iwt in terms of exponentials and then use Eq 217 ZA ei IWl t eilWlt 5A eIWIt j ZAej1 A eIWlt I e11t Similarly Asin IwItj 2j AeIWlt eiHWltj jeIWlt Example 22 The Analytic Signal of st coswlt coswlt For definiteness we take 0 wl w2 Rewrite st in terms of exponentials st coswlt cos w2t 4 eW2t ejW2t eWlt ejWlt 4 eW2W1t e9W2Wlt e9W2W1t eW2Wlt 218 219 220 221 222 223 224 225 The last two terms have negative frequencies and hence only the first two terms re main giving 11 zt 24 eiW2W1t e3W2W1t 2 eyWlt e7lt e72t 226 coswlte 2t 227 Notice that the analytic signal procedure chooses the higher frequency for the in stantaneous frequency The Analytic Signal of an Analytic Signal If we start with an analytic signal then its spectrum is nonzero only on the positive frequency axis and hence there is noth ing to drop Therefore we get back the same signal except for the factor of two Al zt i 2zt if zt is analytic 228 Analytic Signal of the Derivative Suppose we want the analytic signal of the derivative of a function If we have the analytic signal of the function then all we Sec 4 Calculating the Analytic Signal 33 have to do is differentiate it because the analytic signal of the derivative of a func tion is the derivative of the analytic signal More generally the analytic signal of the nth derivative of a function is the nth derivative of its analytic signal To see this consider dtis A dtn 229 AL1 J jwn Sw et dw J 230 00 27r 2 1 fjwr Sw 231 2 A Sw et do 1 TO dtn 271 0 2 1 Sw ejwt J dtn 27r fOCQ But this is just the nth derivative of the analytic signal of st Therefore 1dns do A dtn dtnAs 232 233 We note that in going from Eq 230 to Eq 231 we have used the fact the ana lytic signal operator is linear and hence can be taken inside the integration because integration is a sum Convolution Suppose we have a signal st whose spectrum is Sw and we form a new signal by way of zt 2 1 J Fw Sw et do 234 27r o where Fw is any function It is dear that zt is analytic since we have made it so by construction By specializing the function Fw many important results can be obtained In particular if we consider Fw to be the spectrum of a function f t and recall that the product of two spectra is the convolution of the two signals then we have in general that f Af st f t t dt is analytic for arbitrary s and f 235 That is the convolution of an analytic signal with an arbitrary function results in an analytic signal Imposed Modulation If we have a signal that is bandlimited between two frequen cies it is often necessary to shift the frequencies to higher values for the purpose of transmission For example a speech signal which is limited to a few thousand 34 Chap 2 Instantaneous Frequency and the Complex Signal hertz is raised to the megahertz range because we can build transmitters and re ceivers at those frequencies and because electromagnetic waves propagate well in our atmosphere at those frequencies What is done is to multiply the original wave form by e0 with wo positive to form a new signal snewt 3t et wo 0 236 In such a case one says that e3t is the imposed modulation and wo is the carrier frequency The spectrum of the new signal is the spectrum of the old signal shifted by an amount wo Sneww Sw wo 237 When will the resulting signal be analytic If the spectrum of the original signal extends from minus infinity to infinity then the spectrum Sne will likewise extend from minus infinity to infinity and not be analytic But suppose the spectrum Sw is single sided which means that it is zero below a certain value Then if wo is greater than that value the spectrum of the new signal will be shifted to the positive part of the frequency axis and the signal will be analytic Therefore st e30t is analytic if the spectrum of st is zero for w WO 238 Note that we have not specified whether st the original signal is real or complex It can be complex If it is real then this result implies that it must be bandlimited to the interval wowo Analytic Signal of the Sum of Two Signals Suppose we have a signal and want to add to it another signal so that the resulting sum will be analytic All we have to do is choose a signal whose spectrum for negative frequencies is identical to the first signal but differs in sign That way the spectrum of the resulting signal will be zero for negative frequencies Therefore s s1 82 is analytic if Sl w S2w for w 0 239 Factorization Theorem An important situation is when we want to find the an alytic signal of the product of two signals Let us first address when one of the functions can be factored out That is when does A s1 S21 s1A s21 If the spec trum of sl is nonzero for all frequencies then there is nothing we can do to raise its spectrum to be only positive The only possibility for a factorization theorem is if the spectrum of s1 is zero below some frequency Let us say that frequency is w1 Therefore A S21 must raise the spectrum of s1 so that it is on the positive side of the frequency axis Think of the spectrum of s1 as a sum of exponentials The worst case is the lowest frequency e31 Now A S21 is analytic and again think of it as a sum of exponentials If there is a frequency below wl then the product of this Sec 5 Physical Interpretation of the Analytic Signal 35 exponential with the worst case of si will result in a negative frequency Therefore s2 cannot have any frequencies below wl and hence A 81821 s1A 821 if the spectrum of s1 is zero below the value w1 and s2 is any signal whose analytic function has a spectrum that is zero below w1 240 Notice that the spectrum of s2 does not have to be zero below w1 it is the spectrum of A 521 that has to be zero below w1 If 52 is real then the condition is that the spectrum of s2 must be zero outside the range wi wi These results are due to Bedrosianl641 and Nuttall1402 Product of Two Analytic Signals If si and 82 are both analytic then they are both single sided and we have A81521 s1A s2 2s132 sl and 82 analytic 241 Real Signals Suppose si and 82 are real Their spectra are then symmetric Since the spectrum of Si vanishes below wl it must vanish above wi Therefore the spectrum of s1 must be zero for Iw wl Similarly the spectrum of s2 must vanish for w i wi Hence A sis2 s1A s2 for real signal if spectrum of si is zero for jwj wi 242 and the spectrum of 82 is zero for jw1 w1 25 PHYSICAL INTERPRETATION OF THE ANALYTIC SIGNAL Since the analytic signal is complex it can always be put into polar form A st At eiwt 243 We now ask what the analytic signal procedure has done in terms of choosing the particular amplitude and phase that is what is special about the amplitude and phase to make it an analytic signal Generally speaking the answer is that the spectral content of the amplitude is lower than the spectral content of eiwt We illustrate this first with a simple case where st At c3t 244 Call the spectrum of the amplitude SAw SAW 27r r At eit dt 245 36 Chap 2 Instantaneous Frequency and the Complex Signal The spectrum of st is then SAW wo For SAW wo to be analytic that is zero for negative frequencies SA w it must be zero w wo Therefore At e3 0t is analytic if the spectrum of At is contained within wo w0 246 Hence all the low frequencies are in the amplitude and the high frequency is in the cosine One can generalize this result in the following way Call Se w the spectrum of ejet SW 1 eawt 247 27r The spectrum of At ejet is then SW 12f SAwwSwwdw 248 vlr We can consider this the sum of shifted spectra of At with coefficients Sw Now suppose SAw is bandlimited in the interval w1 wi A sufficient condition to shift Sw to the positive axis is if the lowest value of the range of w is greater than wl That is S w is zero for values less than wl At e7St is analytic if the spectrum of At is contained in wl wl and the spectrum of e3et is zero for w wl 249 Therefore what the analytic procedure does at least for signals that result in the above forms is to put the low frequency content in the amplitude and the high frequency content in the term ejet 26 THE QUADRATURE APPROXIMATION It seems natural that if we write a signal in the form st At cos cpt the com plex signal ought to be sqt At e for st At coscpt 250 at least for some situations The sqt thus formed is called the quadrature model signal This idea was used before the introduction of the analytic signal As we mentioned in the introduction it begs the question since the procedure does not tell us how to write the signal in the form At cos W t to start with Nonetheless in many situations we may think we know the phase and amplitude Or we may have an At and apt and want to construct a signal and ask whether or not it is analytic More importantly since calculating the analytic signal is difficult if we Sec 6 The Quadrature Approximation 37 can approximate it with the quadrature model a considerable simplification would be achieved We therefore want to know when the quadrature model will agree with the analytic signal If the spectrum of the quadrature model is on the positive frequency axis only then we have total agreement In general that will not be the case and we want a measure of the error Energy Criteria The less of the spectrum of the quadrature signal there is on the negative frequency axis the better is the agreement To examine this question more precisely we define the following three signals st At cos cpt real signal 251 sqt At e2mt quadrature model 252 sat 2 f Sw et dw analytic signal 253 27r o where Sw is the spectrum of the real signal Sw J At cos apt a3t dt 254 Using cos W j sin cp the spectrum of the quadrature signal is seen to be Sqw Sw 2 J At sin pt at dt 255 Take the complex conjugate of this equation let w w and add to obtain 2 Sw Sqw SQ w 256 where we have used the fact that Sw Sw since st is real Also the spec trum of the analytic signal is Saw 0 ifw0 2 Sw Sq w Sq w if w 0 257 As a criterion on closeness between the quadrature signal and analytic signal Nuttall02 considered the energy of the difference between the two signals of f I 258 f o 10Sqw12dw f ISawSqw12dw 259 00 0 fpsqw2thjSwi2th o 100 260 38 Chap 2 Instantaneous Frequency and the Complex Signal The two terms are equal and therefore 0 DE2J 00 1Sqw12d 261 which is twice the energy of the quadrature model in the negative part of the spec trum Point by Point Comparison A stronger condition which compares sq and S at each time was given by Vakman and Vainshtein550 Using Eq 257 consider sat sqt 1 f Sawedt dw f Sqw ejt dw 262 27r 1 J 0 Sqw e t dw 1 1 00 S eJt dw 263 27r 27r 00 1 r 2qr Jcc Sq w ajt Sq w t I dw 264 This compares the signal at every time If we take absolute values then I sa t sq t I 1 I 2qrf Sq w ajt Sq w t dw 00 1 265 7 00 I Sq w et Sq w e3 I dw 266 2 and therefore sat sqt 2 f Sqw I 267 27r This gives the absolute possible deviation for any time Imposed Amplitude and Phase Modulation Often we have a carrier and impose an amplitude and phase At and cpt to produce the complex signal st At ejtiwt 268 The spectrum of At ejwt is shifted upward by an amount w0 and therefore using Eq 261 r0 DE 2 J 1 Sq w w0 1 2 dw 269 00 2ISqw12dw f 270 Note that Sqw is still the spectrum of At ejet Sec 7 Instantaneous Frequency Example 23 Chirp Take the real signal The quadrature model is 39 st eat22 coswot f3t22 271 39t a7r14 e02 ejot9Rt22 272 How does this compare with the analytic signal of st Using Eq 142 for the spec trum of sq we have o a2 AE2J ISw12dw1 aA a12dx 273 00 fo This will be close to zero when the upper limit is close to infinity and therefore Sa Sq when a2 Q2 WO w0 is large 274 V 2aW where QW is the bandwidth of sqt as per Eq 140 This is reasonable because for a given wo the energy density spectrum will not spill over much into the negative frequency region if it is relatively narrow 27 INSTANTANEOUS FREQUENCY Instantaneous frequency is one of the most intuitive concepts since we are sur rounded by light of changing color by sounds of varying pitch and by many other phenomena whose periodicity changes The exact mathematical description and understanding of the concept of changing frequency is far from obvious and it is fair to say that it is not a settled question In Chapter 1 we discussed a fundamen tal reason why a good definition of instantaneous frequency is the derivative of the phase If we do define instantaneous frequency that way then its time average with the energy density gives the average frequency Note however that this result does not depend on a specific method of getting the phase it is true for any com plex signal We have in the previous sections given some plausibility arguments for why the phase should be determined by way of the analytic signal In fact in stantaneous frequency is often defined as the derivative of the phase of the analytic signaL This is a bad idea because we should keep an open mind as to whether that is indeed the most suitable definition of the intuitive concept The issue is whether the derivative of the phase of the analytic signal does indeed satisfy our intuition regarding instantaneous frequency Although in many cases it does in other cases it produces results that at first sight seem paradoxical Of course it is the paradoxes and unusual results that lead to abandonment of ideas adjustment of our intuition or the discovery of new ideas 40 Chap 2 Instantaneous Frequency and the Complex Signal Paradoxes Regarding the Analytic Signal There are five paradoxes or difficulties regarding the notion of instantaneous frequency if it is defined as the derivative of the phase of the analytic signal To some extent we will be able to understand and resolve some of these paradoxes when we study the concept of instantaneous bandwidth in Section 132 and with the introduction of the idea that instantaneous frequency is a conditional average frequency It is important to understand these difficulties in the most basic terms because in more complicated situations their re flection may cause difficulties in interpretation First instantaneous frequency may not be one of the frequencies in the spec trum That is strange because if instantaneous frequency is an indication of the frequencies that exist at each time how can it not exist when we do the final book keeping by way of the spectrum Second if we have a line spectrum consisting of only a few sharp frequencies then the instantaneous frequency may be continuous and range over an infinite number of values Third although the spectrum of the analytic signal is zero for negative frequencies the instantaneous frequency may be negative Fourth for a bandlimited signal the instantaneous frequency may go outside the band All these points are illustrated by the following simple example Example 24 Instantaneous Frequency for the Sum of Two Sinusoids Consider st 81t s2t 275 Al ejt A2 ejw2t 276 At e t 277 where the amplitudes Al and A2 are taken to be constants and wi and W2 are positive The spectrum of this signal consists of two delta functions at wi and w2 Sw A16w wi f A26w w2 278 Since we take wi and w2 to be positive the signal is analytic Solving for the phase and amplitude Al sin wit A2 sinw2t apt arctan Ai cos wit A2 cos wet 279 A2 t A2 A z 2A1 A2 cosw2 wi t 280 and taking the derivative of the phase we obtain A2 A2 281 wt cps t 2 W2 wl 2 W2 wi A2 t 1 By taking different values of the amplitudes and frequency we can illustrate the points above This is done in Fig 22 Sec 8 Density of Instantaneous Frequency 41 0 t 2 3 4 0 1 2 3 4 t Fig 22 The instantaneous frequency for the signal st Ai e3lot A2 20t The spectrum consists of two frequencies at w 10 and w 20 In a Al 2 and A2 1 The instantaneous frequency is continuous and ranges outside the bandwidth In b A 12 and A2 1 Although the signal is analytic the instantaneous frequency may become negative One last paradox regarding the analytic signal If instantaneous frequency is an indication of the frequencies that exist at time t one would presume that what the signal did a long time ago and is going to do in the future should be of no con cern only the present should count However to calculate the analytic signal at time t we have to know the signal for all time This paradox has been analyzed by VakmanM91 who makes the following analogy Before Maxwells equations light was considered localized rays We now know that light is electromagnetic waves which are highly nonlocal This discovery forced a fruitful enlargement of the un derstanding of what light is Moreover from Maxwells equations we can actually understand in which circumstances light behaves as rays In the same sense then while we started out thinking of instantaneous frequency as a local concept it may be the case that for a full explanation of the phenomenon we must enlarge the idea and context and accept its nonlocal nature 28 DENSITY OF INSTANTANEOUS FREQUENCY13I We now ask for the density of instantaneous frequency in contrast to the density of frequency that is the energy density spectrum I Sw 12 The method for finding densities is described in detail in Chapter 4 but we use the method here and make it plausible for the case of instantaneous frequency The density of instantaneous frequency Pwi is given by Pwt f bwi pt I st I2 dt 282 This says that for a given w2 choose only the values for which wi cpt If there is only one such value then this simplifies to Pwi I St 2 283 4t 42 Chap 2 Instantaneous Frequency and the Complex Signal where t41 is a function of wi obtained by solving w cp tW Instantaneous Frequency Spread We already know that the average of instanta neous frequency is the mean frequency that is wi w Therefore the spread of instantaneous frequency is QIF fdt wi 2 st 12 dt 284 0t w2Ist12dt 285 This is precisely the first term appearing in the bandwidth equation Eq 1 and therefore we can write that equation as B2 UIF At At 2 A2t dt 286 f But the second term is positive and hence we conclude that the spread in instanta neous frequency is always smaller than the bandwidth aIF B 287 This may seem paradoxical since in the example above we showed that the spread of instantaneous frequency can be outside the range of the frequencies that exist in the spectrum What this must mean is that while the instantaneous frequency can range widely the occurrences when it ranges outside the bandwidth of the signal have small duration and hence do not contribute significantly to QIF Note that the spreads are equal when the amplitude of the signal is constant Example 25 Chirp We calculate the distribution of instantaneous frequency for the signal st air14eat22j t22jot 288 The instantaneous frequency is w wo 3t and its derivative is simply Q Hence Pwi s t 2 289 Wt tWWO a 1 eaWW02A2 V 7r Q On the other hand the energy density spectrum is i Sw 12 a eaWW02a292 Ca2 Q2 290 291 Sec 8 Density of Instantaneous Frequency 43 The spread of the spectrum that is the bandwidth and the spread of the instanta neous frequency density are respectively a2 Q2 Q 2c arF 2 292 The distribution of instantaneous frequency is narrower than the distribution of fre quency in conformity with our discussion Chapter 3 The Uncertainty Principle 31 INTRODUCTION 2 The timebandwidth product theorem or uncertainty principle is a fundamental statement regarding Fourier transform pairs We are going to be particularly care ful in our discussion since the uncertainty principle has played a prominent role in discussions metaphysical and otherwise of joint timefrequency analysis The discovery of the uncertainty principle in physics and chemistry is one of the great achievements of the century Unfortunately it has generated many pseudo ideas which are rivaled only by the number of pseudo ideas generated by relativity the other great discovery of this century The pseudo ideas in relativity are perhaps characterized by everything is relative and the uncertainty principle by every thing is uncertain neither view being remotely related to the physical or mathe matical truth The uncertainty principle was first derived by W Heisenbergin the paperUber den anschhaufichen Inhalt der quantentheoretichen Kinematic and Mechanik On the Conceptual Content of Quantum Theoretical Kinematics and Mechanics in 1927 In that paper he derived the uncertainty relations on the basis of a thought experiment involving resolution of an idealized microscope In fact Heisenberg presented it as an equality rather than as an inequality It was Weyl who subsequently saw that un certainty can be defined by the standard deviation and gave the proof commonly used today which is based on the Schwarz inequality The physicist C G Darwin grandson of Charles Darwin made the connection between the uncertainty principle and Fourier transform pairs E U Condon and H P Robertson extended the uncertainty principle for arbitrary variables In 1930 Schrbdinger grasped the full generality of the uncertainty principle derived it for arbitrary variables and saw the fundamental connection with the commutator and anticommutator see Eq 1587 While the mathematics of the uncertainty principle was settled within five years of the discovery of quantum mechanics 19251926 by Schrbdinger Heisenberg Born and Dirac and within three years of Heisenbergs first paper on the sub ject the interpretation and consequences of the uncertainty principle in quantum mechanics remains a subject of great interest and activity and the number of papers written on it in the intervening sixty years or so is truly phenomenal After all it is one of the most important discoveries of mankind 2The uncertainty principle is discussed further in Sections 69 and 155 44 Sec 1 Introduction 45 For signal analysis the meaning of the uncertainty principle and its importance have been clearly stated often enough although a mystery still persists for some Right on the mark is the statement by Skolnik514 The use of the word uncer tainty is a misnomer for there is nothing uncertain about the uncertainty rela tion It states the wellknown mathematical fact that a narrow waveform yields a wide spectrum and a wide waveform yields a narrow spectrum and both the time waveform and frequency spectrum cannot be made arbitrarily small simultane ously Equally clear is Lerneri331 The uncertainty principle has tempted some individuals to draw unwarranted parallels to the uncertainty principle in quantum mechanics The analogy is formal only AckroydJ41 has emphasized that There is a misconception that it is not possible to measure the t f energy density of a given waveform and that this is a consequence of Gabors uncertainty relation However the uncertainty principle of waveform analysis is not concerned with the measurement of t f energy density distributions instead it states that if the effective bandwidth of a signal is W then the effective duration cannot be less than about 1W and conversely Although we can hardly improve on the above we try Here is the point The density in time is I st 12 and the density in frequency is ISw I2 but st and Sw are related and hence we should not be surprised to find that there is a relation between the densities The relation is such that if one density is narrow then the other is broad Thats it and no more It is not that both time and frequency cannot arbitrarily be made narrow but that the densities of time and frequency cannot both be made narrow If the interpretation is so straightforward in signal analysis why is it one of the most profound discoveries in physics Here is why In classical physics and in every day life it seemed clear that we can choose the position and velocity of objects at will No one imagined that we cannot place a ball at a given spot with a given velocity However present day physics quantum mechanics says precisely that and it is one of the great discoveries regarding the behavior of matter Quantum mechanics is inherently probabilistic When we speak of densities in quantum mechanics we mean probability densities and the word uncertainty is appropriate because we are dealing with probability In signal analysis the word uncertainty is highly misleading Let us be very clear that in both physics and signal analysis the uncertainty prin ciple never applies to a single variable It is always a statement about two variables Furthermore it does not apply to any two variables but only to variables whose associated operators do not commute In this chapter we deal with the uncertainty principle for time and frequency and in Section 155 we consider its generalization to other variables There is a very elegant and simple way to derive the uncertainty principle for arbitrary quantities and this is done in Section 155 Here we use a simple brute force approach One of the reasons that there has been considerable confusion about the uncer tainty principle in signal analysis is that one very often modifies a signal eg filters it windows it etc Once that is done we have two different signals the original and the modified one Therefore we have two uncertainty principles one relating to the 46 Chap 3 The Uncertainty Principle original signal and the other to the modified signal Very often these are confused In Section 34 we will be particularly careful in making this distinction by deriving the uncertainty principle for modified signals 32 THE UNCERTAINTY PRINCIPLE The proof of the uncertainty principle is easy but it is important to understand what goes into the derivation We have defined duration at and bandwidth a and have shown that they are good measures of the broadness of a signal in time and frequency For convenience we repeat the definitions here T2 0i t2 I st 2 ft dt 31 B2 a2 fww2ISl2dw 32 We emphasize that T and B are standard deviations defined in the usual manner and no more The uncertainty principle is TB z 33 Therefore one cannot have or construct a signal for which both T and B are arbi trarily small A More General Uncertainty Principle A stronger version of the uncertainty prin ciple is at or 2 1 4 Covt2 34 where Covt is the covariance as defined by Eq 1124 What Does the Proof of the Uncertainty Principle Depend On It is important to have a clear picture of what the proof depends on so that no misleading interpre tations creep in The proof depends on only four things first on I st 12 being the density in time second on taking I Sw 12 as the density in frequency third that 9 t and Sw are Fourier pairs and fourth on defining T and B as standard deviations of time and frequency Notation Very often the notation used to write the uncertainty principle is OtEw 1 There is nothing wrong with this notation as long we understand that A means standard deviation and nothing more However because A is typically used for the differential element of the calculus or to signify error there is a tendency to think of the uncertainty principle as having something to do with differential elements smallness measurement or resolution The A of the uncertainty principle means only one thing the standard deviation If this is kept in mind then no difficulties arise in interpretation or philosophy Sec 3 Proof of the Uncertainty Principle 47 33 PROOF OF THE UNCERTAINTY PRINCIPLE First let us note that no loss of generality occurs if we take signals that have zero mean time and zero mean frequency The reason is that the standard deviation does not depend on the mean because it is defined as the broadness about the mean If we have a signal sold then a new signal defined by snewt e7ttsoldtt 35 has the same shape both in time and frequency as sold except that it has been trans lated in time and frequency so that the means are zero3 Conversely if we have a signal s t that has zero mean time and zero mean frequency and we want a signal of the same shape but with particular mean time and frequency then soldt Ot snewt t The bandwidth expressed in terms of the signal is as per Eq 135 07W f f 13t I2 dt The duration is at ft2stI2dt 38 and therefore Qt QW f t st I2 dt x f I Wt I2 dt 39 Equation 39 is it no other assumptions or ideas are used The fact that s and S are Fourier transform pairs is reflected in Eq 37 Now for any two functions not only Fourier transform pairs z I fx I2 dx f dx ffxgxdx I f 310 which is commonly known as the Schwarz inequality4 Taking f is and g a gives z 2 at QU f t stst dt 2 311 The constant phase factor e w t is irrelevant and can be left out or in for the sake of symmetry 4There are many proofs of the inequality A simple one is to note that for any two functions f f 12 dx f ffz9xdx z fffzgyfygxI2dxdy which is readily verified by direct expansion of the right hand side Since the right hand side is manifestly positive we have Eq 310 48 Chap 3 The Uncertainty Principle The integrand written in terms of amplitude and phase is istst tAAjtcpA2 312 2 d tA2 A2 j t 01 t 313 The first term is a perfect differential and integrates to zero The second term gives one half since we assume the signal is normalized and the third term gives j times the covariance of the signal Hence Ot2 Or2 f tststdt I 2 j Covt 12 4 Covt2 314 2 Therefore we have the uncertainty principle as given by Eq 34 Since Cov W is always positive it can if we so choose be dropped to obtain the more usual form Eq 33 Minimum Uncertainty Product Signals Since the minimum value for the uncer tainty product is one half we can ask what signals have that minimum value The Schwarz inequality becomes an equality when the two functions are proportional to each other Hence we take g cf where c is a constant and the 1 has been inserted for convenience We therefore have c t st s t 315 This is a necessary condition for the uncertainty product to be the minimum But it is not sufficient since we must also have the covariance equal to zero because by Eq 34 we see that is the only way we can actually get the value of 12 Since c is arbitrary we can write it in terms of its real and complex parts c c jcc The solution of Eq 315 is hence st act22 eaict22 316 The covariance is the average value of t multiplied by the derivative of the phase The derivative of the phase is ct and remembering that we are considering a signal whose time and frequency means are zero we have r ft cit1st12dtwtctJ teet2 317 The only way this can be zero is if c is equal to zero and hence c must be a real number If we take c a2 we then have st a7x14 eat22 318 where we have included the appropriate normalization Reinstating the arbitrary mean time and frequency by way of Eq 36 we have st a7r14 eatt2iwt 319 This is the most general signal that has a timebandwidth product equal to one half Sec 3 Proof of the Uncertainty Principle 49 Example 31 Chirp The standard deviations of time and frequency for the signal 8t a7r 14 eat22jot22jmot are given by Eq 112 and 140 Using those values gives a2ni 1 at Qw 2 a22p2 2 2 1 4CovtW 321 For this case the stronger version of the uncertainty principle yields an equality Example 32 Sinusoid Modulation Using the results of Eq 112 and 150 we have for st a7r14eat22jm emwmtjwOt that 320 322 2 2 m2a QtQW A mm 1e 2 2 323 2 v1 ram 1 eWm2a2 Example 33 Damped Exponential For the signal 9t a2n1 to ec 2n the spectrum is and t 2n1 a 324 t0 n1 325 1 27r2n a2jw wo n1 326 t2 2n 22n 1 02 2n 1 327 a2 t a2 z az 2 az w wo wz wo2n1 2n1 Therefore 1 2n 1 at ow 2 2n1 328 329 50 Chap 3 The Uncertainty Principle 34 THE UNCERTAINTY PRINCIPLE FOR THE SHORTTIME FOURIER TRANSFORM There are many things one can do to signals to study them However if we do something to a signal that modifies it in some way one should not confuse the un certainty principle applied to the modified signal with the uncertainty principle as applied to the original signal One of the methods used to estimate properties of a signal is to take only a small piece of the signal around the time of interest and study that piece while neglecting the rest of the signal In particular we can take the Fourier transform of the small piece of the signal to estimate the frequencies at that time If we make the time interval around the time t small we will have a very high bandwidth This statement applies to the modified signal that is to the short interval that we have artificially constructed for the purpose of analysis What does the uncertainty principle as applied to a small time interval have to do with the un certainty principle of the original signal Very often nothing and statements about the chopped up signal should not be applied to the original signal The process of chopping up a signal for the purpose of analysis is called the shorttime Fourier transform procedure Although we will be studying the shorttime Fourier trans form in Chapter 7 this is an appropriate place to consider the uncertainty principle for it From the original signal st one defines a short duration signal around the time of interest t by multiplying it by a window function that is peaked around the time t and falls off rapidly This has the effect of emphasizing the signal at time t and suppressing it for times far away from that time In particular we define the normalized short duration signal at time t by ntT sr hr t 330 18T hT t I2 dT where ht is the window function t is the fixed time for which we are interested and r is now the running time This normalization ensures that f I7tT 12 dr 1 331 for any t Now 77i T as a function of the timer is of short duration since presumably we have chosen a window function to make it so The time t acts as a parameter The Fourier transform of the small piece of the signal the modified signal is5 Ftw 1 ejT7trdT 332 27r 51f the denominator in Eq 330 is omitted then Ftw would become what is traditionally called the shorttime Fourier transform In calculating quantities such as conditional averages the normal ization must come in at some point and it is a matter of convenience as to when to take it into account In Chapter 7 we use the more conventional definition and omit it from the definition There we use St w to denote the shorttime Fourier transform and the relation between the two is St w FtwI f 18T hT t I2 dT12 Sec 4 The Uncertainty Principle for the ShortTime Fourier Transform 51 Ftw gives us an indication of the spectral content at the time t For the modified signal we can define all the relevant quantities such as mean time duration and bandwidth in the standard way but they will be time dependent The mean time and duration for the modified signal are d frlsThrtI2dr 3 33 T T t fTl77tTI f IsThTtI2dT 2 2 f T 2 d h 2 I2 dT 7T 3 34 T J T T r 77tT I f I 3 I2 dT h t Similarly the mean frequency and bandwidth for the modified signal are w t f w I Ft w I2 dW f nt T dT 17t T dT 335 Bi Jwwt2IFtw2dw 336 TimeDependent and WindowDependent Uncertainty Principle Since we have used a normalized signal to calculate the duration and bandwidth we can imme diately write that BtTt2 337 This is the uncertainty principle for the shorttime Fourier transform It is a function of time the signal and the window It should not be confused with the uncertainty principle applied to the signal It is important to understand this uncertainty prin ciple because it places limits on the technique of the shorttime Fourier transform procedure However it places no constraints on the original signal We know that for infinitely short duration signals the bandwidth becomes in finite Hence we expect that Bt oo as we narrow the window which is indeed the case This is shown in Chapter 7 where we obtain explicit expressions for the above quantities in terms of the amplitude and phases of the signal and window The point that we are making here is that we must be very clear with regard to the uncertainty principle for the original signal and modified signal It is true that if we modify the signal by the technique of the shorttime Fourier transform we limit our abilities in terms of resolution and so forth This is a limitation of the technique The uncertainty principle of the original signal does not change because we have decided to modify it by windowing Example 34 Chirp with a Gaussian Window Consider a chirp and use a Gaussian window st a7x14 et22jt22wot ht a7x14 aate2 338 52 We first find the normalization factor the denominator in Eq 330 12 J 1 ar h7 t 12 d7 as f exp Iaa t2 339 asa and therefore the modified signal is 77 T aal14 exp aaT2aTtjWorjpT22 az t2 1 a 1 aa Chap 3 The Uncertainty Principle from which we get a T a at T T 1 2a a 340 341 z a a at WO Bt z a a 2 a a 342 and therefore z Bt Tt 2 1 a 2 343 For this case the local duration and bandwidth are time independent and so is the uncertainty product This is a peculiarity of the Gaussian window and will not be the case with another window andor signal Now notice that as we narrow the window that is as a oo the bandwidth of the modified signal Bt goes to infinity Of course this has nothing to do with the bandwidth of the original signal Example 35 Example Quadratic FM Signal Consider a signal whose phase is cubic in time with a Gaussian envelope st a7r114 i7t33 ht an114 eat3 2 344 The mean time and durations are the same as in the preceding Example The mean frequency and bandwidth of the modified signal are z M t 2 a a a a2 t2 345 f 2 2y2 BL z a a 2a a2 a a3 t2 346 The uncertainty principle is then Bt Tc 21 a 2 a2 a a4 t2 347 For large times and short windows the time dependent bandwidth goes as Bt t2a2 hence for large times or small windows we get a very large bandwidth Again this has nothing to do with the original signal The bandwidth of the original signal is fixed Chapter 4 Densities and Characteristic Functions 41 INTRODUCTION Densities are the basic quantities we deal with since we seek the energy density of a signal at a particular time and frequency We have already encountered densities in the previous chapters namely the absolute square of the signal is the energy density in time and the square magnitude of the Fourier transform is the energy density per unit frequency In this chapter we develop the basic methods and con cepts used to study and construct densities We present the ideas in a manner that is useful in studying energy densities in time frequency and joint timefrequency densities By density we mean nothing more than the common usage such as the density of mass or the density of trees Many of the ideas such as standard deviation will have a probabilistic ring to them since they are usually associated with probability densities However we emphasize that the methods presented in this chapter ap ply to any density For the one dimensional case we should simply think of a wire that may stretch from minus infinity to infinity and where the mass density varies with location Similarly for the two dimensional case one should think of a sheet where again the density varies with location on the sheet 42 ONE DIMENSIONAL DENSITIES A one dimensional density is the amount of something per unit something else For example the number of people per unit height the amount of mass per unit length the intensity per unit frequency or the intensity per unit time We use Px 53 54 Chap 4 Densities and Characteristic Functions to denote a one dimensional density of the quantity x Px Ox the amount in the interval Ax at x 41 Since PxAx is the amount in the interval Ax the total amount is the sum Total amount f Px dx 1 42 Densities are often normalized so that the total amount is equal to one The effect of doing that is for example if Px represents the actual number of people of a certain height then by normalizing it to one it represents the fraction of people at that height Densities have to be single valued and positive The positive requirement comes from the fact that a density is the amount of something per unit something else and both quantities are positive It should be emphasized that the numerator is the amount of something and while that something can be negative the amount is still positive For example while charge can be positive or negative the amount of positive or negative charge is positive Terminology Density function is universally understood in some fields distribu tion is used interchangeably with density because Px indicates how something is distributed In mathematics however distribution usually denotes the amount up to a certain value that is the integral of the density from oo to x Fx f Px dx 43 00 x which is also called the cumulative distribution function because it accumulates the density up to x We will use density and distribution interchangeably and will have no occasion to use the concept of cumulative distribution function Inciden tally the advantage of defining the cumulative distribution function is that often densities do not exist in the strict mathematical sense but cumulative distributions always do Note that the density is related to the cumulative distribution function by Px Fx when indeed the derivative exists Another reason for introduc ing a cumulative distribution function is that it allows one to handle in a mathe matically smooth manner both continuous and discrete distributions However if we are not particularly interested in mathematical sophistication the delta function provides a very effective way to do this Averages From the density simpler quantities are obtained which sometimes give a gross indication of the basic characteristics of the density The average x is x J x Px dx 44 Sec 2 One Dimensional Densities 55 The average of any function f x can be obtained in two different ways Either from the density of x or from the density of u Pu where u f x Mx J f x Px dx fuPndu 45 How to obtain the density Pu from Px is described in Section 47 Two general properties of averages should be noted First the average of the sum of two func tions is the sum of the average of each function f x gx f x gx Second the average of a constant times a function is the constant times the average of the function cf x c f x Standard Deviation While the average gives the place where things are balanced it gives no indication as to whether the density is concentrated there or not A measure of that is the variance which is the average of x x 2 Variance is a good measure of concentration because if the density is concentrated near x then x x 2 will be relatively small The weighting of that particular value is of course taken with the density The standard deviation am is defined as the square root of the variance and given by Qx Jx x 2 Px dx 46 x2x2 47 Similar considerations apply to functions of x 02 f ffx f x 2 Px dx f2x fx2 Some general properties of the standard deviation are that the standard deviation of a constant is zero v 0 and the standard deviation of cx is given by o cam Moments The moment of order n is the average of x x f x Px dx 410 The significance of the moments are many fold First the first few moments give an indication of the general properties of the distribution Second in some sense the more moments we know the more we know about the distribution Third for well behaved distributions the moments uniquely determine the distribution The procedure for constructing the distribution from the moments is presented in the next section In Chapter 10 we see how these methods can be used for the construction of timefrequency distributions 56 Chap 4 Densities and Characteristic Functions 43 ONE DIMENSIONAL CHARACTERISTIC FUNCTIONS The characteristic function is a powerful tool for the study and construction of den sities It is the Fourier transform of the density M9 f eiexPx dx 9x 411 The characteristic function is the average of 08 where 9 is a parameter By ex panding the exponential we have M9 f e7BxPx dx f 7 nx Px dx n1n xfz 412 which is a Taylor series in 9 with coefficients j x Since the coefficients of a Taylor series are given by the nth derivative of the function evaluated at zero we have 1 8M9 x in 59n Bo 413 The fact that moments can be calculated by differentiation rather than by integra tion is one of the advantages of the characteristic function since differentiation is always easier than integration Of course one has to first obtain the characteristic function and that may be hard Fourier transform pairs are uniquely related and hence the characteristic func tion determines the distribution Px 2 J MO eJex d9 414 Some general properties that a function must posses if it is a proper characteris tic function are easily obtained By proper we mean a characteristic function that comes from a normalized positive density First taking 9 0 we see that MO f Px dx 1 415 Taking the complex conjugate of Eq 411 and using the fact that densities are real we have or M 9 f eiGxP x dx M9 416 MO MO 417 The absolute value of the characteristic function is always less than or equal to one M9 I 1 418 Sec 3 One Dimensional Characteristic Functions 57 This follows from I Me I f dx We know that the characteristic function at the origin is equal to one and therefore I M9 15 M0 Example 41 Gaussian Distribution For the Gaussian a20 jxo MB the characteristic function is M9 2 vz J exp x x022021 hex dx e0282z3x08 The first and second moments can be obtained by differentiation 1 8M9 x 89 xz z1 82M Px 2 exp x xoz2oz 421 f I ejex I I Px I dx J Px dx 1 419 90 90 z a29 jxo z Qz M8 90 422 x0 423 80 xgQz 424 Note that the advantage of pulling out MB is that it is always equal to one for 8 0 Example 42 Exponential Density For the exponential density Px A eaz 0 x 00 425 the characteristic function is A ee29x A MO foo A jO which by differentiation gives for the first and second moments x 1 8M8 j 88 xz 1 82M8 jz 802 420 426 1 JA 1A 427 90 i Aj6z 90 z 1 2A 2A2 428 9o jz AjO9 a0 58 Chap 4 Densities and Characteristic Functions Relation of Moments to Density Generally speaking the knowledge of all the moments determines the distribution This can be seen in the following way From the moments we can calculate the characteristic function as per Eq 412 Once the characteristic function is obtained the density is obtained by Fourier inversion There are cases where the moments do not determine a unique characteristic func tion or density but these exceptions will not concern us Example 43 Distribution from the Moments Suppose we have the set of moments n 1 fin We construct the characteristic function an 429 M8 7B Xn jnen n 1 j9nn 1 a n n an an a j9 Therefore the density is 2 430 2 Px 2 M9 ejaw dO 2 f a a 8 eix d8 431 j a2x aaxs if 0 a oo and zero otherwise 432 When Is a Function a Characteristic Function A characteristic function is a com plex function but not every complex function is a characteristic function since the function must be the Fourier transform of a density The conditions just de rived Eqs 415 417 and 420 are necessary but not sufficient The reason is that nowhere have we used the fact that the density is positive A necessary and sufficient condition for a function to be a characteristic function was given by Khinchinii A function M9 is a characteristic function if and only if there exists another function gO such that f M9 J gBg0 9 d9 433 The function g is to be normalized to one f I g9 12 dB 1 434 This is a very basic result and will be important to our considerations since the gs will turn out to be signals We show the sufficiency here because it produces a result that is revealing As suming that M9 is a proper characteristic function the density is given by Sec 4 Two Dimensional Densities 59 Px 2 f MO ajax dO 2 ff gBg9 9 ajex d8 dO 435 Making a change of variables 9 0 9 dB d9 we have Px 27r f g9 ejex dO 2 436 which shows that Px is a proper density since it is positive By Parcevals theorem we know that Px is normalized to one if g is normalized to one as per Eq 434 Note that while there is a onetoone relationship of the density to the character istic function that is not the case with the function g Many gs produce the same characteristic function and hence the same density 44 TWO DIMENSIONAL DENSITIES Consideration of two dimensional densities that is densities that depend on two variables forces the introduction of a number of new concepts not encountered with one dimensional densities These new ideas such as conditional average and correlation remain intact for densities of more than two variables A two dimensional density Px y is the amount of something per unit x and per unit y at the point x y The total amount is Total amount ffPxydydx 1 437 Again we normalize to one so that for example if Px y is the number of people of a certain weight and height then when normalized to one it becomes the fraction of people of that weight and height Marginals If we have a joint density we may want the density of just one of the variables irrespective of the value of the other variable This is achieved by inte grating out the other variable Px f Px y dy Py J Px y dx 438 These densities are called marginal densities or marginals The origins of the word marginal came about as follows When the basic concepts of probability and statistics were being developed joint densities were written on a piece of paper in columns and rows in the fashion of the modem spreadsheet For example if we wanted to make a table indicating how many individuals there are of a certain height and weight the joint density the weight axis would be on top of the page and the height axis would run down the page as in a typical spreadsheet At the 60 Chap 4 Densities and Characteristic Functions intersection of each row and column one would write the number of people of that weight and height Now suppose one wanted the number of people of a certain weight irrespective of height the weight marginal and the number of people of a certain height irrespective of weight the height marginal Then one adds up the columns and rows A natural place to put the sums is in the margins of the paper that is on the right margin and the bottom of the page hence the terminology marginals Global Averages For a function gx y the global average is gx y JJgxy Px y dy dx 439 Two Dimensional Characteristic Functions and Moments The two dimensional characteristic function M9 r is the average of ejex 7y M9 T efexiT ff eiexiTYPx y d x dy 440 and the distribution function may be obtained from M9 T by Fourier inversion Px y 4x2 Jf M9 T erexhhv d9 d7 441 Similar to the one dimensional case expanding the exponential in Eq 440 re sults in a two dimensional Taylor series 7en7 rm xnym 442 M9 r 00 00I nim n0 m0 and therefore xnym 1 jnjm anm 8097 M9 T 9T 0 443 Relation Between the Joint Characteristic Function and the Characteristic Func tion of the Marginals The characteristic function of the marginal for x is M0 J eBxPx dx r eBxPx y dx dy M9 0 444 Similarly the characteristic function for y is MT f eTPy dy f e2T1Px y dx dy MO T 445 Sec 4 Two Dimensional Densities 61 Therefore if we know the characteristic function of the joint distribution we can trivially obtain the characteristic functions of the marginals by taking zero for one of the variables Example 44 Two Dimensional Gaussian The standard form for a two dimensional Gaussian is Px y 1 27ro oy 1 r2 exp 1 2 x Za2 y 2b2 2r x ay b 446 l 21 r Qs ay Qorv with Irl 1 The characteristic function calculated by way of Eq 440 is MO r exp jaB jbr 2 a9 2rixvyOr a 2T2 447 The characteristic function of the marginal in x is therefore M9 M9 0 exp jaO o 9 2 2 448 This is the same form as the characteristic function of a one dimensional Gaussian Eq 422 and hence the marginal is Gaussian with mean a and standard deviation a Example 45 NonGaussian Joint Density with Gaussian Marginals The distribution given by Eq 446 is the standard joint Gaussian distribution As we just showed it has Gaussian marginals However an infinite number of two di mensional non Gaussian distributions have Gaussian marginals A method for readily constructing such distributions is discussed in Section 142 Example 46 Characteristic Function of the Sum One of the main uses of characteristic functions is that they allow us to obtain new densities in an easy way This is discussed in subsequent sections but a simple example is appropriate here Suppose we want to obtain the density of the sum of two variables x x y 449 The characteristic function of z is M9 M00 450 from which the distribution of x may be obtained by inversion For example for the two dimensional Gaussian discussed above M0 MO0 exp jabOB2 o 2rvvya 2 451 This is recognized to be the characteristic function of a Gaussian Eq 422 with mean a b and standard deviation ay 2ro o Qy 62 Chap 4 Densities and Characteristic Functions Independence If fixing a particular variable has no effect on the density of the other variable then one says that the joint distribution is independent That is the case if the joint distribution is factorable and the factors are the marginals Px Y Px Py 452 For this situation the characteristic function and the joint moments are also fac torable M9T M9MYT xym xym 453 However one should be careful because sometimes a density may appear factorable without being independent For independence the factors must be the marginals The examples below illustrate this point Example 47 Rectangular and Circular Two Dimensional Uniform Distributions Consider first The marginals are y 2 Q2 x IQ Px 4 a4 xy d 474xy if0 xy a 0 otherwise fQ P 3l 4 Q4 xy 2 Q2 y 454 455 Thus Px y PxPy and hence the distribution is an independent one Now consider We have 4 xy if x2 y2 a2 0 C X y Px y T4 456 0 otherwise 2y2 Px a4 x J y dy Q4 x a2 x2 I 0 457 V2y2 8 y J x dx 4 y a2 y2 458 Py a4 o a We see that even though the joint density is a product in some sense it is not inde pendent because the product is not the product of the marginals In fact this joint distribution is not really factorable for all values of x and y Examples likethese have historically been presented as a warning to use caution In reality it is just a failure of notation If we rewrite the joint density as 4x y Ea2 x2 y2 Ex EY Px y W 459 where cx is the step function then it is clear that Px y is not factorable and the issue never arises Sec 5 Local Quantities 63 Covariance and Correlation Suppose we want to know how strongly onevariable depends on the other This is what a joint distribution tells us in precise terms However if we want a gross measure that reflects in a simple way the strength of the dependence we can consider the expected value of the first mixed moment x y If the joint distribution is independent the first mixed moment will equal the product of the moments x y x y x y independent 460 The dependency can be measured by the excess of the first mixed moment over x y That is the covariance Covxy X Y X Y 461 The covariance is a number that can be positive or negative and of any magnitude To standardize the correlation coefficient is defined by r COVxy QxQy 462 where Qx and ay are the standard deviations of x and y The correlation coefficient ranges from 1 to 1 for all densities and hence standardizes the strength of the correlation between x and y The variables x and y are said to be uncorrelated if the correlation coefficient is zero strongly correlated at r 1 and strongly oppositely correlated when r 1 However one should be cautious in applying these ideas because they do not necessarily reflect the intuitive notion as to whether the two variables have something to do with each other One must not confuse dependence with correlation In fact we can have a distribution where it is dear that the two variables are strongly dependent and yet the correlation coefficient is zero The correlation coefficient is a single number and we cannot expect too much from just one number Nonetheless the correlation coefficient is often a good gross indicator of the dependence of two variables The simplest way to calculate the first mixed moment is from the characteristic function x y 492M9 T a00T if it is available 9r 0 463 45 LOCAL QUANTITIES Suppose we have the density of height and weight of a population and want to study the weight of people who are 6 ft tall The density of weight for that sub population is the joint density but we fix the height at 6 ft Specifically if we have two variables then we use the notation Py I x to mean the density of y for a fixed x Such a density is called a conditional density since it depends on the value of x 64 Chap 4 Densities and Characteristic Functions chosen It can be thought of as a one dimensional density with x as a parameter If we insist as we should that the conditional densities be normalized then they are Py I X Px Y Px I Y Px y 464 Px Py where Px and Px are the marginals If Py I x does not depend on x then this is exactly what we would want to mean by independence And if that is the case then Py I x Py which implies that Py x PyPx in conformity with our previous discussion of an independent distribution Conditional Averages and Their Standard Deviation Suppose we want the aver age of y for a given x for example the average weight of those people who are 6 ft tall That is called the conditional average and is denoted by y Since the density of y for a given x is Px I y we have Y X f yPy I x dy px fyPx y dy 465 More generally the conditional average of any function the local average is 9y z px f 9y Px y dy 466 Conditional Standard Deviation Suppose a person weighs 250 lb Is he over weight If he compares himself to the mean of the total population he may be ex ceptional But he should be comparing himself with individuals of the same height Therefore one defines the conditional standard deviation by the deviations from the conditional mean a2 y1x 1 Px f Y 2 Px y dy 467 y2ry2 468 46 RELATION BETWEEN LOCAL AND GLOBAL AVERAGES If we integrate the conditional average y x over all values of x we expect to obtain the global average of y and indeed that is the case since y ffvPxYddx ffPYixPx dy dx 469 The inner integration is precisely the conditional average and therefore y fyPx dx 470 Sec 7 Distribution of a New Variable 65 This equation shows that the global average is the average of the conditional aver age We now seek the relationship between the global standard deviation and the conditional standard deviation The global standard deviation is the not average of the local standard deviation but rather Q2 f vy21x Px dx J Y y 2 Px dx 471 To prove this we average er f7JPxdx JfyYx2 Px y dy dx 472 y2 fyPxdx 473 Now subtract and add y 2 to the right hand side of Eq 473 f QY1x Px dx y2y2 f y2 Pxdx 474 r12f Y y 2 Pxdx 475 which is Eq 471 We see that there are always two contributions to the global standard deviation One is the average of the local standard deviation and the other the deviations of the conditional mean about the global mean This equation is particularly important in our considerations of timefrequency analysis The reader is invited now to com pare this equation with the bandwidth equation derived in Chapter 1 Eq 196 and to draw his own conclusions 47 DISTRIBUTION OF A NEW VARIABLE One Dimensional Case Suppose we have a density Px and want the density of a new variable say u which is a function of x u fx 476 If we knew the characteristic function of it Mu 0 we could obtain the distribution by pu 2 f ejGuMuO d8 477 But the characteristic function of u is the average value of ejef x and therefore can be obtained from Px by MuO eiefx fe9Pxcix 478 66 Chap 4 Densities and Characteristic Functions This solves the problem because from Px we obtain MuO and then Pu by Eq 477 We can actually carry out the procedure explicitly and obtain an interest ing form for Pu Substitute Eq 478 into Eq 477 to obtain Pu 2 ff eiefx e2euPx d9 dx 479 or Pu J 6u f x Px dx 480 The physical interpretation is dear The delta function picks up from Px only the values for which u f x This form avoids the calculation of the characteris tic function Further simplification is possible One of the properties of the delta function is 6gx Igxi 6x xi 481 where xis are the solutions to gTi 0 that is the zeros of g In our case we have gx f x u and since the derivative of g is the derivative of f we have 6f x u 6x xi fx1 i 482 where now the xi values run over the solutions of f xi u Therefore Pu J 6u f x Px dx 483 1 Px dx 484 6x xi I fxi or Pu fxi 485 fxi where xis are the solutions of f xi u Suppose there is only one solution to f xi u which we might as well call x Then Px Pu I Px x f u 486 where x f u signifies that we must solve for x in terms of u by solving f x u Equivalently we note that for this case Pu du Px dx which is the easiest way to do it Sec 7 Distribution of a New Variable 67 Example 48 One Zero As a specific example consider Px 0 x it ir 487 We wish to find the density of u where u f x A cos x 488 Since for the range considered the function is single valued we have f x A sin x and therefore Pu 1 1 7r A sin x j 1 1 if AuA 489 xarccoeuA 7r J42 u Example 49 Quadratic Transformation In the particular case where we transform u x2 we have two solutions xl U x2 u 490 and the derivative of u is 2x Therefore Pu 2uPxuPxu 0uoo 491 Example 410 Scaled and Translated Densities Suppose we wish to scale a variable and also translate it We take u ax b The characteristic function of u is Mu 8 eJaxb Jf eeaxbPx dr eJbMaO 492 where MaO is the characteristic function of x The distribution for u is therefore Pu I f ejauMU0 dO a1a P u a b 493 T7r Alternatively we can use Eq 486 Solving for the zerowe have x u ba and hence 1 1p u b 494 Pu aPx xuba ax a 68 Chap 4 Densities and Characteristic Functions Two Dimensional to One Suppose we have a two dimensional density Px y and wish to find the density of a single variable which is a function of x and y u fxy 495 The characteristic function for u is MU B e3ef xv fJ etxv px y dx dy 496 Taking the inverse transform leads directly to Pu 497 Example 411 Sum of Two Variables Consider the case where u x y 498 The density u is then Pu ff du x y Px y dx dy 499 JPxtLxdx 4100 Two Dimensional to Two Dimensional In the most general case we start with a density of two variables and wish to find the density of two new variables function ally related by u fxy v gxy 4101 Using the characteristic function approach we have for the characteristic function for the variable u and v M e T Jf eJef xvjrgxv Px y d x dy 4102 The density for u v is then Pu v 47r2 ff Muv 9 T ejOu3rv d9 dT 4103 41 ffff jeuf virvgxv Px y dx dy dO dT 4104 The 0 and T integrations give delta functions and hence Pu v if bu yx y bv gx y Px y dx dy 4105 Sec 8 Negative Densities 69 48 NEGATIVE DENSITIES Densities by definition are positive However it is of interest to examine which ideas and results regarding positive densities still hold true if we relax the positiv ity requirement The reason we concern ourselves with this issue is that some of the densities we will be dealing with will not be manifestly positive Perhaps they should not be called densities but nonetheless they will be treated and manipulated as densities As specific cases arise we will examine the consequences of not having a manifestly positive density but we make some general remarks now Most of the concepts of the characteristic function and moments present no dif ficulty Specifically the necessary requirements for a function to be a characteristic function obtained in Section 43 remain valid However they will not have the physical interpretation commonly associated with densities For example the stan dard deviation may turn out to be negative In addition the correlation coefficient may not be bound by 1 1 The results that do not go through are those pertaining to conditions on the characteristic function to assure that it came from a manifestly positive density The only time we have addressed that issue thus far is in Khinchins theorem in Section 43 Hence questions that address the positivity issue must be reexamined and we will do so as the occasions arise Chapter 5 The Need for TimeFrequency Analysis 51 INTRODUCTION The aim of this chapter is to show by way of examples the need for a combined timefrequency representation and to clarify why time analysis and frequency anal ysis by themselves do not fully describe the nature of signals We also describe some of the physical reasons why spectra change in time Suppose we have the individual densities of height and weight of a particu lar type of animal These individual densities tell us everything we want to know about the distribution of height and the distribution of weight From these distri butions can one determine how height and weight are related Can one determine whether the tall individuals are the heavy ones The answer is no Can one de termine whether an individual with weight of 250 lb is exceptional The answer is yes But can one determine whether he is exceptional for his height The an swer is no The individual densities of height and weight are not a full description of the situation because from these densities we cannot ascertain how height and weight are related What is needed is the joint density of height and weight In the same way the time energy density and the frequency energy density are not sufficient to describe the physical situation because they do not fully describe what is happening In particular from the spectrum we know which frequencies were present in the signal but we do not know when those frequencies existed hence the need to describe how the spectral content is changing in time and to develop the physical and mathematical ideas to understand what a timevarying spectrum is We wish to devise a distribution that represents the energy or intensity of a signal simultaneously in time and frequency Time varying spectra are common in ordinary life During a sunset the fre 70 Sec 2 Simple Analytic Examples 71 quency composition of the light changes quickly and dramatically In saying that the sky is getting redder we are conveying a timefrequency description because we are describing how the frequencies are changing in time The pitch which is the common word for frequency of human speech changes as we speak and produces the richness of language Similarly the pitch of animal sounds changes during vo calization Standard musical notation is a timefrequency representation since it shows the player what notes or frequency should be played as time progresses In this chapter we give a number of simple analytic and real examples of time varying spectra so that we may become familiar with the need for a timefrequency description the variety of applications and the language involved We leave aside the methods used to construct the joint timefrequency representations That will be the subject of the subsequent chapters The reader should think of these repre sentations as no different from a joint density of height and weight of a population that gives the relative concentration of people at particular heights and weights Similarly we should think of a timefrequency distribution as telling us the inten sity or energy concentration at particular times and frequencies We first take a real example to illustrate the basic idea before embarking on a more systematic discussion Using a whale sound Fig 51 shows three plots Run ning up the page is the sound the air pressure as a function of time By examining it visually we cannot tell much although we can clearly tell how the intensity or loudness varies with time Below the main figure is the energy density spectrum that is the absolute square of the Fourier transform It indicates which frequencies existed and what their relative strengths were For this sound the spectrum tells us that the frequencies ranged from about 175 to about 325 cycles per second This in formation is interesting and important but does not fully describe what happened because from the spectrum we cannot know when these frequencies existed For example we cannot know just by looking at the spectrum when the 300 Hz sound was made or whether it was made for the total duration of the sound or just at certain times The main figure is a time versus frequency plot that is a joint time frequency distribution From it we can determine the frequencies and their relative intensities as time progresses It allows us to understand what is going on At the start the frequency was about 175 Hz and increased more or less linearly to about 325 Hz in about half a second stayed there for about a tenth of a second and so forth In answer to the question of when the 300 Hz sound occurred we can now see the answer It occurred twice at 06 and 13 seconds The difference between the spectrum and a joint timefrequency representation is that the spectrum allows us to determine which frequencies existed but a com bined timefrequency analysis allows us to determine which frequencies existed at a particular time 52 SIMPLE ANALYTIC EXAMPLES We now examine some simple analytic examples which will progressively develop the main ideas 72 Chap 5 The Need for TimeFrequency Analysis 14 H 04 H 02 150 175 200 225 250 275 300 325 350 FREQUENCY Hz Fig 51 A timefrequency plot of a sound made by a Bowhead whale The wave form is on the left plot with time increasing upwards The energy density spec trum is below the main figure and indicates which frequencies existed for the dura tion The main figure is a joint timefrequency plot and shows how the frequencies change with time From the work of C Rosenthal and L Cohen Finite Duration Sine Waves In Fig 52 we show examples of signals composed of finite duration sine waves at three frequencies All three examples have basically the same spectrum which is reasonable since for all cases we had three frequencies The cases are different in regard to when the frequencies existed and this can be easily ascertained from the timefrequency plots Chirp Consider the signal st 1 eAt22jwot vT 0 t T 51 From our discussion in Chapter 2 the instantaneous frequency is wo 3t Here it ranges from wo to wo QT In Fig 53 a we have plotted the energy spectrum and the timefrequency plot The spectrum is basically flat telling us that these frequencies existed with equal intensity The timefrequency plot tells us precisely when they existed Sec 2 Simple Analytic Examples 1 2 3 a FREQUENCY I A A A 1 2 3 c FREQUENCY A A A 73 Fig 52 Signals composed of finite duration sine waves The energy spectrum of each is essentially the same indicating that three frequen cies existed The time frequency plot shows when they occurred Sinusoidal Modulation Take st ait14 eatz2it22jmsinwtjwot where m w and wo are constants The spectrum is given by co Sw 47r 7ra 14 E Jn3 ewnwWos2a noo where Jn is the nthorder Bessel function Fig 53b plots the signal energy density spectrum and the timefrequency plot There is a considerable difference between what we learn from the two Other Examples The cases of a cubic phase and a log phase are shown in Figs 53 c and d Multicomponent Signals One of the significant advantages of timefrequency analysis is that it allows us to determine whether a signal is multicomponent or not A multicomponent signal is one that has well delineated regions in the time frequency plane A few examples are illustrated in Fig 54 Multicomponent signals are common in nature as will be seen from some of the examples shown subse quently 74 Chap 5 The Need for TimeFrequency Analysis a 0 b 0 d so 04 10 W E 10 Fig 53 The timefrequency plots for various signals In a we have a chirp st eipt22wot The instantaneous frequency m o t is increasing linearly The energy density spectrum is more or less flat indicating that each frequency existed but gives no indication of when it existed The timefrequency plot does In b we have a sinusoidal modulation eOt22msin WmtikJot and the energy spectrum shows a concentration at certain frequencies but gives no indication of the times The reason that there is concentration at those frequencies is that those frequencies last longer as can be clearly seen from the timefrequency plot In c we have a signal with a cubic phase st iyt33Ot22iwot In d we have a hyperbolic signalst e1 Intto which gives an instantaneous frequency ofwt 1tto which is a hyperbola in the timefrequency plane Mn 1I Fig 54 Examples of multicompo nent signals Sec 3 Real Signals 53 REAL SIGNALS We now give some examples of real signals 75 Human speech The analysis of human speech was the main reason for the practical development in the 1940s of timefrequency analysis The main method was and still is the shorttime Fourier transform which we will study in Chapter 7 This approach gave dramatic new understanding of the production and classification of speech An example is shown in Fig 55 Time Whale Sounds In Fig 56 we show a number of whale sounds plus one 32 sec equals zero Fig 55 Tunefregency plot of a speech signal The utterance is eA i 1 0 1 08 06 04 02 0 0 O L 150 50 100 e to the i pi 150 200 Bowhead Whale 200 250 FREOUENCY I ft 300 0 Fig 56 Timefrequency plots of whale sounds From the work of C Rosenthal and L Cohen 76 Chap 5 The Need for TimeFrequency Analysis Propeller Sounds When a propeller or fan turns it pushes the air at a repetitive rate depending on the rotation If the propeller has for example three symmetrical blades then the frequency of repetition will be three times the number of revolu tions per unit time For a constant revolution rate this will show up as an acoustic pressure wave of constant frequency However if the ship decreases or increases its speed by changing the revolution rate this will result in changing frequencies Inelastic Acoustic Scattering When a rigid object is hit with a sound wave the wave gets reflected producing an echo The characteristics of the reflected wave depend on the shape of the object its surface properties the medium of propa gation and location This reflection is called specular reflection However if the object is deformable then the initial wave that hits the object produces waves in the object no differently than if the object had been hit with a hammer The waves travel within the object and get reradiated Hence for an elastic object hit by a wave not only do we get the usual echo but we get additional waves due to the elastic response Fig 57 gives an example of a toy submarine which is a thin metallic shell with internal supporting structures The submerged submarine was irradiated with an acoustic wave sonar The first part of the returned signal is the specular reflec tions The rest of the signal is due to the reradiated acoustic elastic response The advantage of a timefrequency plot is that we can immediately see at what times the reradiation occurred and what frequencies were radiated out W T i t Fig 57 A timefrequency plot of the pressure wave at a particular point in space coming from a model subma rine after it was irradiated by a sonar pulse The first return is the spec ular reflection not shown and the other returns are due to the reradi ation of waves induced in the struc ture by the impeding wave From the work of C Rosenthal and L Co hen Windshield Wipers The sounds that windshield wipers make are often annoying Fig 58 shows a timefrequency plot of a good blade and an annoying one In the annoying case a weblike structure is seen at higher frequencies The frequency band appearing in both is due to the motor sound Sec 3 Real Signals Fig 58 Time frequency plots of windshield blades of two different models of cars The first a is judged to be annoying the second b is not Courtesy of J Feng Ford Motor Co 77 Car Door Slams The slamming of a car door has historically been a major concern for many reasons Madison Avenue notwithstanding We have all experienced the rich sounding door with its solid thump and the tinny door that does not engen der room for confidence in the rest of the construction The sound produced by a slamming car door is quite involved because there are a number of diverse events occurring including the coupling with the latch the hinging and the seals around the frame Fig 59 shows a timefrequency plot for a good sounding car door a and a tinny one b Note that in the poor sounding door there is considerable acoustic energy at a wide spread in frequencies immediately after the contact of door with latch It has been found that this acoustic energy which lasts for about a twentieth of a second is the main reason people say a door sounds tinny and hence poor The main frequency concentration is the vibration of the automobile as a whole caused by the slamming of the door a b t Fig 59 Timefrequency plots of two car door slams The one in a is characterized by a tinny sound and the one in b by a rich sound The tinny sound is caused by the generation of a wide spectrum of frequencies in the initial contact of the door with the latch Courtesy of J Feng Ford Motor Co 78 Chap 5 The Need for TimeFrequency Analysis Fault Analysis It is important to have methods for the earliest possible detection of when a machine is starting to go bad This is particularly important for machines such as in airplanes and ships whose failure may lead to critical situations It is also critical in manufacturing since a bad manufacturing machine may produce defec tive products We illustrate such situations in Figs 510 and 511 a W a 71 I i c19A nilI w 4J f b c Fig 510 A timefrequency plot of a region near a spectral line of a ma chine that is about to go bad a and one that has gone bad b Courtesy of R A Rohrbaugh Fig 511 Drilling of certain metal parts often requires the resulting hole to be of high precision There fore it is of importance to have an ef fective means for the determination as to when a drill is beginning to get dull In a we a have sharp drill in b the drill is somewhat dull and in c the drill is very dull The data is obtained from an accelerometer on the spindle Courtesy of P Lough lin J Pitton L Atlas and G Bernard Data courtesy of Boeing Commerical Airplane Group Sec 3 Real Signals 79 Trumpet In Fig 512 we show a timefrequency plot of the first half second of a trumpeter initiating a note The ridges in the frequency direction are the harmonics of the fundamental To achieve the steady state desired the trumpeter has to adjust in a split second The frequency of each harmonic gets adjusted in the same way that is the percentage change is the same While the absolute change is insignificant for the lower frequencies it is significant at the higher frequencies l00 1100 4800 4800 5000 5200 Fig 512 Timefrequency plots of a trumpet sound at its initiation Courtesy of W J Pielemeier and G H Wakefield Bottlenose Dolphin Fig 513 shows the timefrequency plot of a dolphin sound 10 Fig 513 Bottlenose Dolphin The timefrequency plot shows that the sound consists of linearly increas ing frequency and periodic clicks Courtesy of Williams and Tyack I I f r 1 Time sec 4 80 Chap 5 The Need for TimeFrequency Analysis Heart Sound While the metaphysical importance of the heart has been recognized since ancient times the notion that the heart is responsible for the heart sound was not accepted until the 1700s There was considerable debate as to whether the heart produces a sound at all Indeed the invention of signal analysis can be traced to Hook I have been able to hear very plainly the beating of a Mans heart Who knows I say but that it may be possible to discover the Motions of the Internal Parts of Bodies by the sound they make that one may discover the Works performed in the several Offices and Shops of a Mans Body and thereby discover what In struments or Engine is out of order 16181 Fig 124 shows a timefrequency plot of a heart sound 54 WHY SPECTRA CHANGE There are many causes for timevarying spectra but two broad physical mecha nisms encompass most situations The primary reason is that the production of particular frequencies depends on physical parameters that may change in time For example a string of a fixed length and tension produces a particular frequency if disturbed The mechanical oscillation of the string beats the air at the mechani cal vibration rate If the length or tension changes with time different frequencies will be produced in time because the string will vibrate with different frequencies and beat the air accordingly When violinists play for example they continually change the length of strings to produce different vibrations of the strings which in turn produce different vibrations in the air If a ship is going at constant speed by turning the propeller at a constant speed then the sound of the propeller is at a constant frequency namely the number of times it turns per minute times the num ber of blades If the ship accelerates then the propeller will change the revolution rate beating the water at a changing rate That shows up as increasing frequency Related to this question of changing the properties of the parameters that are caus ing the vibration is the possible changing of the immediate physicalsurroundings Suppose we produce sound waves at the end of a pipe If the length of the pipe is constant then the frequency spectrum of the output will be constant in time Butif the shape and length of the pipe change with time then the output will also vary with time That is the case with human speech As we speak we are continually changing the physical shape of our tongue mouth nose etc The other broad reason for changing spectra is that the propagation of waves in a medium is generally frequency dependent That is why we can see through glass but not through wood while if our eyes where sensitive to Xrays we would be able to see through wood but not so well through glass The propagation of waves is governed by a wave equation If the velocity of propagation v does not in textbooks Robert Hook is remembered for Hooks law However he was one of the greatest scientists of his century Hook discovered that plants were made up of cells the red spot of Jupiter the wave nature of light and Boyles law of gases Boyle published this law in a book on the subject and gave full credit to Hook but Boyles name stuck among many other discoveries He was about ten years older than Newton and their lives were intertwined both positively and negatively Sec 4 Why Spectra Change 81 depend on frequency for example electromagnetic waves in a vacuum then the disturbance obeys v aUx taxe aux tat where u is the physical quantity that is changing pressure electric field etc and x t are the position and time The unique feature of this wave equation is that if we start with a signal at t 0 given by ux 0 then at a later time the wave will be the same function but displaced according to ux vt 0 The shape remains the same and so does the frequency content That is why a person 5 feet away sees and hears pretty much the same as a person 50 feet away from the source It is because light and sound propagate in air without any substantial change in the frequencies that our ears and eyes can detect However it is generally the case that the propagation of waves in a medium is a frequency dependent phenomenon This is reflected in the wave equation by addi tional terms that do not admit solutions of the form ux vt The functional form changes in time and we have distortion or filtering This phenomenon that waves of different frequencies propagate with different velocities is called dispersion The reason for the name is that a prism disperses light and that was the earliest dis covered manifestation of the effect The velocity may decrease or increase with frequency depending on the materials and the two situations are described by the phrase normal and anomalous dispersion the former being the most common In addition absorption dying out and attenuation all meaning the same thing are also generally frequency dependent In normal conditions here on earth there is almost no attenuation of sound waves at the frequencies we hear and that is why we can hear from far away On the other hand light gets attenuated when passing though water and the attenuation is dependent on frequency Similarly high fre quency electromagnetic waves are damped within a short distance when entering the surface of a conductor Also as a wave propagates from one medium to another part of it gets reflected and part of it is transmitted which is generally frequency dependent The above situations are generally described by filtering which usually means filtering of frequencies Thus glass allows the visible portion of the spectrum togo through but is a pretty good filter of the Xrays Paper on the other hand filters visible light out but lets through Xrays The suns maximum output is at frequen cies we call the visible spectrum which may seem to be the reason why evolution gave us the ability to see at those frequencies But what is important for evolution is not what the sun produces but what gets through our atmosphere It is a re markable coincidence that our atmosphere lets through the visible part If we lived on a planet whose atmosphere filtered everything but the Xrays we might have developed Xray vision assuming that a good fraction of the light coming from the star was in the Xray region Chapter 6 TimeFrequency Distributions Fundamental Ideas 61 INTRODUCTION The basic objective of timefrequency analysis is to devise a function that will de scribe the energy density of a signal simultaneously in time and frequency and that can be used and manipulated in the same manner as any density If we had such a distribution we could ask for the fraction of the energy in a certain frequency and time range we could calculate the density of frequency at a particular time we could calculate the global and local moments of the distribution such as the mean conditional frequency and its local spread and so on We now begin our study of how to construct such distributions and in this chapter we describe the main ideas To crystalize our aim we recall that the instantaneous power or intensity in time is I st I2 intensity per unit time at time t or I st I2 At the fractional energy in the time interval At at time t and the density in frequency the energy density spectrum is Sw 12 intensity per unit frequency at w or Sw I2 Ow the fractional energy in the frequency interval AU at frequency w What we seek is a joint density Pt w so that Pt w the intensity at time t and frequency w or Pt w At Aw the fractional energy in the timefrequency cell At Aw at t w 82 Sec 1 Introduction 83 Do there exist joint timefrequency distributions that satisfy our intuitive ideas of a timevarying spectrum How can they be constructed Can they be interpreted as true densities Do they represent the correlations between time and frequency What reasonable conditions can be imposed to obtain such densities The hope is that they do exist but if they dont in the full sense of true densities what is the best we can do Are there inherent limitations to such a development This is the scope of timefrequency analysis Marginals Summing up the energy distribution for all frequencies at a particular time should give the instantaneous energy and summing up over all times at a particular frequency should give the energy density spectrum Therefore ideally a joint density in time and frequency should satisfy f Pt w d w I st 12 61 f Pt w dt I Sw l2 which are called the time and frequency marginal conditions 62 Total Energy The total energy of the distribution should be the total energy of the signal E Jf Pt w dw dt f I st12 dt f I Sw12 dw 63 Note that if the joint density satisfies the marginals it automatically satisfies the total energy requirement but the converse is not true It is possible that a joint density can satisfy the total energy requirement without satisfying the marginals The spectrogram that we study in the next section is one such example The total energy requirement is a weak one and that is why many distributions that do not satisfy it may nonetheless give a good representation of the timefrequency struc ture Characteristic Functions We have seen in Chapter 4 that characteristic functions are a powerful way to study distributions The joint characteristic function of a timefrequency density is M8r eatTW ffPtwei9t7Tdtdw 64 However many of the distributions that we will be studying are not proper By proper we mean a welldefined manifestly positive density function Hence for a particular distribution the characteristic function may not satisfy all the standard attributes of characteristic functions that we discussed in Chapter 4 84 Chap 6 TimeFrequency Distributions Fundamental Ideas 62 GLOBAL AVERAGES Global Averages The average value of any function of time and frequency is to be calculated in the standard way 9t w ff gt w Pt w dw dt 65 and of course the answers should be meaningful and reasonable If the marginals are satisfied then we are guaranteed that averages of the form 91t 92w ff91t 92wPtwdwdt 66 fitl st I2 dt f 92 w I Sw I2 dw 67 will be correctly calculated since the calculation requires only the satisfaction of the marginals But what about averages that are mixed that is average values of arbitrary time frequency functions An example is the covariance which involves the calculation of t w For mixed averages we do not know what we want and can only be guided by intuition and by the plausibility of possible guesses That is not surprising Math ematics cannot answer what is essentially an issue of science Consider for example the question of height and weight of an unknown species and suppose we knew the distribution of height and weight that is the marginals Can we know how height and weight are related how the correlation coefficient behaves No Similarly from knowing the timefrequency marginals we cannot know how time and frequency should be related and mathematics cannot help since mathematics allows all pos sibilities In Chapter 1 we discussed the covariance of a signal and showed how that is related to the first mixed moment t w We argued that a plausible guess to t w is the average of t gyp t and we showed that this quantity behaves reasonably well in the sense that it meets our intuition for the cases considered Therefore a plausible requirement of a joint distribution is t w JftwPtLidwdt f t cpt I st 12 dt 68 However for other mixed moments we have nothing to guide us 63 LOCAL AVERAGES Treating a joint timefrequency density as any other density we can immediately argue as was done in Chapter 4 that the density of frequency for a given time and the density of time for a given frequency are respectively given by Pw It Pt W Pt I W Pt W Pt Pw Sec 4 Time and Frequency Shift Invariance where Pt and Pw are the marginal distributions 85 Pt fPtwdw Pw fPtwdt 610 Notice that we have used Pt and Pw rather than I st 12 and Sw 12 to allow for the possibility that the marginals are not satisfied as will be the case with the spectrogram discussed in the next chapter The conditional average value of a function at a given time or frequency is Pt w dw f 6 11 gw t gw p t t t Pt dt f 6 12 g w 9 Pw In Chapter 2 we saw that a good candidate for the average frequency at a given time is the derivative of the phase the instantaneous frequency Similarly we argued that the average time for a given frequency should be the derivative of the spectral phase If we accept these results then a joint timefrequency density should satisfy w t Pt J w Pt w dw cPt 613 t PW ft Pt w dt iiw 614 From this point of view instantaneous frequency is an average the average fre quency at a particular time We should also be able to define the spread of fre quencies at a given time and the spread of time for a given frequency that is the conditional standard deviations These are 2 1 2 P t d 15 it Pt w w w w 6 2 1 2 at1w Pw Pt w dt t t 616 What should these quantities be These quantities will play a crucial role in the development of the basic theory so we hope they turn out to be sensible For some signals we know what the answer should be For example for a pure sinusoid we should get vIt 0 since a pure sinusoid has one definite frequency for all time 64 TIME AND FREQUENCY SHIFT INVARIANCE Suppose we have a signal s t and another signal that is identical to it but translated in time by to We want the distribution corresponding to each signal to be identical in form but that the one corresponding to the time shifted signal be translated by to That is 86 Chap 6 TimeFrequency Distributions Fundamental Ideas if st f st to then Pt w Pt to w 617 Similarly if we shift the spectrum by a constant frequency we expect the distribu tion to be shifted by that frequency if Sw Sw wo then Pt w Pt w wo 618 Both of these cases can be handled together If st is the signal then a sig nal that is translated in time by to and translated in frequency by wo is given by eOtst to Accordingly we expect the distribution to be shifted in time and frequency in the same way if st e0tst to then Ptw Pt tow wo 619 65 LINEAR SCALING For a signal st the signal given by sct asat is a scaled version of st The new signal is blown up or reduced depending on whether a is less or greater than one The square root factor keeps the normalization the same as the original signal The spectrum of the scaled signal is S W I Swa if ssct f sat 620 Vfa We see that if the signal is compressed then the spectrum is expanded and con versely If we want these relations to hold for the joint distribution then we must have P t w Pat wa The scaled distribution satisfies the marginals of the scaled signal 621 f 1 sct w dw Isect I2 a Jsat J2 622 J Psct w dt J Sscw J2 a 1 Swa 12 623 66 WEAK AND STRONG FINITE SUPPORT Suppose a signal doesnt start until t1 We want the joint distribution also to not start until t1 Similarly if the signal stops after time t2 we expect the distribution to be zero after that time If that is the case we say the distribution has weak finite time support The reason for the word weak will be apparent shortly Similarly if Sec 7 Uncertainty Principle 87 the spectrum is zero outside a frequency band then the distribution should also be zero outside the band In such a case we say that the distribution has weak finite spectral support We can express these requirements mathematically as Pt w 0 fort outside tl t2 if st is zero outside tl t2 624 Pt w 0 for w outside wl w2 if Sw is zero outside wl w2 625 Now suppose we have a signal that stops for a half hour and then starts again We would expect the distribution to be zero for that half hour Similarly if we have a gap in the spectrum then we expect the distribution to be zero in that gap If a distribution satisfies these requirements namely that it is zero whenever the signal is zero or is zero whenever the spectrum is zero then we say the distribution has strong finite support Pt w 0 if st 0 for a particular time 626 Pt w 0 if Sw 0 for a particular frequency 627 Strong finite support implies weak finite support but not conversely Distributions Concentrated in a Finite Region A signal cannot be both of finite duration and bandlimited in frequency Therefore if a distribution satisfies the weak finite support property it cannot be limited to a finite region of the timefrequency plane If it were it would be both time and frequency limited which is impossible If it turns out that a distribution is limited in a finite region then it does not satisfy the finite support properties andor the marginals 67 UNCERTAINTY PRINCIPLE In Chapter 3 we emphasized that the uncertainty principle depends on only three statements First and second are that the time and frequency standard deviations are calculated using I st 12 and I Sw I2 as the respective densities T2 J t t 21 st 12 dt 628 B2 fww21Sw12dw 629 and the third is that st and Sw are Fourier transform pairs From a joint distri bution the standard deviations are obtained by at t 2 Pt w dt dw f t t 2 Pt dt 630 0 Jfww2Ptwdtdw fw2Pwdw 631 88 Chap 6 TimeFrequency Distributions Fundamental Ideas When do we get the correct uncertainty principle When the standard deviations calculated using the joint distribution give the same answer as when calculated by Eqs 628 629 This will be the case when the marginals are correctly given Pt I st 12 and Pw I Sw 12 for uncertainty principle 632 Therefore any joint distribution that yields the correct marginals will yield and is totally consistent with the uncertainty principle 68 THE UNCERTAINTY PRINCIPLE AND JOINT DISTRIBUTIONS We have emphasized that the uncertainty principle depends only on the time and frequency marginals that is on 1 s t12 and I Sw12 being the density in time and fre quency Any joint distribution that has these marginals will satisfy the uncertainty principle For any two marginals there are an infinite number of joint distributions but the fact that the marginals are related imposes certain constraints on the possi ble joint distributions For example since the marginals are the absolute square of Fourier transform pairs and since we know that Fourier transform pairs cannot both be of finite extent we cannot possibly have a joint distribution that is confined to a finite region in the timefrequency plane We now examine how the uncertainty principle constrains the possible timefrequency distributions Let us crystalize the essence of the issue in the following way Suppose that someone has arranged the world so that no matter what marginals the two variables x and y have they are rigged so that the standard deviations of the marginals satisfy Qyox r 633 where r is a universal constant This type of situation is easy to set up So as not to get too abstract let us con centrate on the following example Suppose x and y have the following densities P r z x 6 34 x expL J P 1 277 z 5 6 y 27r7712 k2 1 exp 1 2 7712 k2l where l and k are any two positive numbers at our disposal The universal constant rl is fixed and not at our disposal These two marginals are perfectly good densities But no matter what we do or try to do we have the uncertainty principle ayQx rl That is the case since The more general case is where is a functional of the signal In the case of timefrequency it is a constant equal to half For time and scale for example it is functional of the signal See Section 155 To keep things simple we taken as a constant here but our discussions apply generally Sec 8 The Uncertainty Principle and Joint Distributions 89 Qx rl cry 712 k21 636 and therefore oyQy rl 1 l2k2 rl 637 No matter how we choose l and k we will always have ri Related Marginals Independence and Joint Distributions The reason we have an uncertainty principle is that the marginals are functionally related In this case that appears by way of the parameters l and k which occur in both distributions Be cause the marginals are functionally related if we change one marginal we change the other However one must not conclude from this that the variables are de pendent or that there is any correlation between them Functional relationship between the marginals does not mean that the variables are dependent That can be decided only by the joint distribution and not the marginals In Section 142 we show how to construct for any two marginals an infinite number of joint distributions Here we want to examine what constraints marginals that satisfy the uncertainty principle put on the possible joint representations Or phrasing the question another way what are the properties of joint distributions that produce marginals that satisfy the uncertainty principle Let us first consider some examples of joint densities that satisfy the above marginals Example 1 Consider the following perfectly well behaved joint distribution which satisfies the marginal and hence the uncertainty principle Px y 1 exp x a2 y b2 Px Py 638 27r77 V1 12k2 2r71 2112 k2l This distribution is the independent joint distribution Even though the marginals are related the variables are not correlated For any value of x the density of y is the same Example 2 Another joint density which satisfies the same marginals and has cor relations is 1 Px y 2ir771 12k21 r2 639 1 x a2 rx ay b y b2 ex 2 6 40 p 21 r2 77 1 l2k2 rll 77l2 k2l where r is any number between 1 and 1 For this case there is correlation be tween the variables The correlation is positive or negative depending on the sign of r There is a rich variety of possibilities but the marginals are always satisfied 90 Chap 6 TimeFrequency Distributions Fundamental Ideas Fig 61 The uncertainty principle depends only on the marginals It constrains the type of joint distribution that is possible A joint distribution cannot be narrow in both variables but this still leaves an infinite number of possibilities The marginals give no indication as to whether or how two variables may be correlated and so is the uncertainty principle In particular when one marginal is broad the other is narrow A proper joint distribution has no difficulty accommodating such marginals and although we have just demonstrated two of them there are an infi nite number of them 11771 Lest the reader think this is an artificial example we point out that it is precisely the Wigner distribution for a chirp with a Gaussian envelope Also constructing other joint distributions is very easy and we do so in Section 142 In Fig 61 we plot the distribution for two different sets of marginals and dif ferent values of l and k For a given set of marginals one can always find an infi nite number of joint distributions with various correlations between the variables From the marginals nothing can be concluded about the correlation between the variables 69 UNCERTAINTY PRINCIPLE AND CONDITIONAL STANDARD DEVIATION What general and specific statements can be made regarding joint distributions having marginals that satisfy the uncertainty principle Basically the only general property is that the joint distribution cannot be concentrated in both directions If it was then it would produce narrow marginals which would be a violation since both marginals cannot be narrow The fact that both marginals cannot be narrow eliminates a number of possible joint distributions but still leaves an infinite number to choose from We can fix this idea somewhat better by considering the conditional standard deviations of a joint distribution First let us recall that the uncertainty principle relates the global standard deviations But we know from Section 46 that Sec 10 The Basic Problems and Brief Historical Perspective 91 there is a relationship between the global standard deviation and the conditional standard deviation and the conditional average In particular writing Eq 43 for both x and y we have 02 JPxdx J Y y 2 Px dx 641 a f 2Iv Py dy f x v x 2 Py dy 642 Note that for each variable there are two ways to control the standard deviationand hence when we multiply ci by o we will have four terms the sum of which must be greater than r72 We can satisfy the uncertainty principle by making any one of those terms greater than 772 and have freedom with the rest of them Or in general we can choose any value for each term as long as the sum adds up to greater than 772 These four terms are a gross characterization of a joint distribution Because they are not individually constrained we can have a wide variety of qualitatively different joint distributions and still satisfy the uncertainty principle Signal Analysis The case of signal analysis is no different although historically the uncertainty principle was viewed as presenting unsurmountable difficulty for the construction of joint distributions Quite the contrary it is trivially easy to construct joint distributions consistent with the uncertainty principle as the above consider ations have shown All we have to do is satisfy the marginals In fact the real problem is that there are an infinite number of such joint distributions and we do not fully understand what other considerations besides the uncertainty principle are required to construct a sensible and comprehensive theory Satisfying the un certainty principle is easy the rest is challenging 610 THE BASIC PROBLEMS AND BRIEF HISTORICAL PERSPECTIVE The fundamental issue is how to construct joint distributions that satisfy the rather mild conditions we have set forth From a mathematical point of view there are an infinite number of joint distributions satisfying these requirements However writing them down explicitly is not easy and which ones out of the infinite number are right is not obvious Mathematics cannot solve the problem although it can guide us in constraining what is possible and what is not The basic issue is not one of mathematics since the conditions do not define the problem uniquely The issue is one of science engineering and desire to achieve particular goals The methods that have been developed to obtain distributions are not strict mathematical deriva tions but are based on physical considerations mathematical suggestiveness and the time honored scientific method of guessing intelligent and otherwise As of this writing the method of characteristic function operators which is discussed in the later chapters is the only consistent method for deriving distributions 92 Chap 6 TimeFrequency Distributions Fundamental Ideas The current state of affairs is that we do not have a complete theory Nonethe less the ideas and techniques that have been developed thus far are powerful give us considerable insight into the nature of signals meet our intuition to a very large extent and have been applied with immense success to practical problems This is demonstrated in the subsequent chapters Often some of the predicted results are clearly not plausible and that is what makes the subject challenging and fascinat ing Because we do not have a complete and comprehensive theory it is important to understand what is known with certainty what is speculation and what hidden assumptions go into any particular proof Since we have not yet studied any distributions it would not be appropriate at this juncture to discuss the mathematical or physical issues in any great detail To appreciate these issues fully we should first see how they arise We do mention one idea that has a checkered history Because many distributions have been proposed over the last fifty years something that should be considered in a positive light there have been attempts to prove that a particular one is best This has been done by listing a set of desirable conditions and trying to prove that only one distribution usually the writers favorite fits them Typically however the list presented is not complete with the obvious requirements because the author knows the added de sirable properties would not be satisfied by the distribution he or she is advocating Also these lists very often contain conditions that are taken out of thin air and are obviously put in to force a particular result Moreover there are usually hidden as sumptions in the proofs that are glossed over or hidden in the methodologies of the proofs However there is one benefit to these impossibility and uniqueness proofs The searching of the errors and hidden assumptions sometimes leads to a reexam ination of the basic ideas There are timevarying spectra in nature and therefore their description and properties can be described and understood Someone will eventually come up with the right theory and when that happens we will not have to prove it mathematically correct it will be obviously correct Terminology For the type of distributions we will be studying a rich variety of ter minology has arisen We first address the general usage of density and distribution When we say that a function represents the number of things per unit something that is called a density and hence density function In certain fields density func tions are called distributions because they indicate how things are distributed For example one says the Maxwell distribution of velocities As discussed in Chapter 4 we shall use the words density and distribution interchangeably These types of distributions first arose in quantum mechanics where the term probability den sity or distribution is properly applied since quantum mechanics is inherently probabilistic For deterministic signals where no probabilistic considerations enter the reader should think of distributions as intensities or densities in the com mon usage of the words or simply as how the energy is distributed over the timefrequency plane As we will see many of the known distributions may be come negative or even complex Hence they are sometimes called quasi or pseudo distributions Also a joint timefrequency distribution is of course dependent on the signal and can be said to represent the signal in time and frequency hence the phrase timefrequency representation Chapter 7 The ShortTime Fourier Transform 71 INTRODUCTION The shorttime Fourier transform is the most widely used method for studying non stationary signals The concept behind it is simple and powerful Suppose we listen to a piece of music that lasts an hour where in the beginning there are violins and at the end drums If we Fourier analyze the whole hour the energy spectrum will show peaks at the frequencies corresponding to the violins and drums That will tell us that there were violins and drums but will not give us any indication of when the violins and drums were played The most straightforward thing to do is to break up the hour into five minute segments and Fourier analyze each interval Upon examining the spectrum of each segment we will see in which five minute intervals the violins and drums occurred If we want to localize even better we break up the hour into one minute segments or even smaller time intervals and Fourier analyze each segment That is the basic idea of the shorttime Fourier transform break up the signal into small time segments and Fourier analyze each time segment to ascertain the frequencies that existed in that segment The totality of such spectra indicates how the spectrum is varying in time Can this process be continued to achieve finer and finer time localization Can we make the time intervals as short as we want The answer is no because after a certain narrowing the answers we get for the spectrum become meaningless and show no relation to the spectrum of the original signal The reason is that we have taken a perfectly good signal and broken it up into short duration signals But short duration signals have inherently large bandwidths and the spectra of such short duration signals have very little to do with the properties of the original signal This should be attributed not to any fundamental limitation but rather to a limitation of the technique which makes short duration signals for the purpose of estimating 93 94 Chap 7 The ShortTime Fourier Transform the spectrum Sometimes this technique works well and sometimes it does not It is not the uncertainty principle as applied to the signal that is the limiting factor it is the uncertainty principle as applied to the small time intervals that we have created for the purpose of analysis The distinction between the uncertainty principle for the small time intervals created for analysis and the uncertainty principle for the original signal should be clearly kept in mind and the two should not be confused We should always keep in mind that in the shorttime Fourier transform the properties of the signal are scrambled with the properties of the window function the window function being the means of chopping up the signal Unscrambling is required for proper interpretation and estimation of the original signal The above difficulties notwithstanding the shorttime Fourier transform method is ideal in many respects It is well defined based on reasonable physical principles and for many signals and situations it gives an excellent timefrequency structure consistent with our intuition However for certain situations it may not be the best method available in the sense that it does not always give us the clearest possible picture of what is going on Thus other methods have been developed which are discussed in subsequent chapters 72 THE SHORTTIME FOURIER TRANSFORM AND SPECTROGRAM To study the properties of the signal at time t one emphasizes the signal at that time and suppresses the signal at other times This is achieved by multiplying the signal by a window function ht centered at t to produce a modified signal stT sr hT t 71 The modified signal is a function of two times the fixed time we are interested in t and the running time T The window function is chosen to leave the signal more or less unaltered around the time t but to suppress the signal for times distant from the time of interest That is stT sr for 7 near t 0 for T far away from t The term window comes from the idea that we are seeking to look at only a small piece of the signal as when we look out of a real window and see only a relatively small portion of the scenery In this case we want to see only a small portion Since the modified signal emphasizes the signal around the time t the Fourier transform will reflect the distribution of frequency around that time Stw 7stT dT 1 727 aCWT sT hT t d7 74 Sec 2 The ShortTime Fourier Transform and Spectrogram 95 The energy density spectrum at time t is therefore 2 Pspt U I Stw I2 2n J eTsr hr t d7 75 For each different time we get a different spectrum and the totality of these spectra is the timefrequency distribution Psp It goes under many names depending on the field we shall use the most common phraseology spectrogram Since we are interested in analyzing the signal around the time t we presumably have chosen a window function that is peaked around t Hence the modified signal is short and its Fourier transform Eq 74 is called the shorttime Fourier trans form However it should be emphasized that often we will not be taking narrow windows which is done when we want to estimate time properties for a particular frequency When we want to estimate time properties for a given frequency we do not take short times but long ones in which case the shorttime Fourier transform may be appropriately called the longtime Fourier transform or the shortfrequency time transform The ShortFrequency Time Transform In motivating the shorttime Fourier trans form we emphasized the desire to study frequency properties at time t Conversely we may wish to study time properties at a particular frequency We then window the spectrum Sw with a frequency window function Hw and take the time transform which of course is the inverse Fourier transform In particular we de fine the shortfrequency time transform by s t 2rr f e3t Sw Hw w dcu 76 If we relate the window function in time ht with the window function in fre quency Hw by Hw 27r J ht e3t dt 77 Vr then St w ejt Swt 78 The shorttime Fourier transform is the same as the shortfrequency time transform except for the phase factor ejt Since the distribution is the absolute square the phase factor eit does not enter into it and either the shorttime Fourier transform or shortfrequency time transform can be used to define the joint distribution Pt W I Stw I2 I S t 12 79 This shows that the spectrogram can be used to study the behavior of time proper ties at a particular frequency This is done by choosing an Hw that is narrow or equivalently by taking an ht that is broad Narrowband and Wideband Spectrogram As just discussed if the time window is of short duration the frequency window Hw is broad in that case the spec 96 Chap 7 The ShortTime Fourier Transform trogram is called a broadband spectrogram If the window is of long duration then Hw is narrow and we say we have a narrowband spectrogram Characteristic Function The characteristic function of the spectrogram is straight forwardly obtained MsPBT ff I Stw I2 ejetiTdtdw 710 A8O T Ah6 r 711 where A0T fst 2r st 2T e3et dt 712 is the ambiguity function of the signal and Ah is the ambiguity function of the window defined in the identical manner except that we use ht instead of st Note that AO T A 0 r a relation we will use later Notation The results we will obtain are revealing when expressed in terms of the phases and amplitudes of the signal and window and their transforms The nota tion we use is st At Owt ht Aht eawnt 713 Sw Bw Hw BHw erLXW 714 In the calculation of global averages eg mean frequency bandwidth we will have to indicate which density function is being used We will use the superscript e for example to indicate that the signal is being used In particular the mean of frequency with respect to the spectrogram signal and window will be indicated respectively by wSP f715 w8 J w I Sw I2 dw 716 w h J w Hw I2 dw 717 and similar notation will be used for other quantities When it is clear from the context which density is being used the superscript notation will be omitted 73 GENERAL PROPERTIES Total Energy The total energy is obtained by integrating over all time and fre quency However we know that it is given by the characteristic function evaluated at zero Using Eqs 711 and 712 we have Sec 3 General Properties 97 Esp JfPstwdtdw MspO 0 718 A 0 0 Ah 0 0 719 r I st 12 dt x f I ht I2 dt 720 Therefore we see that if the energy of thelwindow is taken to be one then the energy of the spectrogram is equal to the energy of the signal Marginals The time marginal is obtained by integrating over frequency Pt f I Stw I2 dw 721 f sr hT t9r hr t ejcTTdrdr dw 722 2f 3T hT tsT hT t 6r r dT dT 723 fsr2IhrtI2dr 724 fA2rA2hrtdr 725 Similarly the frequency marginal is Pw B22Iw w dw 726 As can be seen from these equations the marginals of the spectrogram generally do not satisfy the correct marginals namely I st I2 and I Sw 12 Pt A2t I st 12 727 Pw 0 B2w I Sw I2 728 The reason is that the spectrogram scrambles the energy distributions of the win dow with those of the signal This introduces effects unrelated to the properties of the original signal Notice that the time marginal of the spectrogram depends only on the magni tude of the signal and window and not on their phases Similarly the frequency marginal depends only on the amplitudes of the Fourier transforms Averages of Time and Frequency Functions Since the marginals are not satisfied averages of time and frequency functions will never be correctly given 98 Chap 7 The ShortTime Fourier Transform 91t 92w if 91t 92w Psptwdwdt 729 fgitstI2dt fg2wS2dJ 730 never correctly given This is in contrast to other distributions we will be studying where these types of averages are always correctly given Finite Support Recall from our discussion in Chapter 6 that for a finite duration signal we expect the distribution to be zero before the signal starts and after it ends This property was called the finite support property Let us see whether the spectro gram satisfies this property Suppose one chooses a time t before the signal starts Will the spectrogram be zero for that time Generally no because the modified signal as a function of t will not necessarily be zero since the window may pick up some of the signal That is even though st may be zero for a time t sr hr t may not be zero for that time This will always be the case for windows that are not time limited But even if a window is time limited we will still have this effect for time values that are close to the beginning or end of the signal Similar considera tions apply to the frequency domain Therefore the spectrogram does not possess the finite support property in either time or frequency Localization Tradeoff If we want good time localization we have to pick a narrow window in the time domain ht and if we want good frequency localization we have to pick a narrow window Hw in the frequency domain But both ht and Hw cannot be made arbitrarily narrow hence there is an inherent tradeoff be tween time and frequency localization in the spectrogram for a particular window The degree of tradeoff depends on the window signal time and frequency The uncertainty principle for the spectrogram quantifies these trade off dependencies as we discussed in Section 34 One Window or Many We have just seen that one window in general cannot give good time and frequency localization That should not cause any problem of principle as long as we look at the spectrogram as a tool at our disposal that has many options including the choice of window There is no reason why we cannot change the window depending on what we want to study That can sometimes be done effectively but not always Sometimes a compromise window does very well One of the advantages of other distributions that we will be studying is that both time and frequency localization can be done concurrently Entanglement and Symmetry Between Window and Signal The results obtained using the spectrogram generally do not give results regarding the signal solely be cause the shorttime Fourier transform entangles the signal and window Therefore we must be cautious in interpreting the results and we must attempt to disentan Sec 4 Global Quantities 99 gle the window That is not always easy In fact because of the basic symmetry in the definition of the shorttime Fourier transform between the window and sig nal we have to be careful that we are not using the signal to study the window The mathematics makes no distinction The distinction must come only from a ju dicious choice of window which is totally under our control The basic symmetry between the window and signal does have a mathematical advantage in that results obtained should show a symmetry and therefore act as a mathematical check 74 GLOBAL QUANTITIES The mean time and frequency in the spectrogram are given by tSP JW f f Stw12dtdu 731 Direct evaluation leads to tSP t8 th W SP Wa W h 732 If the window is chosen so that its mean time and frequency are zero which can be done by choosing a window symmetrical in time and whose spectrum is symmetri cal in frequency then the mean time and frequency of the spectrogram will be that of the signal The second conditional moments are calculated to be W2SP W2 a W2 h 2 W e W h 733 t 2 SP t2 8 t2 h 2 t 8 t 734 By combining these with Eqs 732 we find that the duration and bandwidths are related by TspTa Th B2PB8 Bh 735 which indicates how the duration of the windowed signal is related to the durations of the signal and window Covariance and Correlation Coefficient For the first mixed moment t U SP fftwStw2dtdd t a th tg Ph h 736 737 100 Chap 7 The ShortTime Fourier Transform Subtracting t SP w SP as given by Eqs 732 from both sides we have that the covariance of the spectrogram is Cov t w SP t SP w 5P Cove CovtW 738 We know from Chapter 1 that the covariance of a real signal is zero and hence if we take real windows the covariance of the spectrogram will be the covariance of the signal CovtWP Covtw for real windows 739 75 LOCAL AVERAGES Method of Calculation Much of the analysis and calculations using the spectro gram is simplified significantly if we keep in mind that the modified signal sh and the shorttime Fourier transform St w form a Fourier pair with respect to the run ning time T sT hT t b Stw f Fourier pair between r w 740 The real time t is considered a parameter The modified signal expressed in terms of the phases and amplitudes is stT sr hT t Ar AhT t eiwTcphTt 741 A fruitful way to look at the situation is that we are dealing with a signal in the variable r whose amplitude is AT AhT t and whose phase is WT YhT t However stT is not normalized in the variable T and therefore we define as in Section 34 the normalized modified signal by 77t 0 ST hr t sT hr t 742 f sT hT t j2 dT Pt Everything we have done in Chapter 1 can be translated to the calculation of condi tional values for the spectrogram by replacing the signal s with 7 and considering r as the time In particular the conditional average of any function is 9w t pI f 9w I Stw 12 f 17 7 9j d7 ntT d7 743 Local Frequency Estimate of Instantaneous Frequency Look at Eq 190 and simply let A2 b A2 T Ah T t and W cp T ch T t to obtain fw Stw 12 dw firitTdT 744 w t Pt 1 JAT AhT t 9r p r t dr 745 Pt Sec 6 Narrowing and Broadening the Window Local Square Frequency Similarly transcribing Eq 198 yields w2t f1iT 2rtrdr I 1 dTiitT12dT 12 P1 l AT AhT t dr T dr 101 746 P1 IArArtfwrwh7tld7 747 The Conditional or Instantaneous Bandwidth Using a w2 t w t with w2 t and wt as given by Eqs 745 and 747 we can obtain the conditional or instantaneous bandwidth However the resulting expression does not explicitly show the manifestly positive nature of the standard deviation A way to obtain an alternative but manifestly positive expression is to use Eq 196 the bandwidth equation and do the appropriate replacement as above Bt QWIt Pt Jwwt2Stwl2dw Pt I dT Ar Ahr t dT 748 I t If A2rl A2r2 Ahr1 t Ah72 t 2P2 X dr1 dr2 749 For convenience we define z w2 0 P1 1 dd7 Ar AhT t dT 750 76 NARROWING AND BROADENING THE WINDOW All the above results show that the physical quantities obtained using the spectro gram result in an entanglement of signal and window We can view these quantities as estimates which unfortunately depend on the window function chosen How ever one would hope that in some sense or other the basic results are window independent and that when the window is narrowed we will get better and bet ter estimates Let us first consider the conditional frequency and suppose that we 102 narrow the window so that A t approaches a delta function A2 t 5t Chap 7 The ShortTime Fourier Transform 751 Also we consider real windows so as not to introduce phase due to the window In this limit Eq 745 goes as w t f A27AhrtcpTdr f A2T Ah r t dr f A2r 6T t cor dT A2t cpt f A2 r 6T t dr A2t 752 753 or wtcpt 754 That is if the window is narrowed to get increasing time resolution the limiting value of the estimated instantaneous frequency is the derivative of the phase which is the instantaneous frequency This a very pleasing and important result but there is a penalty to pay As we narrow the window we broaden the standard deviation and therefore as the estimate goes to the right answer it becomes harder to see and determine The reason for this can be seen by direct mathematical substitution of the delta function in Eq 749 which results in Qwit 4 00 for Aht 5t 755 A simple explanation is available we have taken a signal and made it of narrow duration at time t in fact we have made it infinitely narrow by letting the magni tude square of the window go to a delta function The standard deviation of the modified signal must therefore go to infinity because of the timebandwidth prod uct theorem Even though the average is correctly predicted it is hard to discern A compromise is discussed in Section 711 77 GROUP DELAY We now give the average time for a given frequency Because of the symmetry we can immediately write t Pw f B2w Bhw w Pw Oh w w dw 756 where Pw is the marginal in frequency given by Eq 726 If the window is nar rowed in the frequency domain then a similar argument as before shows that the estimated group delay goes as t for H2w 6w 757 Sec 8 Examples 103 The second moment and conditional standard deviation are given by t2W 2 Pw f dw Bw B w w dw Pw f B2w Bhw w iiw 0 w w 2 dw 758 2 Q j w fBwBhwdw 2P2w ffB2wi B2w2 Bh2w wl B2w W2 x Ywl IPw2 4 W wl Oh w W2 2 dwl dw2 759 The standard deviation of time at a given frequency may be interpreted as the duration of the signal for that frequency The physical significance can be thought of in the following way We envision a signal as composed of many frequencies each frequency having an envelope The duration at each frequency is the time width of the envelope for that frequency Equation 759 is an estimate of that squared dura tion All the remarks we made about the estimated instantaneous frequency apply to the group delay except we now have to narrow the window in the frequency domain 78 EXAMPLES Example 71 Sinusoid As a first example we consider a sinusoid and use a Gaussian window st eWot ht ax114 eat22 760 The shorttime Fourier transform is 2 Stw Q jl4 eiWWOt exp 2ao J 761 which yields the following timefrequency distribution PSPtw I Sew I2 d l2 eXp w a 02 Using it we have 762 M t Wo o e ia 763 The average value of frequency for a given time is always wo but the width about that frequency is dependent on the window width 104 Chap 7 The ShortTime Fourier Transform Example 72 Impulse For an impulse at t to with the same window as above st 27r 6t to ht a7r14 aat22 we have Stw a7r 14 awt etto22 PsPtw I Stw I2 a7r12eatt02 764 765 766 Example 73 Sinusoid Plus Impulse Now consider the sum of a sinusoid and impulse st e0t 27x6t to ht a7r 14 eat22 767 We have St W a 114 eWwt eXp w 2a o2 1 a7r 14 aito eatto22 l J 768 and PsPt W I Stw I2 a 112 eww02a a7r 12 eatt2 2 eWwo2aatto2 coswt to wot 769 7r This example illustrates one of the fundamental difficulties with the spectrogram For one window we cannot have high resolution in time and frequency The broadness of the self terms the first two terms of Eq 769 depends on the window size in an inverse relation If we try to make one narrow then the other must be broad That will not be the case with other distributions In Fig 71 we plot the spectrogram for different window sizes The cosine term is an example of the so called cross terms We discuss these in greater detail later Note that they essentially fall on the self terms in the spectrogram Example 74 Linear FM Consider a chirp with a Gaussian envelope and a Gaussian window st Q7x14 eet22jt22jwot ht a7r 14 aate 770 A straightforward but lengthy calculation gives Pt PsPw I Stw I2 w tex p JI 771 Pw exp t 2 w2 772 7 021w 2ortlw 20 0 0 b 20 10Q W010 00 20 10 10 0 0 Fig 71 Spectrograms of a signal composed of a constant frequency plus an im pulse st ej1ot 6t 10 In a we use a short duration window which gives a good indication of when the impulse occurred but gives a broad localization for the frequency In b we use a long duration window which gives the opposite effect In c a compromise window is used For a comparison with the Wigner distribution see Fig 85 where Pt and Pw are the marginal distributions Pt V 7ra a exp f as t2 773 and as as 21 PW ex p J 774 aa2 aa2 02 p aa2 aa2 32 a 2 w t a a pt wo Uwlt 2 a a 1 ap a 775 tom aR T2 w 776 aa2 aa2 2 1 a a2 Q2 7 t 2 aa2 aa2 N2 The concentration of energy for a given time is along the estimated instantaneous fre quency and for a given frequency it is along the estimated time delay As the window becomes narrow that is as a oo the estimate for the instantaneous frequency ap proaches pt wo However in this limit the estimated group delay approaches zero which is a very bad estimate In fact this answer is just the average time for the sig nal It is an understandable answer because in that limit we have a flat window in the frequency domain Conversely if we want to focus in on time properties for a given frequency we must take a broad window in the time domain which is achieved by taking a oo In that case t 0a2 p2 which is the correct answer see Eq 106 Chap 7 The ShortTime Fourier Transform 1121 On the other hand at this limit the estimate of the instantaneous frequency approaches wo which is not the instantaneous frequency at all but just the average frequency Again the reason is in that limit the window in the time domain is flat for all time This example shows that for the spectrogram one window cannot give good localization in both time and frequency In contrast other distributions will allow us to do that We also note that for this case the local bandwidth is a constant in time That would not be the case if a different window were chosen Example 7S Single Sided Sinusoid with Rectangular Window Take st e0 t 0 ht T2 t T2 778 VfT A straightforward calculation yields 7 eiWwotT2 I T2 t T2 t 27r w wo 1 2ejot sinw woT2 T2 t M U 779 and n w wot T2 T2 t T2 sin22 Stw 12 2 w 1 2 780 w sin2w woT2 T2 t Even though the signal is zero before t 0 the spectrogram indicates that there is a value in the range T2 t T2 The reason is that for that range the window is picking up values from times greater than zero The longer we make the window the sharper the spectrogram will be around w wo but then we will have poor lo calization in time If we take a small window then the spectrum will be broad This is illustrated Fig 72 This example shows that the spectrogram does not satisfy the finite support property Example 76 Chirp With Rectangular Window For the signal of Eq 770 if a rectangular window is used instead of a Gaussian we obtain Pt ZT erf t4 ZT erf t IT 781 wt Pt a T etT4 sinh atT 782 where erfx is the error function It can be readily verified that wt w as the window size becomes small If we now attempt to calculate the second conditional moment and the standard deviation infinity is obtained This is due to the sharp cutoff of the rectangular window Hence a new measure of spread is needed for some windows or the window has to be modified to avoid divergences Sec 8 Examples 5 eif Sd n 10 107 Fig 72 Spectrograms of a sine wave that starts at t 5 In b a very broad window is used to get good frequency localization This spectro gram does not give a good indication of where the signal is zero In c a very narrow window is used The spectrum is very broad because it is the spectrum of a short duration sig nal the modified signal Eq 71 Example 77 Sinusoidal FM SignalChirp with a Gaussian Window Consider st ce7r14 eate2jOt2jwotjmsinwmt ht a7r14 aat22 783 The instantaneous frequency is wii m wm COS wmt 3t wp 784 which indicates that the frequency oscillates sinusoidally around the straight line given by 3two To calculate the estimated instantaneous frequency we use Eq 745 to ob tain wt a s 3twomwme 4a acos a u t 785 eCOS t11 786 The standard deviation is calculated from atjw w2 t W l In the limit as a p 0 w t p wi as expected The shorttime Fourier transform can be done exactly Substituting the signal and window in the definition we have ack Stw 14 aat2 787 ejmsinwmr E Jnm ejnwmr 788 noo 2 w 2 108 Chap 7 The ShortTime Fourier Transform where J are the Bessel functions Substituting this into Eq 787 and evaluating the Gaussian integral we have 14 1 eate2 Stw ir2 a a as jp 2 79 INVERSION 00 X Am exp f l at jw WO rudm 2 789 L 2aaj n00 Can the signal be recovered from the spectrogram Since the spectrogram is the absolute square frequency of the shorttime Fourier transform it is often stated that we have lost the phase and hence the signal cannot be recovered This argument is not correct because the phase we have lost is the phase of the shorttime Fourier transform not the signal In fact the phase and amplitude of the signal appear in both the phase and amplitude of the shorttime Fourier transform Therefore hav ing the amplitude of the shorttime Fourier transform may be sufficient to recover the signal We now see under what conditions this is the case From Eq 711 the ambiguity function of the signal is A89r MsP0 790 AhB Ah 19 I st p I2ejetiT4 dt dw 791 T But the ambiguity function determines the signal To see this we Fourier invert the ambiguity function of the signal Eq 712 to obtain s8t 2T st 2T 21 J A0T aBt d9 792 and taking a particular value t T2 we have 88 0 st 2a J A8 0 t a3et2 dg 793 where s 0 is a number Therefore st 1 MsPB t ejet2 d9 794 27rs 0f Ah 0 t which shows that the signal can be recovered from the spectrogram However there is an important caveat The division by the ambiguity function of the window must be possible and that will be the case if it has no finite regions where it is zero in the 0 T plane That depends on the window function Sec 10 Expansion In Instantaneous Frequency 109 710 EXPANSION IN INSTANTANEOUS FREQUENCY We now show that the local quantities have interesting expansions in terms of the instantaneous frequency and its derivatives For simplicity we consider real win dows Local Mean Frequency We rewrite Eq 745 as wt fA2fA2tAd tdT Expanding goT t as a power series in r T t Wn1 t L n n0 and substituting into Eq 795 we have w t M7 t gnl t n0 where Mnt ff A t Ahr dr 795 7 797 798 For n 0 the denominator and numerator are equal and hence Mot is always equal to one Writing out the first few terms for w t we have wt goMItWit 21 M2tgo 799 Similar considerations lead to the following expansions for the mean square and conditional standard deviation 00 n w2 t w2 t E E no k0 nk1 go k1 n k k Mnt w2 0 go2 2 gol go M1 go Writ gor2 7100 M2 7101 1 00 n 1 i 1 gonk1 gokl Mn Mnkk 7102 or 21t U20 k n2 k1 n k ti w2 t M2M12go112 M3M2M1Wit g 7103 110 Chap 7 The ShortTime Fourier Transform Note that the derivative of the phase of the signal is absent from the expansion of o t Also M2 M1 is the local duration as defined by Eq 334 2 2 Tt M2Ml Example 78 Arbitrary Frequency with Gaussian Window We take an arbitrary phase but impose a Gaussian amplitude modulation st a7r14 eQt229wt A2 t ax12 aate 7104 7105 in which case the integrals can be readily done to obtain 2 W t 4P a aV 2 a a pi rr 7106 2 1 1 112 at it Ill 1 1 a2t2 rr rrrr It 2aa 2aa aa2 2aa 2 aa 7107 711 OPTIMAL WINDOW To obtain the finest resolution of the instantaneous frequency we minimize the spread about the estimated instantaneous frequency That is a window should be chosen so that the local standard deviation QIt is a minimum Since the local standard deviation depends on the signal the optimal window will also depend on the signal How the dependence comes in may be seen in rough terms as follows As an approximation we drop all terms higher than the second in the expansion of QIt in Eq 7102 Q21t w2 t T2 2t 7108 Now roughly by the time dependent uncertainty principle we have that w2 c T2 i t t 4 and therefore 1 c I t N 4T2 T2 lp12t The minimum is achieved at TtzlII t N 2 cp t At that value the local standard deviation is 7109 7110 rr I t Pt 7112 Sec 11 Optimal Window 111 We emphasize that 7 is not the width of the window It is the width of the modified signal at time t which involves both the window and the signal However if the signal is purely frequency modulated then T2 is the width of the window in which case we have Optimal width of window N 1 7113 21 V11t for purely frequency modulated signals Example 79 Exactly Solvable Equation 775 gives the exact answer for a It and it is revealing to do the minimiza tion exactly To minimize Quit we set the derivative with respect to a to zero da2 WIt da which gives I I 0 2 2 a a 2 7114 amm 101 a 7115 Note that am does not have to be positive If it is negative the window will not be normalizable but nonetheless the STFT will be normalizable as long as aom a is pos itive Hence divergent windows may be used and indeed are desirable and necessary to obtain a high resolution The standard deviation of the window is ow 12a and we have that the optimal window is Ow 2am 2161 a 7116 If we have a constant amplitude signal a 0 then aoi 13 1 in accordance with Eq 7113 Example 710 Gaussian Envelope with Arbitrary Phase Let st a7r 14 eat223wt and suppose we use the window We obtain where amm 2 t222 Aht 2ittrw e 1 2C162 3c26 2Qw N a p4 6c1cp 12c2 7117 7118 7119 Cl 4 c i 8 2 at 40 will 7120 C2 1 a2t2 Wit i 112 2 1 7121 J For a chirp where WP pt c1 and c2 are both zero and we have that am I R a which agrees with Eq 7115 Q2 112 Chap 7 The ShortTime Fourier Transform Example 711 Cubic Phase For the cubic case where the phase is 0t ryt33 we get the following approxima tion dun N 27213 a 2cx2 S r213 3 13 J t1 2ryt3a47t4b t t 00 t0 7122 Chapter 8 The Wigner Distribution 81 INTRODUCTION The Wigner distribution is the prototype of distributions that are qualitatively dif ferent from the spectrogram The discovery of its strengths and shortcomings has been a major thrust in the development of the field Although it has often been studied in contrast to the spectrogram we will see in Chapter 9 that both are mem bers of a general class of representations Wigner was aware that there were other joint densities but chose what has now become the Wigner distribution because it seems to be the simplest Nonetheless Kirkwood30 a year later and Terletsky15411 a few years after came up with a sim pler candidate that is commonly called the Rihaczek or MargenauHiU distribution The Wigner distribution was introduced into signal analysis by Ville161 some 15 years after Wigners paper Vile gave a plausibility argument for it and de rived it by a method based on characteristic functions Remarkably the same type of derivation was used by Moyal390l at about the same time These derivations appeared to be from first principles however they contained an ambiguity and Wigners original motivation for introducing it was to be able to calculate the quantum correction to the second virial coefficient of a gas which indicates how it deviates from the ideal gas law Classically to calculate the second virial coefficient one needs a joint distribution of position and momentum So Wigner devised a joint distribution that gave as marginals the quantum mechanical distributions of position and momentum The quantum mechanics came in the distribution but the distribution was used in the classical manner It was a hybrid method We discuss the analogy of quantum mechanics to signal analysis in Section 1313 Also Wigner was motivated in part by the work of Kirkwood and Mar genau who had previously calculated this quantity but Wigner improved on it Kirkwood subsequently developed what is now the standard theory for nonequilibrium statistical mechanics the BBGKY Hier archy the theory was developed independently by Bogoliubov Born and Green Kirkwood and Yvon Kirkwood attempted to extend the classical theory to the quantum case and devised the distribution commonly called the Rihaczek or MargenauHill distribution to do that Many years later Margenau and Hill derived the MargenauHill distribution The importance of the MargenauHill work is not the distribution but the derivation They were also the first to consider joint distributions involving spin 113 114 Chap 8 The Wigner Distribution the generalization of the method which we shall call the characteristic operator method can be used to derive all distributions This is discussed in Chapters 10 and 17 In a comprehensive study MarkJ pointed out the cross term issue of the Wigner distribution and also showed the relation of the Wigner distribution to the spectrogram In 1980 in an important set of papers Claasen and Meck lenbraukeri11e1191201 developed a comprehensive approach and originated many new ideas and procedures uniquely suited to the timefrequency situation Calculating properties of the Wigner distribution is fairly straightforward How ever they can be easily determined by the methods we will develop to study the general class of distributions in Chapter 9 Hence in this chapter we do not delve into detailed proofs for every property but emphasize the interpretation of the re sults 82 THE WIGNER DISTRIBUTION The Wigner distribution in terms of the signal st or its spectrum SW is W t w 2 J s t 2rr st 2 r ajr dr 81 1 S w 10 Sw 29 aj8 dB 82 21r The equivalence of the two expressions is easily checked by writing the signal in terms of the spectrum and substituting into Eq 81 The Wigner distribution is said to be bilinear in the signal because the signal enters twice in its calculation Notice that to obtain the Wigner distribution at a particular time we add up pieces made up of the product of the signal at a past time multiplied by the signal at a future time the time into the past being equal to the time into the future There fore to determine the properties of the Wigner distribution at a time t we mentally fold the left part of the signal over to the right to see if there is any overlap If there is then those properties will be present now at time t If this simple point is kept in mind many issues and results regarding the Wigner distribution become clear For example suppose we have a finite duration signal with noise appearing only for a small part of the time We use noise only for the purpose of illustration and our remarks hold for any other property of the signal Now let us pick a time and ask whether noise will appear at that time Fold the signal at that time and if in the folding the noise is overlapped then noise will appear at the present time even though there is no noise in the signal at this time This is illustrated in Fig 81 Everything we have said for the time domain holds for the frequency domain because the Wigner distribution is basically identical in form in both domains An other important point is that the Wigner distribution weighs the far away times equally to the near times Hence the Wigner distribution is highly nonlocal Range of the Wigner Distribution The above argument shows that for an infinite duration signal the Wigner distribution will be nonzero for all time since no matter Sec 2 The Wigner Distribution w ta tb Fig 81 An easy way to ascertain the behavior of the Wigner distribution is to men tally fold over the signal about the time being considered and determine whether the overlap includes properties of interest We have illustrated a finite duration sig nal where noise appears for a small time interval At time to no noise will appear in the Wigner distribution because if we fold over the signal about to there is no noise in the overlap Now consider time tb Folding over does result in overlap with the noise and thus noise will apear at time tb even though there is no noise at that time Although we have used noise for illustration the same arguments apply to any other property tl t2 to t3 Fig 82 The Wigner distribution is not generally zero when the signal is zero and similarly it is not zero for values of frequency at places where the spectrum is zero To see this consider the signal illustrated which is zero for the interval t2 t3 Now focus at t ta Mentally fold over the signal and note that we have an overlap Hence the Wigner distribution will not be zero at to even though the signal is zero then 115 what time we choose the folding of the right with the left parts will result in a nonzero answer Now let us consider a signal that has a beginning in time and calculate the Wigner distribution for a time before the signal starts Mentally folding over the right with the left we will get zero since there is nothing to the left to fold Hence the Wigner distribution will be zero for times before the signal starts Also for a signal that stops the Wigner distribution will be zero after that time For a finite duration signal the Wigner distribution will be zero before the start of the signal and after the end The same arguments in the spectral domain show that for a bandlimited signal the Wigner distribution will be zero for frequencies outside the band Therefore the Wigner distribution satisfies the finite support properties in time and frequency Wt w 0 fort outside t1 t2 if st is zero outside t1 t2 83 Wt w 0 for w outside w1i w2 if Sw is zero outside w1i w2 84 116 Chap 8 The Wigner Distribution Now consider a signal that is turned off for a finite time and then turned on and let us focus on a time for which the signal is zero as illustrated in Fig 82 Will the Wigner distribution be zero No because if we fold over the right side with the left side we do not obtain zero Similar consideration applies to the spectrum Therefore generally the Wigner distribution is not necessarily zero at times when the signal is zero and it is not necessarily zero for frequencies that do not exist in the spectrum Manifestations of this phenomenon have sometimes been called inter ference or cross terms and the cause for this behavior has very often been attributed to the fact that the Wigner distribution is bilinear in the signal It is not bilinearity as such that is doing it since there are other joint distributions that are bilinear satisfy the marginals but are always zero when the signal is zero Particular illustrations of this effect are presented in the examples and also when we discuss multicomponent signals The Characteristic Function of the Wigner Distribution We have MO T if ejot7Td W t w dt dw 85 2zrfff 2rst 86 Jf eot8r Tst 2T st 2T dT dt 87 J s t 2T st 2T eot dt 88 AB T 89 This function and variants of it have played a major role in signal analysis This particular form is called the symmetric ambiguity function It was first derived by Vile and Moyal and its relation to matched filters was developed by Woodward We have previously discussed it in the calculation of the characteristic function of the spectrogram Eq 712 In terms of the spectrum the characteristic function is M9 T f S w 2e Sw Z9 e3 dw 810 Nonpositivity We mentioned in Chapter 6 that a bilinear distribution that satis fies the marginals cannot be positive throughout the timefrequency plane it must go negative somewhere As we will see the Wigner distribution does satisfy the marginals and hence we expect it to always have regions of negative values for any signal That is indeed the case with one glaring exception the signal given by Eq 843 How is it possible that there can be an exception The reason is that the Wigner distribution for that signal is not really bilinear and it belongs to the class of positive distributions that are not bilinear These distributions are considered in Section 142 Sec 3 General Properties 117 83 GENERAL PROPERTIES We now discuss the basic properties of the Wigner distribution Reality The Wigner distribution is always real even if the signal is complex This can be verified by considering the complex conjugate of W t w Wtw 2rJ stZTst2Tedr 811 1 OO 1 st2Tst 2TajTdT 812 1 O 27rf st IT st 2T aT dr 813 Wt w 814 The fact that the Wigner distribution is real for any signal can also be seen from the characteristic function Recall that M 0 T M0 T is the condition for a distribution to be real But the characteristic function of the Wigner distribution is the ambiguity function A0 T Eq 88 which does satisfy this property Symmetry Substituting w for w into the Wigner distribution we see that we ob tain the identical form back if the signal is real But real signals have symmetrical spectra Therefore for symmetric spectra the Wigner distribution is symmetrical in the frequency domain Similarly for real spectra the time waveform is symmetrical and the Wigner distribution is symmetric in time Therefore Wt w Wt w for real signals symmetrical spectra Sw Sw 815 W t w Wt w for real spectra symmetrical signals st st 816 Marginals The Wigner distribution satisfies the timefrequency marginals f Wtw dw I st I2 817 f W t w dt I Sw I2 818 Both of these equations can be readily verified by examining M8 0 and M0 T By inspection of Eq 88 and Eq 810 we have M0 0 f 819 118 Chap 8 The Wigner Distribution But these are the characteristic functions of the marginals and hence the marginals are satisfied To do it directly Pt J Wt w dw 21113 t 2rst2rejrdrdw 820 fstTstr6rdr 821 I st 12 822 and similarly for the marginal in frequency Since the marginals are satisfied the total energy condition is also automatically satisfied E f W t w dw dt f I st 12 dr 1 823 Tune and Frequency Shifts If we time shift the signal by to andor shift the spec trum by wo then the Wigner distribution is shifted accordingly if st eJotst to then W t w W t to w wo 824 To see this we replace the signal by ejwot st to in the Wigner distribution and call Wah the shifted distribution 1 eiotr2 st to 2r Wshtw 21r J x e wotr2 st to 2r aJr dr 825 1 1 1 7ro 27r s t to 2r st to 2r a d7 826 Wt tow wo 827 84 GLOBAL AVERAGES The global average of a function g is gtw if W t w dw dt 828 Since the Wigner distribution satisfies the marginals it will give the correct answer for averages that are only functions of time or frequency or the sum of two such functions 91t 92w fJ91t g2wWtwdwdt 829 Sec 5 Local Averages 119 f91t I st I2 dt J 92w I Sw 12 dw 830 always correctly given since averages are calculated with correct marginals Mean Time Mean Frequency Duration Bandwidth and the Uncertainty Princi ple All these quantities are automatically satisfied since all these quantities depend on the marginals only which are correctly given by the Wigner distribution Correlation and Covariance The first mixed moment of the Wigner distribution is tw fftwWtwdtdi Jtwt I st12dt 831 Therefore the Wigner distribution gives the covariance of a signal as defined by Eq 1124 85 LOCAL AVERAGES The first conditional moments of time and frequency are w t l st 12 Jw W t w dw t Sw I2 ftWtwdt 832 When these are evaluated we obtain w t WV t 0w 833 where 0 and 0 are the phase and spectral phase of the signal But Wt and 0W are the instantaneous frequency and group delay These are important results be cause they are always true for any signal Recall that for the spectrogram they were never correctly given although we could approximate one or the other Further more if we tried to make one of the two relations approximately true by narrowing the window the other relation would become very poor Local Spread The result just obtained shows that instantaneous frequency is an average the conditional average for a particular time We now consider the spread about that average the conditional standard deviation First consider the second conditional moment in frequency w2 t st 12 fw2 W t w dw 834 1 CAI t 2 At 2t 835 2 At At 120 Chap 8 The Wigner Distribution where A is the amplitude of the signal The conditional spread in frequency is O 2 It w2t wi At 2 At 2 At At 836 837 This expression for o may go negative and hence cannot be properly interpreted In fact for such a case wIt is imaginary Therefore while the Wigner distribution gives an excellent result for the average conditional frequency it gives a very poor one for the spread of those frequencies Are there distributions that give a plausible answer The answer is yes and we consider this question in Section 132 86 EXAMPLES Before proceeding with the general properties of the Wigner distribution we con sider some examples Example 81 Sinusoid and Impulse For the sinusoid and impulse the Wigner distribution gives an answer totally consis tent with intuition 8t ejWOt Wt w 6w wo 838 st 2a6t to W t w 6t to 839 For a sinusoid the Wigner distribution is totally concentrated along the frequency of the sinusoid and for the impulse it is totally concentrated at the time of occurrence Example 82 Sinusoid with Gaussian Amplitude If we modulate a pure sinusoid with a Gaussian envelope then st a7r 14 eat22jIot Wtw 1IreateWWOa 840 We see that the Wigner distribution is still concentrated around the single frequency of the sinusoid but now there is spreading In addition for any particular value of frequency it falls off as the squared amplitude of the signal To calculate the conditional spread Eq 837 we have AA at AA a2t2 a QW t a2 841 Note that this expression for the conditional spread as given by the Wigner distribu tion is always the case for a Gaussian envelope since the conditional spread does not depend on the phase of the signal However other distributions will give different answers See Section 132 Sec 6 Examples Example 83 Pure Chirp For the chirp with no amplitude modulation 3t e29t22jWot 121 Wtw 6w 3t wo 842 The instantaneous frequency is w pt wo and therefore for this case the Wigner distribution is totally concentrated along the instantaneous frequency Example 84 Chirp With Gaussian Amplitude Now place a Gaussian envelope on the chirp The Wigner distribution is st a7r 14 eat2j9t22jWOt Wtw 1 eateW9tWOa 843 7r This is the most general case for which the Wigner distribution is positive and this case encompasses all previous examples If a is small then the distribution is concentrated along the w wo Ot which is the instantaneous frequency In Fig 83 we plot the Wigner distribution 5 5 Fig 83 The Wigner distribution for the signal st a7r 14 eat2 2j8t2 2jwot This is the most general signal for which the Wigner distribution is positive through out the timefrequency plane As the envelope becomes more and more flat the Wigner distribution gets more and more concentrated along the instantaneous fre quency as shown in b In a and b a 1 and a 0001 respectively In both figures wo8andf1 Example 85 The Signal t eat2jat22jOt Now take the signal considered above and multiply it by time st 4a37r14teat2j9t22jWot where we have also normalized it The Wigner distribution is Wt w a at2 w Qt wo2a z eatW9tWOa 844 845 122 Chap 8 The Wigner Distribution and is plotted in Fig 84 It is negative whenever at2 w pt w021tY 846 This is the typical situation The Wigner distribution goes negative somewhere for every signal with the exception of the signal given by Eq 843 5 5 Fig 84 The Wigner distribution for the signal st t Ev erything below the plane is negative Example 86 Cubic Phase i For t e s gnal h st eiwt i Vt y3t33 0t2 2 wot 847 we have 2r pt 17 yr312 yt2 pt wor 848 r12 Wp tr 849 d therefore the Wigner distribution is an w 1 e2r3127wtwrdT Wt 850 2zr 2 4 13 f r J dt 27r 851 r 13 Ai 4y l 13 P1 t 852 where Aix is the Airy function 00 cosu33 xu du Aix 1 853 J 7 o Sec 6 Examples 123 Unlike the quadratic case this distribution is not a delta function of wcp t However it is concentrated along w cp t Example 87 Finite Duration Chirp For the finite duration chirp st ejat22awot T2 t T2 854 the Wigner distribution is calculated to be sinw jt wot T2 T2 t 0 Wtw l sinw 3t woT2 t 0 t T2 7rwQtwo 0 otherwise Note that the Wigner distribution goes to zero at the end points Example 88 Sinusoidal Frequency Modulation The signal 9t D7r14 eat22jOt223msmwmt7wot 855 856 has an instantaneous frequency given by w 3t wo mwm cos wt The Wigner distribution can be obtained analytically Substituting the signal into the definition of the Wiper distribution we obtain W t w tar Ct 12 CIO f earea2am coswmt sinwmr27rwwopt dr Now 857 00 e2jmcoswmt smwmr2 E Jn2m coswmt ejnwr2 858 nm where J is the Bessel function of order n Substituting this into equation Eq 857 and integrating we have W t w 1 eate 7r 00 1 Jn2mcoswmt ww0Otnwm22a noo 859 This can be interpreted as the Wigner distribution of the sum of Gaussians each with instantaneous frequency w wo nm 2 3t Complex Signals and Real Signals The reader may have noticed that all the ex amples presented thus far have been for complex signals The reason is that there is an effective way to think of the Wigner distribution for the sum of two signals A 124 Chap 8 The Wigner Distribution real signal can always be thought of as the sum of a complex signal plus its complex conjugate We therefore leave the consideration of real signals until we develop some simple results regarding the Wigner distribution for the sum of two signals 87 THE WIGNER DISTRIBUTION OF THE SUM OF TWO SIGNALS The Cross Wigner Distribution Suppose we express a signal as the sum of two pieces 8t 81t 82t 860 Substituting this into the definition we have Wtw Wlitw W22tw W12tw W21tw where 861 W12tw 2 fst 2T82t 2Te77dr 862 This is called the cross Wigner distribution In terms of the spectrum it is W12 t w 2 fSi z 9 S2 w z 0 ate d9 863 The cross Wigner distribution is complex However W12 Wzi and therefore W12 t w W21 t w is real Hence Wtw W11twW2Ztw2Re W12tw 864 We see that the Wigner distribution of the sum of two signals is not the sum of the Wigner distribution of each signal but has the additional term 2 Re W12 t w This term is often called the interference term or the cross term and it is often said to give rise to artifacts However one has to be cautious with the images these words evoke because any signal can be broken up into an arbitrary number of parts and the socalled cross terms are therefore not generally unique and do not characterize anything but our own division of a signal into parts Sometimes there is a natural decomposition where the self terms and cross terms take on special meaning This question is addressed in the next section and in Section 133 Example 89 Sum of Two Sinusoids We take st Ai eWt A2 ejm2t 865 The Wigner distribution is Wtw Ai6wwiAZ6ww22A1A26wzwiW2 cosw2wit 866 The three terms correspond to the ones just discussed for Eq 864 Besides the con centration at wi and w2 we also have nonzero values at the frequency z wi W2 Sec 7 The Wigner Distribution of the Sum of Two Signals 125 This is an illustration of the cross term It is sharp at w 1 wi w2 because that is the only value for which there is an overlap since the signal is sharp at both wi and W2 Example 810 Cosine For a cosine wave st cos wot we specialize Example 89 by taking A2 Al 12 and wi w2 wo which gives W t w 6w wo 6w wo 2 6w cos 2wot st cos wot 867 Example 811 Sinusoid Plus Impulse Now consider the sum of a sinusoid and impulse st eO 27r6t to Straightforwardly we obtain 868 W t w 6w wo 6t to 2 cos2w wot to woto 869 Vr7r Comparing this equation to the spectrogram Eq 769 we see that the self terms are infinitely sharp something the spectrogram cannot achieve However the cross terms in the Wigner distribution are not hidden by the self terms as is the case in the spectrogram as per the discussion after Eq 769 In Fig 85 we plot this distribution We also give the ambiguity function for this case AO r 27re060 27reBt06r 2a eiotoTietoT2 cc 870 where cc stands for the complex conjugate of the preceding terms 10 10 Fig 85 The Wigner dis tribution of st elot 6t 10 It is infinitely peaked in time and fre quency at the indicated val ues The ripples are the cross terms Compare this with the spectrogram Fig 71 Note the spectrogram also has cross terms but they are mostly hidden un der the self terms 0 0 126 Chap 8 The Wigner Distribution Example 812 Sum of Two Tones with Amplitude Modulation Now if we put amplitude modulation on each part 8t Al a17r14 ealt227w1t A2 Q2 7r14 E a222IW2t 871 where the modulation extends over all time the cross terms will extend over all times and frequencies The Wigner distribution can be calculated exactly Wtw A C altWWlal A Ca2t2 WW22 a2 7r It A1A2 2alaa12 r 2t 1 2 a a1 a2 cos L a1 az w w2a1 W 1 0 2 x exp 2 al a2t2 w j w1 W22 872 C11 a2 Cross Terms Interference Terms Ghosts Artifacts Multicomponent Signals From the examples we have just considered we see that the Wigner distribution sometimes places values in the middle of the two signals both in time and in fre quency Sometimes these values are in places in the time frequency plane at odds with what is expected A typical case is illustrated in Fig 86 Because of this a language has arisen regarding the cross terms and very often phrases like artifacts ghosts and interference terms are used words chosen to describe something ap parently undesirable The implication is that since these effects are due to the cross terms the cross terms themselves are undesirable It is important to clarify these issues First let us make clear that it is not generally true that the cross terms produce undesirable effects Any signal can be broken up into parts in an infinite number of ways If we have a signal for which we are pleased with the answer the Wigner distribution gives and someone comes along and expresses the signal in terms of two parts will the cross terms be undesirable Quite the contrary they are highly desirable for without them we would not get the pleasing answer In fact since any signal can be broken up into a sum of parts in an arbitrary way the cross terms can be neither bad nor good since they are not uniquely defined they are different for different decompositions The Wigner distribution does not know about cross terms since the breaking up of a signal into parts is not unique There are decompositions for which we think the parts are special in the sense that we think they should be concentrated in certain regions of the timefrequency plane hence if we have two such signals we also sense that they should be well delineated in the timefrequency plane and that no other terms should appear If a signal is indeed well delineated in a region we shall call it monocomponent and if there is more than one well delineated region we shall call it a multicomponent signal Very often in the literature a signal has been called multicomponent when it is expressed as the sum of two parts This is not appropriate since a signal can be broken up into parts in an infinite number of arbitrary ways We shall reserve the Sec 8 Additional Properties 127 phrase multicomponent signal to mean a signal that in the timefrequency plane is well delineated into regions and not simply to mean that it has been expressed as the sum of parts One of the strengths of the Wigner distribution is that it indeed does indicate when a signal is multicomponent We emphasize that when we feed a signal into the Wigner distribution and others we do not tell it that it is multi component it tells us and that is one of the accomplishments of timefrequency analysis However these considerations do not answer the question as to what type of signals produce well delineated regions in the timefrequency plane This is a fundamental question which is addressed in Section 133 Fig 86 The Wigner dis tribution of the sum of two chirps illustrating the cross terms 88 ADDITIONAL PROPERTIES Inversion and Uniqueness The Wigner distribution is the Fourier transform of s t ar st zr with respect to r Inverting st2r 8t2r fWtweTLdi 873 and taking the specific value t 72 letting k 1s 0 and then setting r t we have st k f W t2 w eat dw 874 which shows that the signal can be recovered from the Wigner distribution up to a constant The constant can be obtained from the normalization condition up to an arbitrary constant phase factor A constant phase factor in the signal clearly drops out when we calculate the Wigner distribution since we are multiplying the signal times its complex conjugate Therefore it can never be recovered Representability Not every function of time and frequency is a proper Wigner dis tribution because there may not exist a signal that will generate it If a two dimen sional timefrequency function is generated from some signal we shall say that it is 128 Chap 8 The Wigner Distribution representable or realizable It is not always easy to ascertain whether a function is representable or not although sometimes it is For example any manifestly positive two dimensional function is not a representable Wigner function unless it happens to be of the form given by Eq 843 This is because the Wigner distribution always goes negative somewhere the only exception being Eq 843 Generally a way to ascertain whether a two dimensional function is a Wigner distribution is to assume that it is use the inversion formula to find the signal and then calculate the Wigner distribution from the derived signal If we recover the same function then it is a representable Wigner function Example 813 Representability of the Wigner Distribution Take the two dimensional function Gt w etaW2s2 t 875 For what values of a 0y can this be a Wigner distribution Assume that it is and obtain the signal using the inversion formula Eq 874 st k J Gt2 w eitW dw e1a6Tit24 876 Now calculate the Wigner distribution of this signal to obtain Wtw expci2y2ct2w2c2rypwt c 21aQAry2 When does W t w Gt w Only when c i or 877 1 aQ 1 ry2 878 Therefore the three constants must be chosen according to Eq 878 if Gt w is to be a Wigner distribution Example 814 Wigner Nonrepresentability Now consider the function Gtw t2w2eat2 22ywt 879 If the above calculation is repeated we see that there is no way to choose the constants appropriately There is a much easier way to ascertain that Note that this G is always nonnegative We know that cannot be unless the distribution is given by Eq 842 and this G is not of that form Hence it is not a representable Wigner function Overlap of Two Signals Moyal Formula In many problems the overlap of two signals enters It can be expressed as the overlap of their respective Wigner distri butions fsit s t dt 2 27r f f W1 t W W2 t w dt dw 880 Sec 8 Additional Properties 129 This is easily verified and was first shown by Moyal13901 Of course if wehave two signals it would make no sense to calculate their overlap by first calculating their respective Wigner distributions and then doing the integrals indicated However this expression is of theoretical interest and also has been applied to problems of detection by Flandrin11911 Range of the Cross Terms We discussed the range of the Wigner distribution in Section 82 Similar considerations establish the range of the cross Wigner distribu tion In particular for two functions Si t and 32 t which arerespectively zero outside the intervals t1 t2 and t3 t4 the cross Wigner distribution satisfies the followingl2531 W12tw0 fort outside 2t1t32t2t4 881 if s1t is zero outside t1 t2 and s2t is zero outside t3 t4 In the frequency domain for two bandlimited functions Sl w and S2 u which are zero outside the intervals wl w2 and w3i w4 respectively we have W12t w 0 for w outside 2 w1 W3 z w2 W4 882 if Sl w is zero outside w1 w2 and S2 u is zero outside w3 w4 Product of Two Signals and Product of TWo Spectra The Wigner distribution of the product of two signals st si t s2t can be written in terms of the Wigner distribution of each signal The way to do it is to substitute S1 t 82 t into the defi nition of the Wigner distribution and then use the inversion formula for each signal This results in W t w J Wl t w W2 t w w dw for st 81032W1 883 Similarly if we have a spectrum that is the product of two spectra in which case the signal is the convolution of the two signals then Wtw J W1twW2t t w dt for Sw SlwS2w 884 Analytic Signal Since the analytic signal eliminates the negative frequencies the Wigner distribution of an analytic signal will be zero for the negative frequencies Also use of the analytic signal will eliminate the cross terms between the negative and positive parts of the spectrum Relation to the Spectrogram If we convolve in time and frequency the Wigner 130 Chap 8 The Wigner Distribution distribution of the signal with the Wigner distribution of a window we get the spec trogram PSP t w JfWatwWht t w wdt dw 885 1 e3Wrsr hr t dr 2 886 This relation is interesting but no particular significance should be attached to the fact that we are convolving the Wigner distributions because all timefrequency dis tributions satisfy a similar relation Indeed the best way to prove this result will be from our general considerations in the next chapter Noise The Wigner distribution is very noisy and even places noise at times where there was no noise in the signal To understand this consider an infinite duration signal where there was noise for only five minutes Take any particular time and using our usual argument fold the future with the past The five minutes of noise no matter when it occurred will contribute and therefore noise will appear even though there was no noise at the time being considered This will hold true for any time For a finite duration signal it will appear at those times when the folding over procedure includes the noise Fig 87 illustrates these effects An analysis of additive white noise and the Wigner distribution has been done by NuttallI43 t1 W1 w2 FREQLENCY 89 PSEUDO WIGNER DISTRIBUTION Fig 87 The Wigner distribution is noisy because we have cross terms between the noise and between the noise and signal Note that noise ap pears at times when there is no noise in the signal Windowing the Lag For a given time the Wigner distribution weighs equally all times of the future and past Similarly for a given frequency it weighs equally all Sec 9 Pseudo Wigner Distribution 131 frequencies below and above that frequency There are two reasons for wanting to modify this basic property of the Wigner distribution First in practice we may not be able to integrate from minus to plus infinity and so one should study the effects of limiting the range Second in calculating the distribution for a time t we may want to emphasize the properties near the time of interest compared to the far away times To achieve this note that for a given time the Wigner distribution is the Fourier transform with respect to r of the quantity s t 1 r st 2 r The variable r is called the lag variable Therefore if we want to emphasize the signal around time t we multiply this product by a function that is peaked around r 0 hr say to define the pseudo Wigner distribution WPStw J hrst 2rst1rajT dr 887 The Wigner distribution is highly nonlocal and the effect of the windowing is to make it less so One of the consequences of this is that the pseudo Wigner distribu tion suppresses to some extent the cross terms for multicomponent signals This is because we have made the Wigner distribution local While windowing the lag does suppress the cross terms it also destroys many of the desirable properties of the Wigner distribution For example the marginals and instantaneous frequency properties no longer hold Example 815 Pseudo Wigner Distribution for a Sine Wave Take as an example a pure sine wave and a Gaussian ht st eWOt ht eat22 888 The pseudo Wigner distribution can be calculated analytically Wpstw 1 eo2a 889 vr27ra In the non windowed version Eq 836 the Wigner distribution was totally concen trated at w wo It was Wtw 6w wo That is no longer the case and the broadness depends on the window size Example 816 Windowed Wigner for the Sum of TWo Sine Waves For the sum of two sine waves st Alejlt A2ejw2t 890 with the same ht as above we have Wt w 1 A2 2a A2 27ra 1 2 2AiA2 cosw2 wlt e12222a 891 27ra If we choose a small a the cross terms can be made small However note that the self terms get spread out 132 Chap 8 The Wigner Distribution 810 MODIFIED WIGNER DISTRIBUTIONS AND POSITIVITY Historically the main motive for modifying the Wigner distribution was the attempt to achieve a positive distribution One way to approach this problem is to attempt to smooth it by convolving it with a smoothing function Lt w to obtain the smoothed Wigner distribution WSM t w fLt t w w W t w dtdw 892 The first example of a smoothing function was devised by Cartwright11091 Ltw et2W2R 893 Substituting this L and the definition of the Wigner distribution into Eq 892 and integrating over w results in WSMtw 27r ff etta9T2 2Tr st 2rrdtdr 894 Making the change of variables y t 1r t x t i r t we obtain WSM t w f f eA1y2fY 1x dxdy 1x e8110x4iwxj 895 Expanding e010xy2 in a Taylor series we have WSM t w Y V 2n 0 1an 27r n0 n f fxxn dx 2 8 If each term is positive then the sum will be positive To assure that we must take 1a 0 or equivalently aQ 1 897 Notice that if af3 1 then only the first term survives But with this condition L is a Wigner distribution as we showed in Eq 878 In that case we know from Eq 885 that the resulting distribution is the spectrogram In the general case Eq 896 is a sum of spectrograms Nuttall1403 404 has generalized this by considering the smoothing function Ltw et2W2A2ywt 898 The resulting distribution is positive if a3 1 y21 Smoothing Transforming and Convolution Smoothing or blurring is sometimes desirable as for example to eliminate the wrinkles of a portrait More often though Sec 11 Comparison of the Wigner Distribution with the Spectrogram 133 it is undesirable as exemplified by the blurring of celestial objects by the atmo sphere Since these phenomena are so common there is a vast methodology on the subject One way to study blurring is to model it as a convolution of the original function with a blurring function Generally convolution is not reversible which can be interpreted as our inability to unblur an image that has been blurred How ever convolution may be reversible We mention this here because it will turn out that all bilinear distributions may be obtained from the Wigner distribution by a convolution No significance should be attached to this since all other distribu tions may be obtained by convolving with any distribution not only the Wigner A distribution that we will be studying is the ChoiWilliams distribution which can be obtained by convolving the Wigner distribution with a certain function It would be wrong to think of the ChoiWilliams as a smoothed Wigner distribution because it is also true that the Wigner distribution can be obtained from the ChoiWilliams by a convolution Hence one could argue that the Wigner is a smoothed version of the ChoiWilliams It is better in such situations to think of the process as a reversible transformation See section 97 811 COMPARISON OF THE WIGNER DISTRIBUTION WITH THE SPECTROGRAM It has often been said that one of the advantages of the Wigner distribution over the spectrogram is that we do not have to bother with choosing the window This viewpoint misses the essence of the issue The spectrogram is not one distribution it is an infinite class of distributions and to say that an advantage is that one does not have to choose makes as much sense as saying one book is better than a library because we dont have to choose which book to read Here is the point The Wigner distribution in some respects is better than any spectrogram It is not that we do not have to bother about choosing a window it is that even if we bothered we wouldnt find one that produces a spectrogram that is better than the Wigner In particular the Wigner distribution gives a clear picture of the instantaneous frequency and group delay In fact the conditional averages are exactly the instantaneous fre quency and group delay This is always true for the Wigner distribution it is never true for the spectrogram We could search forever and never find a window that will produce a spectrogram that will give the instantaneous frequency and group delay although sometimes a good approximation is achieved One of the advantages of the spectrogram is that it is a proper distribution in the sense that it is positive Because it is manifestly positive the results obtained from it can be interpreted although they may be wrong or poor The Wigner distribution is with one exception never manifestly positive which sometimes leads to results that cannot be interpreted and indeed go against our sensibilities For example the conditional standard deviation may be negative Finally the Wigner distribution and spectrograms allow us to ascertain in most cases whether a signal is multicomponent But the Wigner distribution suffers from the fact that for multicomponent signals we get confusing artifacts On the other 134 Chap 8 The Wigner Distribution hand the spectrogram often cannot resolve the components effectively We men tion here that other distributions we will consider keep the desirable properties of the Wigner distribution and have considerably better behavior with regard to the undesirable properties In Figs 88 89 and 810 we give a few examples contrast ing the Wigner distribution and spectrogram Contrast Regarding Global Averages and the Uncertainty Principle Since the Wigner distribution satisfies the marginals it always gives the correct answers for averages of functions of frequency or time and always satisfies the uncertainty prin ciple of the signal On the other hand the spectrogram never gives the correct an swers for these averages and never satisfies the uncertainty principle of the signal T W Fig 88 Comparison of the Wigner distribution bottom with the spectrogram top for two multicomponent signals Sec 11 Comparison of the Wigner Distribution with the Spectrogram 135 a Hz 450 300 150 0 35 70 105 140 0 Time ms U 35 70 105 140 Fig 89 When muscles contract they make a sound In a and b we have the Wigner distribution and spectrogram of a typical case Courtesy of J Wood N M Cole and D I Barry N b 0 160 260 360 460 560 606 Fig 810 Aneurysm signal About 5 of the population has a small spherical en largement of the wall of the cerebral artery aneurysm Some aneurysms emit a sound In the above figure are the Wigner distribution a and spectrogram b of such a sound Courtesy of M Sun and R J Sclabassi Chapter 9 General Approach and the Kernel Method 91 INTRODUCTION The Wigner distribution as considered in signal analysis was the first example of a joint timefrequency distribution that was qualitatively different form the spec trogram The idea of the spectrogram crystallized in the 1940s Independent of that development there was considerable activity in the 1940s 1950s and 1960s devising distributions which were similar in spirit to the Wigner distribution in the sense that they satisfied the marginals the instantaneous frequency condition and other desirable properties Among the distributions proposed then in signal analy sis and quantum mechanics were the Rihaczek Page and MargenauHill In 1966 a method was devised that could generate in a simple manner an infinite number of new ones 1 The approach characterizes timefrequency distributions by an aux iliary function the kernel function The properties of a distribution are reflected by simple constraints on the kernel and by examining the kernel one readily can as certain the properties of the distribution This allows one to pick and choose those kernels that produce distributions with prescribed desirable properties This gen eral class can be derived by the method of characteristic functions In this chapter we explain the method and the general ideas associated with it and give the deriva tion in the next chapter 92 GENERAL CLASS All timefrequency representations can be obtained from Ct w 471 JJJ s u 2 r su 2 r 09 r eyetjTwjeu du dr dB 91 where q50 r is a two dimensional function called the kernel a term coined by 136 Sec 2 General Class 137 Claasen and Mecklenbraukerl1201 and whom with Janssenl2761 made many impor tant contributions to the general understanding of the general class particularly in the signal analysis context The kernel determines the distribution and its properties In Table 91 we list some distributions and their corresponding kernels For the Wigner distribution the kernel is one however no particular significance should be attached to that since it is possible to write the general form so that the kernel of any distribution is one in which case the kernel of the Wigner distribution would be something else Spectrum In terms fff ofthe spectrum the general class is Ct w S u 29 Su 2 B 09 T iTU dB dr du 92 as may be verified by expressing the signal in terms of the spectrum and substituting in Eq 91 Alternate Forms There are a number of alternative ways of writing the general class of timefrequency distributions that are convenient and add considerable physical insight Characteristic Function Formulation Recall that the characteristic function is the double Fourier transform of the distribution By inspection of Eq 91 we see that Ct w 47x2 ff MB T eietjTdB dT 93 where MOr 00 T J s u 2 T su 2r a ou du 94 09 r A0 r 95 and where AO T is the symmetrical ambiguity function The characteristic func tion may be appropriately called the generalized ambiguity function Time Dependent Autocorrelation Function In Chapter 1 we showed that the en ergy density can be written as the Fourier transform of the deterministic autocor relation function Now we generalize this idea by thinking of the timefrequency energy distribution as the Fourier transform of a local autocorrelation function 11171 Rt 7 Ct W Z f R r e2 dr By comparing with Eq 91 we have RtT JJ 8u 2T su 2T 09 r eet0 d9 du 97 We shall call Rt T the deterministic generalized local autocorrelation function 138 Chap 9 General Approach and the Kernel Method Table 91 Some Distributions and Their Kernels Name General class Cohen125 Wigner584 MargenauHill358 Kirkwood 305 Rihaczek4 BornJordan Cohen1251 Page4191 ChoiWilliams117 Spectrogram ZhaoAtlasMarks 626 Kernel dB T 0Br 1 cos Z BT jr2 sin 1Or 20r e7elrl eB2r2 f h u 1r eeu J hu T du 2 9TITI aB 8r eSetfirjeu 00r au 2Tsu 2Tdudrd9 1 27r eWst 1rst 1rdr 2 2 Re 27 stSwe 1 t 27r tlr V 1 r 2IrI a 1ITI2 su 1rsu1rdudr 2 2 8 at 47x32 f T2Q a u 2Tsu2Tdudr if e r sThT t dT 1 f grer 47ra 2 su Zrsu 2rdudr Positive2 Cohen Posch Zaparovanny127 128 see Chapter 14 I Sw12 8112 c1u v i 1 Derived in reference 125 using the ordering rule of Born and Jordan The derivation is repeated in Section 106 2 These distributions and the spectrogram are manifestly positive but only the Positive ones satisfy the margins Distribution Ct w stS w e 2 Sec 2 General Class 139 Fourier Transform of the Kernel Notice that in the general form 9 does not appear in the signal Hence if we define rt T 1 09 r ait9 d9 98 27r the general class can be written as Ct w 1 JJ rt u r s u 2T su 1T aj du dT 99 The local autocorrelation becomes RtT frtuTsuTsurdu 910 2 Bilinear Transformation f585 137 If in Eq 91 we make the transformation xu 2T xuZT or u 2xx Tx x 911 then we have Ct W Jf Kt w x x s x sx dx dx 912 with Kt w x x 2 rt 2 x x x ejwxx 913 As we discuss in the subsequent sections properties of the distribution are reflected in the kernel These constraints may be imposed on 0 or K but are generally sim pler when expressed in terms of 0 This is a bilinear form only when K is signal independent Sum of Two Signals For st slt s2t 914 the distribution is Ct W C11t W C22 t W C12 t w C21 t w 915 where the cross distribution between two signals is Ckt w j p JfJ 00 T sku 2T sLu 1T e9Bt9Tw4JBu du dT d8 916 140 Chap 9 General Approach and the Kernel Method 93 THE KERNEL METHOD There are three basic advantages for characterizing timefrequency distributions by the kernel function First we can obtain and study the distributions with certain properties by constraining the kernel For example suppose we want all distribu tions that satisfy the marginals We will see that to assure the satisfaction of the marginals the kernel must have the simple property 0 r 9 0 1 There fore if we want to study the distributions that satisfy the marginals then we con sider only kernels that satisfy these conditions This still leaves an infinite number of choices but we are assured that the marginals are satisfied Second the properties of a distribution can be readily ascertained by a simple examination of the kernel For example if the kernel satisfies the condition just discussed then we know the distribution satisfies the marginals and we do not have to do any further calculations Third given a kernel a distribution is easy to generate Before we discuss the conditions on the kernel that guarantee that a distribution will have particular prop erties it is worthwhile to classify and discuss the various possibilities and depen dence of the kernel Functional Dependence of the Kernel The kernel can be a functional of frequency and time and explicitly of the signal If we were to be notationally explicit we would write the kernel as 49 r t w s to signify the possible dependencies However we write it as a function of 9 and r and discern from the context whether it depends on other variables Types of Kernels An important subclass are those distributions for which the ker nel is a function of the product Or 00r cbpRBr O9r product kernels 917 For notational clarity we will drop the subscript PR since we can tell whether we are talking about the general case or product case by the number of variables at tributed to 00r Kernels that are a product of a function of each variable 00r 19 t2r separable kernels 918 are called separable kernels Bilinearity If the kernel does not explicitly depend on the signal then the signal enters into the general form twice hence one says that the distribution is bilinear a phrase used by WigneL All the distributions listed in Table 91 are bilinear since the kernel does not depend on the signal explicitly Distributions obtained by taking the kernel to be signal dependent are discussed in Sections 135 and Chapter 14 In this chapter we will assume that the kernel is bilinear and time and frequency independent However the methods developed for the bilinear case often apply directly to the nonbilinear case We point that out when appropriate Notice that Sec 4 Basic Properties Related to the Kernel 141 the marginals are bilinear in the signal However that in no way reflects on the functional dependence of the joint distribution Having bilinear marginals does not imply that the joint distribution must be bilinear Determination of the Kernel If a distribution is given we can usually pick out the kernel by putting the distribution in one of the standard forms If that is not readily doable we can calculate it by finding the characteristic function of the distribution and using Eq 95 Explicitly Ct w ejetj1 W dt dw M9 T ff A6T I s u 2T su 2r ejeu du 919 Example 91 The Spectrogram In Section 72 we derived the characteristic function for the spectrogram Eq 710 We showed that MspBr Ah9TA9T Therefore OSPBT Msp9T Ah9 r A9T Ah0 T 920 A07 A8 T We see that the kernel for the spectrogram is the ambiguity function of the window with 9 replaced by 9 94 BASIC PROPERTIES RELATED TO THE KERNEL We now discuss some of the basic properties of distributions that have been deemed desirable and show how they get reflected as constraints on the kernel Two im portant properties are left for consideration in other chapters the conditions for positivity and the condition on the kernel to minimize the cross terms for multi component signals Marginals Instantaneous Energy and Energy Density Spectrum Integrating the basic form Eq 91 with respect to frequency gives 21r6T and therefore J Ct w dw 2fff br s u 2 T su 2T 09 T eJ8ut d9 du d7 921 ff 0e 0 1 su 12 9ut dO du 922 27r If we want this to equal I st 12 then the 9 integration must be made to give 1 But 27r B 0 e dB bt u 923 142 Chap 9 General Approach and the Kernel Method and the only way we can do that is to insist that 09 0 1 for time marginal 924 This is the condition for the time marginal to be satisfied Similarly for the fre quency marginal to be satisfied f Ct w dt Sw 925 we must take 007 1 for frequency marginal 1 926 Total Energy If the marginals are correctly given then of course the total energy will be the energy of the signal However we can retain total energy conservation normalization to one without insisting on the marginals Integrating the general form with respect to time and frequency shows that for the total energy to be pre served Jf Ct w dw dt 1 for total energy 927 the kernel must satisfy 00 0 1 for total energy 928 Uncertainty Principle As we have discussed in Section 69 any joint distribution that yields the marginals will yield the uncertainty principle of the signal Thus the condition for the satisfaction of the uncertainty principle is that both marginals must be correctly given In terms of the kernel this is 00 0 1 and 00 r 1 for uncertainty principle 929 Reality For a distribution to be real the characteristic function must satisfy M8 T M0 r Look at Eq 95 for the characteristic function and note that the ambiguity function satisfies A9T A9 T Therefore the only way that M9 T M 0 T is for the kernel to also satisfy the identical condition 08T 0e T 930 Time and Frequency Shifts A signal that is shifted by a time to and frequency wo is sah ei0tst to where to is the amount of time translation and wo is the translation in the frequency domain Substituting this into the general form we have using Cah for the translated distribution Sec 4 Basic Properties Related to the Kernel 143 Cahtw 1 ejwour2to T7r2 f 1 1 yet2rvra0 19uto 1 4 ffJ oors u 2rsu ZTa dBdrdu 932 fJf o rsu 2T su 2T ejBttojro ie do dr du 933 47r2 Ct to w wo 934 In going from step 931 to step 932 we assumed that the kernel is not a function of time and frequency Therefore Ct w is time shift invariant if 0 is independent of t 935 Ct w is frequency shift invariant if 0 is independent of w 936 Note that the kernel can be signal dependent Scaling Invariance In Section 65 we showed that if a signal is linearly scaled then the spectrum is inversely scaled and we argued that a plausible requirement on the joint distribution is that it scale the same way in time and frequency In particular if C t w is the distribution of the scaled signal s then the requirement is Cct w Cat wa for s t f sat 937 For the scaled function Ct w 4a2 N sf au 2r sau 2r 00 rr eset3rw 9u do d7 du 938 fransforming by u ua T Ta and o ao we have jOu do dT du CCt w 47r2 fJf sf u 7 au 2r Oao 7a We see that 939 Cct w Cat wa if 00 ra 00 z 940 The only way that a function of two variables can satisfy 0a9 za 00 r is if it is a function of the product of the two variables Therefore for scale invariance the kernel must be a product kernel 00r Oorr for scale invariance 941 144 Chap 9 General Approach and the Kernel Method Note that we have implicitly assumed that the kernel is signal independent Weak Finite Support Recall from Section 66 that weak finite time support means that for a finite duration signal we want the distribution to be zero before the signal starts and after the signal ends The condition for that to be the case is J O0rajetdO 0 fori7I 21tj 942 for weak finite time support Similarly for a bandlimited signal we want the distribution to be zero outside the band Then condition for that to be so is J 00 T e7 d r 0 forJOI 2 1 w l 943 for weak finite frequency support Strong Finite Support Strong finite support means that the distribution is zero whenever the signal or spectrum is zero The conditions for this to be the case were derived by Loughlin Pitton and Atlas for the case of bilinear distributions The conditions are f 9 T ejet dO 0 f 09 7ejT dT 0 Example 92 ChoiWilliams Kernel for 171 21t1 944 for strong finite time support forIBI 21wl 945 for strong finite frequency support To see how the above results can be used to quickly ascertain the behavior of a partic ular distribution consider the ChoiWilliams kernel Ocw0 r e9220 946 where v is a constant It is clear that for either 0 or r being zero the kernel is one and hence the ChoiWilliams distribution satisfies the marginals Also since it is time and frequency independent it satisfies the shift properties For the finite support property we have J 00 r ajet dO T exp 947 Therefore it does not satisfy the weak finite support properties but does so approxi mately for v sufficiently large Sec 4 Basic Properties Related to the Kernel 145 Example 93 Wigner Distribution Since the kernel for the Wigner distribution is one it satisfies the marginals Also since it is signal independent it satisfies the shift properties The finite support property is f B T eB d9 27r6t 948 and this dearly satisfies the weak finite support property Eq 942 Example 94 Spectrogram In the kernel for the spectrogram Eq 920 set 9 0 giving osP0 r J jht 12 aB dt 0 1 949 which cannot be equal to one unless we take a delta function for I h 12 Therefore the spectrogram cannot satisfy the marginals Example 95 Sinc The smc kernel is OOT since T 950 This kernel satisfies the marginals since sin 00 1 Consider now the weak finite support property Using Eq 942 we have sina9T aBt dt f aBT I aT 0 ti or otherwise 951 Comparing to the condition for weak finite support Eq 942we see that the condi tion is satisfied only if a i Inversion From Eq 95 we have MO T A0T OeT fsu T su T eB du 952 and taking the Fourier inverse yields 1 2irfM0T cb9T Lettingt u ZTandt u 1r gives ej0 d9 953 8 t st 1 M9 t t eiett2 d9 954 27r f 00t t 146 Chap 9 General Approach and the Kernel Method and taking the specific value t 0 we have st 1 Mt7 t ejOt2 d9 27rs0f 00 t 1 Ct w ejtwjett2 dt 0 d9 27rs 0N B t 955 956 Hence the signal can be recovered from the distribution if the division by the kernel is uniquely defined for all values of 0r If the kernel is zero in regions we cannot recover the signal uniquely For the spectrogram this will depend on the window Example 96 Wigner For the Wigner distribution fff st 1 Wt w eatBtt2 dt dw dO 957 2as 0 W t2 w et dw 959 which agrees with the answer obtained in Eq 874 95 GLOBAL AVERAGES For a function gt w the global average is gt w fJ gt w Ct w dw dt 960 If the marginals are satisfied then averages of the form gt g1t 92 w will be automatically satisfied 91t 92w f f91t g2wCtwdwdt 961 f 91t I st12 dt J 92w 18w 12 dw 962 correctly given if 09 0 00 T 1 Correlation Coefficient and Covariance Let us now consider the first mixed mo ment wt 82M0T 00197 0r 0 963 Sec 6 Local Averages 147 02 00 rr A2t dt t 0t A2t dt 964 f J 9T o If we want the covariance to be equal to the form given in Section 18 then we must have that tw t cpt which can be achieved by taking the mixed partial derivative of the kernel to be zero at the origin t w f t 01tISt 12 dt if 809 r 0 965 89r B To Example 97 ChoiWilliams For the Choi Williams kernel a09r 403r3o2 407ue02T2b0 0 966 aOar eT 0 9T o and we see that the ChoiWilliams distribution gives the proper expression for the covariance 96 LOCAL AVERAGES The density of frequency for a given time Cu I t and the density of time for a given frequency Ct I w are Cw It Ct Ct I w Ct 967 where Pt and Pw are the marginal distributions Thus the expectation value of a function g at a given time or frequency is gw t pt fw Ct w dw 968 gt Pw ft Ct w dt 969 Instantaneous Frequency and Its Spread For the average frequency at a given time we calculate wt pt f wCtw dw 970 1 2 1aOe T aut27rPt ff A u e 0 u O d0 du 971 T 1 pt 00 A2t t 2 00 At At product kernel 972 148 Chap 9 General Approach and the Kernel Method Suppose we satisfy the time marginal in which case Pt A2 t Then w t t i Pit 2 f f A2u 1900 T r0 6ut de du 973 If we want this local average to be equal to the derivative of the phase we must take the integrand to be zero Therefore w t Wt w2 t Second Conditional Moment of Frequency For simplicity we give the result for product kernels w2 tPt z 00 4 0 A2t 2 00 40 0 1 At A t 00 A2t 2t 20 2 At At pt A2t P11 t 976 If we impose the conditions to obtain the derivative of the phase for the first con ditional moment that is the marginal conditions and 00 0 this becomes Nt 2 1 A11t 2 w2 t 2 1 400 At 1 Z At P t 977 It w2 t Standard Deviation The local standard deviation is obtained from a2 w 2 From Eqs 977 and 974 we have Q2 a 1 400 At J 2 Z 1 400 A t O t Positive Spread In general the conditional second moment and standard devia tion become negative and cannot be interpreted However for the choice 0 4 979 we obtain 806 T 974 if 8z T0 0 if 00 0 product kernel 975 CAt 2cp2t 980 2 At fl 2 GO At 981 Sec 7 Transformation Between Distributions 149 which are manifestly positive Are these plausible results We discuss this issue in detail in Section 133 where we develop the idea of instantaneous bandwidth and connect it to the nature of multicomponent signals Example 98 Wigner Distribution For the Wigner distribution 00 0 and hence 2 1 At 2 At 982 Q 1t 2 At At which agrees with the expression given in Section 85 Group Delay If we want the conditional moment of time for a given frequency to be the derivative of the spectral phase then the identical approach as above gives 983 0111 iw if a0e T Lo 0 where i w is the spectral phase 97 TRANSFORMATION BETWEEN DISTRIBUTIONSI 21 We now address the relationship between distributions First let us discuss whether indeed there should be a procedure that allows us to transform one distribution into another In Section 94 we showed that a signal can be recovered from a particular distribution if the kernel is not zero in a finite region Given a distribution for which the signal can be recovered we may take the recovered signal and calculate any other distribution so in these cases we do expect a relationship to exist between them To obtain that relationship suppose we have two distributions C1 and C2 with corresponding kernels 01 and 02 Their characteristic functions are M19T O1 6T J s u 2T su 2T ejeu du 984 M20T 020T J s u 2T su ZT e3eu du 985 Divide one equation by the other to obtain M0r 01 0r M29T 02eT 986 This is an important relationship because it connects the characteristic functions Note that we have a division by the kernel For the division to be proper the kernel cannot be zero in regions which is consistent with our discussion above 150 Chap 9 General Approach and the Kernel Method To obtain the relationship between the distributions take the double Fourier transform of both sides and use Eq 93 to obtain jet 7 d8 d C M 0 1 t w 1f 9 87 l r a r 2 4 2 J 0 T 0 Now express M2 in terms of C2 to obtain f jBttjrWWdBd C dtd P w ti C t I 0 9 88 e 1 r w w 2 2 47r r J JJ 02 This relationship can be written as C1 t w 912ttwwC2twdtdw 989 with t w 1 r 019 r 912 bet2TW dB dr 990 J 2 0 r Relation of the Spectrogram to Other Bilinear Representations We now specialize to the case where we transform from an arbitrary representation to the spectrogram In Eq 989 we want Ct to be the spectrogram and C2 to be arbitrary To simplify notation we set qsp 41 and r4 02 and gsp 912 and write CsPt w JJ gspt t w w Ct w dt dw 991 The kernel for the spectrogram with window ht is Ah 8 r and therefore 9spt w 1 Ah8 r e3etfr d8 dr 992 47r2 f f 007 JJJ 9 r hu 2r hu 2r dudrdB 993 47r2 ffJhu 2rhu 2r X 00r eSetiTWf eu du dr d0 994 00r 00 r If we take kernels for which 09 r 00r 1 we see that gspt w is just the distribution of the window function except that it is evaluated at w Therefore 9sP t w Ch t w 995 for kernels that satisfy tB r B r 1 Sec 7 Transformation Between Distributions 151 and CsPt w JJCatwCht t w w dt dw 996 for kernels that satisfy 09 r 09 r 1 This was first shown by Janssen12761 For the case where 09 r 09 r does not equal one then t t w w w dt dt dw dwlf CsP t w JfffGtwCstCht 997 where dB dr 998 Gt w 4a2If 0 9 r 00r Positive Bilinear Distributions In Section 810 we saw that if the Wigner distribu tion is convolved with certain functions a positive distribution is obtained We have also just seen that when the general class is convolved with certain gs the spectro gram is obtained The general question as to when one obtains a positive bilinear distribution b convolving the general class has been comprehensively addressed by JanssenJwho derived a number of important results relating to this question Chapter 10 Characteristic Function Operator Method 101 INTRODUCTION In the last chapter we showed that an infinite number of joint timefrequency dis tributions can be generated We now describe a method for deriving them from first principles In this chapter we develop the ideas for time and frequency in Chapter 17 we generalize to arbitrary variables and subsequently specialize to the physical variable scale 102 CHARACTERISTIC FUNCTION METHOD Recall that a distribution and its characteristic function are connected by M9 T ff Ptw eiOtiTw dt dw e3eair 101 and Ct w fJ M8 T dO dT 102 The aim is to find the distribution If it were possible to find the characteristic func tion without first knowing the distribution our problem would be solved because we would then obtain the distribution by way of Eq 102 Correspondence Rules In Chapter 1 we showed that the average of a function of frequency can be directly obtained from the signal In particular the average of 152 Sec 2 Characteristic Function Method 153 gw can be calculated by gw I gW I Sw i2 dw f sit gW st dt 103 where W is the frequency operator We generalize this by assuming that we can find the average of any function of time and frequency Gt w by Gt w J s t 9 T W st dt 104 where G T W is an operator as yet unknown that corresponds to the ordinary function G A rule that associates an ordinary function with an operator is called a correspondence rule a rule of association or ordering rule Characteristic Function Operator The characteristic function is an average the average of ejetjr and hence it seems plausible that we can find an operator M 0r T W so that M8 r M 8 r T W JstM8rTW st dt 105 We shall call M the characteristic function operator What can we take for M One possible choice is to substitute the time and frequency operators in the expression ejetjT for the time and frequency variables eastjrw Mw e2BTjrW Weyli501 106 This is called the Weyl correspondence Equally plausible is eetjT MN e OT ejrW Normal 107 which is called the normal correspondence Now for ordinary variables ejstjrw ejet ejr but for operators it is not true that ejsTjrW equals eOT ejrw because the operators do not commute Since these two choices are not equal they will give different answers for the characteristic function and therefore produce different distributions In addition to the above two choices the following are also equally plausible ejOtjr elrw ejOT 4 2 e3OT e3rW eJrW eyOT 1 sin 0r2 e3rWjoT J 8r2 As we will show we can generalize these rules by eetjr 00r 0TTW Antinormal 1 108 Symmetrization 109 Cohen1J 1010 Coheni1211 1 1011 154 Chap 10 Characteristic Function Operator Method There is one basic reason why all these correspondence rules are plausible and equally correct in some sense Recall from Eqs 444445 that the joint characteris tic function of two variables x y is related to the individual characteristic functions M9 M r by M9 M90 My7 M0T 1012 and therefore the characteristic function operator is constrained by M 0 0 T W ejOT 1013 M 0r T W ejrw 1014 if we are to be assured of the right marginals Moreover this assures that averages of functions of time or frequency only are correctly given All the correspondences given above satisfy these constraining equations To obtain a distribution we choose a characteristic function operator calculate the characteristic function by way of Eq 105 and then find the distribution via Eq 102 These equations can be combined Substituting Eq 105 into Eq 102 we have Ctw 412 fff suM9rTuWu sueyetidOdr du 1015 where Tu and Wu signify the time and frequency operators operating on functions of u We now address the evaluation of expressions like Eq 105 for the various choices of the characteristic function operator 103 EVALUATION OF THE CHARACTERISTIC FUNCTION First let us note that the evaluation of the characteristic function can be done using the signal or spectrum M9 r M fstMOrTW st dt 1016 f SwM07TWSwdw 1017 If we are using the signal we will say that we are in the time domain and if we are using the spectrum we will say we are in the frequency domain We recall from Section 14 that the explicit forms of the time and frequency operators are T t W 4 dt in the time domain 1018 T j W w in the frequency domain 1019 Sec 3 Evaluation of the Characteristic Function The fundamental relationship between these operators is TWWTj 155 1020 which is discussed and derived in Section 153 Three important results are helpful for the simplification and evaluation of char acteristic function operators Two of them already discussed in Chapter 1 are atT ej9 Sw Sw 9 1021 The third is an important result of operator theory which is well known but far from trivial ejOTjrW ejO72 eOT ejer2 eOT ejrW 1022 1023 We will not prove this here because we return to it in our considerations of a more general case in Chapter 17 In general e9r2 ejOT ejrWSt 9rl2 etst r 1024 We are now ready to see by way of examples how to derive distributions using the characteristic function approach Example 101 The Wigner Distribution For the characteristic function operator take the Weyl ordering in which case M9 r r s t erwi8Tst dt 1025 fste9n2e8TeJ1Wstdt 1026 f s t e012 eiOtst r dt 1027 fau 1 r ejesssu 1r du 1028 2 2 The distribution for this ordering is Pt w 42 ff M8 r ajet jr d9 dr 1029 4jr2 JJJ s u 2 r e2Ousu a r alet r d9 dr du 1030 156 Chap 10 Characteristic Function Operator Method T7rff u z 6u t 8u 2 r dr du 1031 I 1 J st 1re st Zrdr 1032 which is the Wigner distribution Example 102 Margenau Hill Distribution Using the ordering given by Eq 107 we have for the characteristic function M9 T Jst eisT ev W 8t fsteJ9tstrdt 1033 The distribution is CIO T d9 d Pt r f jeu d 10 34 w J s u e su r u r 47r2 1 tS t 10 35 e w 27r s If the correspondence of Eq 108 is used the complex conjugate of Eq 1035 is ob tained and if the symmetrization rule is used Eq 109 we obtain the real part 104 THE GENERAL CLASS All possible distributions are obtained if we consider all possible correspondence rules A way to approach this problem is to start with any one say the Weyl rule and any other correspondence is expressible as MG T W 00r ejOTu1W 1036 For Eqs 1013 and 1014 to be satisfied we must have 00 r 9 0 1 1037 The general characteristic function is the average of the characteristic function op erator Mc9 r Mc 0 0 r dt 1038 00r J s t 17 e3etst 2r dt 1039 Sec 5 Averages 157 and the distribution is Ct w 41 1040 r r BtT8u JJ s u T su r B r e du dr dB 1041 which is the general class of timefrequency distributions developed in Chapter 9 105 AVERAGES If we have a function of time and frequency gt w its average can be calculated using the timefrequency distribution or directly from the signal gtw ff gtwCtwdtdw 1042 f s t 9 T W st dt 1043 where Q is the operator corresponding to the ordinary function g The relation between g and 9 is r r 9T W JJ 79r48rej9TjTw dOdr 1044 where 70 r 47r2 f f g t w ejetjT dt 1045 Or equivalently 9T W 412 JfJJ gt w 09 r eJeTt jTww dO dr dt dw 1046 To prove the equivalence of the two methods of calculating averages Eqs 1042 and 1043 consider gt w st dt 1047 1118 t y0 r 09 r eOT3T4st dO dr dt 1048 JJJ s u 1r y9 r 00 r ejeusu ar d9drdu 1049 47r2 fJJJJ g t w ejetj 0e r eu 7su 1rd9drdtdudw 1050 2 2 fftwCtwdtdw x su 1 1051 158 Chap 10 Characteristic Function Operator Method 106 THE MOMENT METHOD The moment method is another way to construct distributions It is equivalent to the characteristic function method just described but presents a number of new features both conceptually and mathematically In Section 43 we showed how the characteristic function can be constructed from the moments by way of M9 T V 7e7T twm 1052 n0 m n If we had the moments we could calculate the characteristic function and the distri bution Moments are average values and hence should be calculable directly from the signal by an operator that corresponds to the moments We symbolically write tnwm CnmTW 1053 where Cnm T W indicates a correspondence We expect to be able to calcul moments from ate the tnwm J s t CnmT W st dt For the nonmixed moments to be correct we want to assure that 1054 to Some obvio 4 Cn us a oT 0 nd not Tn and wm Co n0 so obvious choices for the corresp W Wm ondences are 1055 tnwm TnWm Normal 1056 WmTn Antinormal 1057 TnWm WmT Symmetrization 1058 2n E InBT ntWTt 1059 to J 2m E C W mtvwl Weyll 8 1060 to In manipulating these types of expressions the following relations are useful mmmn WnT L 7t 1 7 T Wm I t 10 mmmn TWm E 7l t p e Wm nt 10 They were given in the quantum mechanical context by Born and Jordan 1911 Sec 6 The Moment Method 159 1 TlWm i n1 10 1061 1 WmtTW BornJordant911 1062 m1 t0 1 anm n m m 09 T ej9TjTW Cohen125i 1063 j Fn 8T 19r0 Making a correspondence to the exponential ej9tjTw as we have done in Section 103 is equivalent to making a correspondence to the mixed moments This is why we have given the same name to the correspondences although we have not yet proven that they are equivalent To see this we start with 00 CO 7er n r ej9tjTW m tnWn 1064 n0mLL0 and take the correspondence of each side cO 0 e39tjTw M L jenjTm C nn T W 1065 n0 m0 and take average value M9T M 1 jonj rm CnmT W CO 0 nm n0 m0 If we consider Eq 1065 as a Taylor series the coefficients are CnmT W anm 7jm aenOTm M97 1 1 anm eJ9T ejTW jn jm ognaTm Or 0 1066 1067 1068 However we know that the most general characteristic function operator can be expressed in terms of the kernel Eq 1036 and therefore CnmT W 1 anm jn jm aenarm OO T 9r 0 1069 1 anm 007 8n2 p39T eTW 10rO 1070 jnjm 90n8Tm To see how the procedure indicated by Eq 1070 is carried out in practice we con sider a few examples 160 M 00 Example 103 Normal Ordering Take 09T ej0T2 which gives 1 anm CnmT W jnjm aenaTm TnWm Example 104 Weyl Ordering EE 1071 1072 We show that indeed Eq 106 and Eq 1060 are equivalent This was shown by McCoy For the Weyl rule the kernel is 1 and CnnT W n0 m0 1 anm ej8r2 BT eyrW T7 a9na m Chap 10 Characteristic Function Operator Method 1 anm e39TjrW BBnr drW 8T 0 9T0 0r 0 1073 1074 2m M Wmtmw 1075 l0 The step from Eq 1074 to Eq 1075 is not trivial but can be shown by using the relations in the footnote of Section 106 Let us now consider an example where we calculate the characteristic function directly from the moments Example 105 BornJordan Distribution Cohen1251 We consider the rule of BornJordan Eq 1062 and calculate the characteristic func tion M9 T 78imm C T W 1076 n 00 00 n Y E E n 1nm fstT11twmrstdt 1077 n0 m0 t0 m n 8n fsittte2TWtstdt 1078 m 1n0 10 1Bn s t toI t r1 st T dt n 1n J n0 10 1079 Sec 6 The Moment Method 161 where in the last step we have used the fact that erw is the translation operator Now n 1 rt 1 E to and further to jen 1 rtn1 1 eietlrt e Ot a n 1n rt j0 eJBr21Bt sin Or2 Or2 1080 1081 1082 Hence MB r s nO 22 ever2 J stCetstrdt 1083 Therefore the kernel is sin Br2 8 u 2 r ejesu 2 r du Br2 sin Br2 8r2 For a further discussion of this distribution see Section 124 1084 1085 Chapter 11 Kernel Design for Reduced Interference 111 INTRODUCTION In our study of the Wigner distribution we saw that it sometimes gives values in the timefrequency plane that do not correspond to our intuitive sense of how the energy should be distributed This is particularly the case when the signal is mul ticomponent These spurious values are sometimes called interference terms be cause they happen whenever we have two or more components that interfere It must be understood that the word interference as used in optics is a real effect but its usage here connotes something artificial and undesirable A major develop ment in the field occurred when the conditions on the kernel that minimized these spurious values were understood and implemented by Choi and Williams1117 and Zhao Atlas and Marks11211 Loughlin Pitton and L Atlasf4 Jeong and Williams11 and Cunningham and Williams In this chapter we use a simple but general ap proach to see what those conditions are and also describe how this has led to the concept of kernel design 112 REDUCED INTERFERENCE DISTRIBUTIONS We now derive the conditions on the kernel so that the cross terms for multicompo nent signals are small in some sense These kernels produce reduced interference distributions a term coined by Williams Williams Jeong and Cunningham in troduced the concept of kernel design and formulated the general methodology to produce reduced interference distributions We shall approach the problem by first considering a simple case and subse quently show that the result is general Consider the simplest multicomponent signal st AejWt A2 eiW2t 111 162 Sec 2 Reduced Interference Distributions 163 where Al and A2 are constants We calculate the general timefrequency distribu tion leaving the kernel totally arbitrary The generalized characteristic function Eq 94 expressed in terms of the self and cross terms is M9T M11M22M12M21 112 with M110r 27rJA1J2OOTe 60 113 M1207 27rAiA2 00 reT 60 wl w2 114 and 12 2wl w2 115 The terms M22 and M21 are obtained from the above by interchanging the sub scripts 1 and 2 The distribution is Ct w 412 fJMorei9tdodr Cl1 C22 C12 C21 116 where C11 I Al I2 21r 117 C12 AA2 e1W12tKw 118 with Kw 2n f Owl W27 ejTWrV12 d7 119 To make the cross term small we must make K small But K is a Fourier trans form with respect to r and to make a Fourier transform relatively flat the integrand mustbe a peaked function Therefore as a function of r we must take 00 r highly peaked Everything we have said for the sum of two sine waves applies to the case of a signal composed of two impulses st A 6t t1 A2 6t t2 1110 because there is total symmetry in the general equation with regard to the signal and spectrum The effects discussed will be the same except that now we focus on the 0 variable Therefore if we dont want cross terms for the case of two impulses then the kernel must be peaked as a function of 0 For the general case when we have two components that are not necessarily parallel to the time or frequency axis the kernel must be peaked in both 9 and r We also know that the characteristic function is given by M8 r 09 7A9 r and that the maximum is at the origin 00 T e7Twwl dr 164 Chap 11 Kernel Design for Reduced Interference Hence the maximum of the kernel is also at the origin and we may take it to be one at the origin since that is the requirement for the preservation of total energy If we also want the marginals to be satisfied the kernel must be one along the 0 and r axes Therefore we seek kernels whose values away from either the 0 or r axis are small relative to the value at the 0 and r axes A way to describe this region is to observe that the product O r is large when we are awayfrom either axis We conclude that for cross term minimization 00rr 1 for Or 0 1111 Product Kernels Product kernels are characterized by functions of one variable If we let x BT then the condition for cross time minimization becomes Ox 1 for x 0 1112 For this case 1 1 j OX ajxv dx K I w2 wl I 27r WGJ12 1113 Let us now see how some of the distributions we have considered thus far behave with regard to the cross term property derived above Example 111 Wigner Distribution For the Wigner distribution the kernel is one throughout the B r plane and hence it does not satisfy the cross term minimization property Example 112 Spectrogram For the spectrogram the kernel depends on the window ht For a Gaussian window we have using Eq 920 00 r ecr24e24a ht a 14 ect22 1 1114 We see that the kernel is peaked at the origin and furthermore for reasonable values of a it falls off appropriately That is the one of the reasons the spectrogram some times has good cross term behavior Note that the kernel is not one along the axes and therefore does not satisfy the marginals Example 113 ChoiWilliams Distribution For the ChoiWilliam distribution which we will be studying in detail later the kernel is a8212 We can control the relative fall off by varying the parameter If o is very large the function is relatively flat and cross term suppression is not achieved For small values of a the function is peaked at the origin is one along the axis and falls off rapidly away from the axis Therefore this kernel satisfies the cross term minimization property Sec 3 Kernel Design for Product Kernels 165 Example 114 Further Generalization of ChoiWilliams kernel Once the basic conditions for cross term minimization are understood it is easy to find kernels that satisfy the basic properties and give finer control for certain signals One extension has been introduced by Diethorn11751 and Papandreou and Boudreaux Bartelsl1 t9 r e8aT6m This gives individual control in the 0 T directions 113 KERNEL DESIGN FOR PRODUCT KERNELS An effective means for designing product kernels has been formulated by Williams et all The constraints on product kernels transcribe themselves as very simple constraints on the Fourier transform of the kernel In addition since product ker nels treat time and frequency on an equal footing the two constraints one for each domain collapse into one For example to satisfy the time and frequency marginals for an arbitrary kernel we must have that 00 rr 1 and 00 0 1 However for product kernels both of these constraints are met by simply requiring that 00 1 The Fourier transform of a product kernel is defined by ht f Ox eit dx 007 fhtei9tdt 1116 In Table 111 we list the constraints for obtaining desirable properties on 0 and ht Table 111 Conditions on the kernel and ht Condition 00 r ht Reality of distribution 0 real ht ht Finite support See Eq 942 ht 0 for I t I 2 Cross term minimization 0x 1 for x 0 Smooth tapering at 2 Marginals 00 1 f ht dt 1 The main advantage of formulating kernel design in ht is that we have been able to collapse a number of diverse conditions into a few rather simple ones In addition many methods that have been developed for filter design can be used to produce such functions 166 Chap 11 Kernel Design for Reduced Interference 114 PROJECTION ONTO CONVEX SETS Finding a kernel with all the properties we have enumerated may not be straight forward indeed as the number of constraints increases the problem becomes that much harder There is a method that automatically finds a kernel that satisfies a list of properties if those properties are convex functions A collection of functions is said to be a convex set if for any two functions f and g and a real number a a new function h constructed by way of h of 1 ag for O a 1 1117 is also a member of the collection For the case of kernels suppose that 019 r and 2 0r are kernels that satisfy a particular constraint Forming a new kernel by 439T a09T1a029T 1118 we ask whether for any a between zero and one 03 0r also satisfies that con straint If it does then the set of kernels satisfying this constraint form a convex set For example consider the time marginal constraint which is 00 0 1 For any two functions that satisfy this constraint will 03 satisfy it Consider 039 0 a19 0 1 a 029 0 a 1 a 1 1119 and we see that 03 9 0 satisfies the time marginal constraint Hence such functions form a convex set The other constraints we have considered also form convex sets The method called projection onto convex sets automatically picks out the functions if they exist that satisfy all the conditions Furthermore if such a function does not exist then the method picks out the best function in the mean square sense This method was devised by Oh Marks II Atlas and Pittonli 115 BARANIUKJONES OPTIMAL KERNEL DESIGN In Chapter 9 we mentioned that the kernel can be signal dependent and most of the constraints on the kernel we have derived apply to signal dependent kernels Once we choose a signal dependent kernel we are no longer dealing with bilinear distributions Signal dependent kernels are important for many reasons We shall see in Section 142 how manifestly positive distributions may be obtained by choos ing signal dependent kernels In Section 112 we have seen how to choose kernels that minimize the cross terms the criterion being that the kernel must be peaked near the origin However it must be peaked in such a way it that encompasses the self terms which is clearly a signal dependent condition A method to obtain ker nels that do that effectively by adapting the kernel to the particular signal at hand has been devised by Baraniuk and Jones 1481 First the ambiguity function A0 T is calculated and from it we form the generalized ambiguity function M0 r and Sec 5 BaraniukJones Optimal Kernel Design 167 define a functional of the kernel by f r f ff I M9 r12 dr dB JJ A0 r qi9 r I2 dr d9 1120 Since the self terms are concentrated around the origin we want this functional of the kernel to be maximized The maximum of the characteristic function is at the origin and therefore one imposes in the maximization the constraint that the kernel never increases in any radial direction Furthermore we want to control how concentrated around the origin the kernel should be This is achieved by controlling ff O9 r 2 dr db a1121 One chooses aand constrains the maximization of the functional f with this con straint The bigger a the more concentrated the kernel will be In addition to the above we may also want to constrain the solution to give the correct marginals and other quantities An illustration is given in Fig 111 0 Time msec 25 Fig 111 An example of the effect of the pa rameter a in the BaraniukJones approach The signal is a bat sound In a a is very large and hence we basically have the Wigner distri bution For b and c a 20 and 4 respec tively Courtesy of R Baraniuk and D Jones Data of C Condon K White and A Feng Chapter 12 Some Distributions 121 INTRODUCTION We have already discussed in detail the Wigner distribution and the spectrogram In this chapter we study additional distributions some new some old and de scribe their properties and the physical and mathematical reasons for their intro duction 122 CHOIWILLIAMS METHOD As discussed in the last chapter Choi Jeong Cunningham and Williamst117 2s4 1651 developed the theory of reduced interference distributions and the ideas that allow one to design kernels to accomplish that Their first example was the distribution determined by the kernel 0eT e02 la 121 where a is a parameter We have already used this kernel in a number of examples for illustrative purposes It is a product kernel and 00 T 18 0 1 which shows that both marginals are satisfied Furthermore it is straightforward to see that it satisfies the instantaneous frequency and group delay properties If we take a to be large then the ChoiWilliams distribution approaches the Wigner distribu tion since the kernel then approaches one For small a it satisfies the reduced inter ference criteria as discussed in the previous chapter Substituting the kernel into the general lass and integrating over 0 we obtain 1 Pcw t U 47r3 1 2 jf T7 v exp 4T2 2 jT J 122 x su 2T su 2Tdudr 123 To better understand the behavior of this distribution we consider a number of ex 168 Sec 2 ChoiWilliams Method 169 amples For the sum of two sine waves st Al e3Wlt A2 e3Wt 124 the distribution can be calculated exactly Ccwt w A6u w A26W w2 2A1A2 cosw2 wl t 77w w1 W2 a 125 where rw w1 w2 a For large or 1 ex w 2wiw221 126 47rwl W2 2a p 4wl W2 2a J lim llwW1W2a 6w 2wl w2 127 o00 and therefore the distribution will be infinitely peaked at w 2 wl W2 which is precisely the Wigner distribution As long as a is kept finite the cross terms will be finite and not concentrated at one point They spread out with a lower maximum intensity In Fig 121 we illustrate the effect of different choices of a where the delta functions are symbolically represented Notice that the self terms are still delta functions along the frequencies wl and w2 In general this will not be true for other signals and it is not generally true of the Wigner distribution either It is true for the Wigner distribution of a chirp only For a chirp the ChoiWilliams distribution is Ccwtw 27r f ep2T4v e3TWpt dT st ept22 128 The concentration is along the instantaneous frequency and the width depends on both 3 and a Figs 121126 illustrate the ChoiWilliams distribution for a number of analytic and real signals and in some cases we compare it to the spectrogram andor Wigner distribution WI FREQUENCY Fig 121 The Wigner distribution a and ChoiWdUams distribution b and c for the sum of two sine waves Both distributions are infinitely peaked at the fre quencies of the two sine waves In the Wigner distribution the cross terms also go to infinity In the ChoiWilliams distribution their intensity is a function of the parameter Q In b we have used a large a while in c a small a is used The ChoiWilliams distribution becomes the Wigner distribution for a oo 170 FREQUENCY Chap 12 Some Distributions Fig 122 A comparison of the Wigner left and ChoiWilliams right distribution for the sum of two chirps Courtesy of W Williams w Fig 123 Comparison of the Wigner distribution a ChoiWilliams distri bution b and the spectrogram c for a signal composed of a chirp and a sinusoidal phase modulation signal Courtesy of W Williams Sec 2 ChoiWilliams Method 171 240 N T c 120 Q7 a LL Spectrogram 0 a 120 Time ms Binomial Transform 240 0 b 120 Time ms 240 Fig 124 Comparison of the spectrogram and the binomial distributions for the part of a heart sound due to the closing of the valve The binomial distribution is a reduced interference distribution devised by Jeong and Williams Courtesy of J Wood and D I Barry a I 38 kh i c 0 Time ms 70 Fig 125 A comparison of the narrowband spectrogram a wideband spectro gram b and ChoiWilliams distribution c for a dolphin sound Courtesy of W Williams and P Tyack 172 Chap 12 Some Distributions Time Fig 126 In some individuals the jaw makes a click when speaking These sounds are called TMJ sounds Because of their very short duration the spectrogram left does not properly reveal the timefrequency structure On the right is the Choi Williams distribution Courtesy of S E Widmalm W J Williams and C Zheng 123 ZHAOATLASMARKS DISTRIBUTION16m In the ChoiWilliams distribution the cross terms are spread out in the time fre quency plane with low intensity There is something else we can do with them We can obscure them by placing them under the self terms as shown by Loughlin Pitton and AtlasEli Consider the case of the sum of two sine waves as in Section 112 The intensity of the cross terms are proportional to Kw as given by Eq 119 For a kernel of the form t9 r f 8 r sin a8r 129 where a is a constant we have K ff w Wl w2i r 4 W STEWa jTWa 1 W a W a 1 W I 2 l 2 I 2 2 e 2 2 e dr 1210 Therefore the cross terms will be functions of w a 2 w1 a 2 w2 and w a 2 w1 a 2 w2 Something unique happens at a 2 At that value we have f fwi W2T f ejrWW21 dr 1211 Kw 4n7 J and we see that Kw is a function of w w1 and w w2 which are the location of the self terms Loughlin Pitton and AtlasM11 have shown that for K to fall exactly on the self terms the kernel f 8 r must produce a distribution whose cross terms lie exactly midway between the self terms Depending on the choice of f 0r these kernels may or may not satisfy other desirable properties such as the marginals But nonetheless they can give a good Sec 3 ZhaoAtlasMarks Distribution 173 indication of the timefrequency structure of a signal Note that for two impulses cross terms will be produced However for multicomponent signals that are mostly parallel in the frequency direction the cross terms will not be prominent because they will be hidden under the self terms A distribution with this property is the Zhao Atlas and Marks distribution OZAM 0 7 9r 1 T j s a9 OT 1212 which gives r rtaIr Ct 4a J gT eir J s u 17 su 2T du dr 1213 tairl for the distribution In the original work of Zhao Atlas and Marks g was taken to be equal to one and a to one half This distribution has many interestin properties and a comprehensive analysis of it has been given by Oh and Marks In Figs 127 and 128 we show two examples a b C a b c d Fig 127 A comparison of the spec trogram b and ZAM distribution c for a signal a that changes fre quency abruptly Courtesy of Y Zhao L Atlas and R Marks Fig 128 The signal a changes from constant frequency to a higher fre quency linearly in a finite time In b is a narrowband spectrogram in c we have a wide band spectrogram and in d we have the Zhao Atlas Marks distribution Courtesy of P Loughlin J Pitton and L Atlas 174 Chap 12 Some Distributions 124 BORNJORDAN DISTRIBUTION In Section 106 we derived the sinc distribution using the correspondence rule of Born and Jordan This distribution was first derivedm in 1966 although itsprop erties were not understood until the work of Jeong and WilliamsM who studied its general properties and pointed out that it satisfies the properties of a reduced interference distributions Loughlin Pitton and Atlas40 and Cohen and Leei134 have also considered its properties For the kernel take 08 T sina9T aer 1214 and the distribution is r Ctw 1 J 1 ejrw fsu 2T su zTdudT 1215 47ra r alrl For the case of two sine waves as per Eq 111 Kw 1 1 if a 2 wl a Z w2w a 2 wl a Z w2 2a W2 wl 1216 and zero otherwise In deriving this expression we have assumed that w2 w1 Thus the cross terms are uniformly spread in the region indicated For the case a i we have Kw 1 if w1 w w2 1217 W2 w1 in which case the cross terms are totally restricted between the two frequencies 125 COMPLEX ENERGY SPECTRUM In Section 103 we derived the MargenauHill distribution by specifying a particular ordering of the characteristic function operator Rihaczek484 gave a plausibility argument based on physical grounds Suppose we have a time dependent voltage and decompose it into its Fourier components Vt J Y et dw 1218 vr2 7r We can think of Vweit as the voltage at a particular frequency If we assume a unit resistance the current for that frequency is iW Velt Hence the total current in the frequency band w to w is rWn 1 o it J iW t dw 2x V et dw 1219 1 Born and Jordan did not consider joint distributions This distribution was derived in reference 125 and called the BornJordan distribution because the derivation is based on the BornJordan rule The derivation is repeated in Section 106 Sec 6 Running Spectrum 175 Power is the product of voltage and current V t i t and is the energy per unit time Hence the energy in the time interval At and frequency interval Aw is f tt 1 r JrEtw Vtitdt VVtetdwdt 1220 27r Jt L and therefore the energy density in time and frequency is et w otlim o Qt 2 VIVt ai Jt 1221 2 st Sw ajt 1222 wherein the last step we have identified the signal with the voltage st V t in which case V Sw Equation 1222 is the MargenauHill distribution de rived in Section 103 also called the Rihaczek distribution and complex energy spec trum Although it does satisfy the marginals it does not have many other desirable properties In particular it does not satisfy the instantaneous frequency condition although it does satisfy the strong finite support property 126 RUNNING SPECTRUM The running spectrum introduced by Page1419 is a method to define a time fre quency distribution The Fourier transform of a signal considers the signal for all time Suppose we consider it only up to a time t St w 1 rt st a7 dtt 1223 27r This is the running Fourier transform defined by Page If the signal was indeed zero after time t then the marginal in frequency is St w I2 If a distribution f Pt w satisfies this marginal then t Pttw dt I St w I2 1224 00 Differentiating with respect to time gives 2 t St w ewt 1225 P t w a I St w I2 2 Re s This is the Page distribution Its kernel is 00r eiO I T I 2 By inspection we see that it satisfies the marginals The main characteristic of this distribution is that the future does not affect the past and hence the longer a frequency exists the larger the intensity of that frequency as time increases Once the particular frequency stops the distribution at that frequency remains constant as time increases The Page distribution satisfies the weak and strong finite support properties 176 Chap 12 Some Distributions Fig 129 A comparison of the Wigner distribution and Page distribution for a finite duration sine wave The Page distribution does not go to zero at the end of the signal Example 121 Finite Sinusoid For the finite duration sinusoid st eW0t 05 t T the running transform and distribution are 1 St w 2 J e3wot aiwt dt 7 e 1 o w WO Ptw sincw wot 0 t T 0 otherwise 1226 1227 1228 As time increases the distribution becomes more and more peaked at wo This is illus trated in Fig 129 Variations of the Page derivation were given by Turner 1M31 Levin1332 and others Levin defined the future running transform by 1 oo St w st ait dt 27r Jt and using the same argument 00 P t w dt I St w 2 t and 1229 1230 19 Pt w at I Si w 12 2 Re 2 s t St w ej 1231 Note that if the two distributions are averaged we get the Rihaczek distribution Sec 6 Running Spectrum 177 Similar to the running Fourier transform we can define the running signal trans form for frequencies up to w by 8w t 2foo Sw Jt du 7n 1232 which gives the distribution P t w I s t 12 2 Re 2 s t S w awt 1233 Fig 1210 compares the Wigner Page and MargenauHill distributions for a finite duration signal that has been turned on and off as illustrated FREQUENCY Fig 1210 A comparison of the Wigner a MargenauHill b and Page c distri bution for a finite duration signal that is turned off and on as illustrated Chapter 13 Further Developments 131 INTRODUCTION In this chapter we address a number of topics that are central to the description of signals in time and frequency 132 INSTANTANEOUS BANDWIDTH 1351451 In previous chapters we have given a number of arguments to indicate that in stantaneous frequency is the derivative of the phase In Chapter 9 we showed that many timefrequency distributions give the derivative of the phase for the first con ditional moment w t Vt 131 From this point of view instantaneous frequency is an average the average of the frequencies existing at a particular time If instantaneous frequency is a conditional average it is natural to then ask for the conditional standard deviation about that average Because it is the spread of frequencies at a particular time we will call it the instantaneous bandwidth and use Bt or a it to denote it We now discuss a number of arguments that point to taking At Bt wlt At 132 as a plausible expression for instantaneous bandwidth This expression does not depend on the phase it depends only on the amplitude of the signal This is rea sonable since standard deviation is the spread about the mean and therefore the location of the mean is immaterial We will present three arguments for its reason ableness 178 Sec 2 Instantaneous Bandwidth 179 Bandwidth Equation In Section 15 we expressed the bandwidth of a signal in terms of its amplitude and phase B2 f A2t dt ft w 2 A2t dt 133 The second term in this expression averages all the deviations of the instantaneous frequency from the average frequency and certainly corresponds to our intuitive understanding of spread But what is the meaning and origin of the first term In Section 46 we showed that for any joint density Px y the global spread Qi is related to the conditional spread oy x in the following manner Qb o22 Px dx f y x y 2 Px dx 134 The similarity of this general result with the bandwidth equation Eq 133 is sug gestive Associating x with time and y with frequency and comparing we can im mediately infer Eq 132 for the conditional standard deviation Of course one cannot unambiguously equate quantities under the integral sign since we can al ways add a quantity that integrates to zero However if we added anything that integrates to zero the expression for the standard deviation would not remain man ifestly positive Joint TimeFrequency Representations In Section 96 we showed that for distribu tions that satisfy the marginals and give the derivative of the phase for the instan taneous frequency the standard deviation for a given time is 0 A 2 z 1 4 At 2 1 400 AA t 135 11 As mentioned in Section 96 the choice 460 a leads to a positive spread and in particular to Eq 132 There are an infinite number of densities that satisfy the condition 00 14 Furthermore it is not surprising that there are an infinite number of distributions that could produce a positive standard deviation What is surprising is that they all produce the same result All possible distributions that always give a positive result for the conditional standard deviation give the same result namely IAtAtI Instantaneous Bandwidth for the Spectrogram In Chapter 7 we derived for the spectrogram the conditional first moment of frequency Eq 745 That expression depends on both the signal and window We then showed that as we narrow the window the conditional moment approaches the instantaneous frequency of the signal We also obtained an exact expression for the instantaneous bandwidth for the spectrogram Eq 748 and discussed that it is reasonable that the expression goes to infinity as we narrow the window The reason is that progressively nar rowing the window is tantamount to progressively making shorter duration signals 180 Chap 13 Further Developments with a corresponding increase in bandwidth While the spectrogram can be used to estimate the instantaneous frequency it appears that it cannot be used to estimate the instantaneous bandwidth However as the window gets narrower it can be shown 1311 that the expression for the instantaneous bandwidth of the spectrogram Eq 748 breaks up into a number of terms one of them window independent and the rest window dependent In particular olt Ast windowdependent terms spectrogram 136 Thus we see that the window independent term is precisely the instantaneous bandwidth of the signal Purely Frequency Modulated Signals The above three plausibility derivations all lead to AtAtI for the instantaneous bandwidth An important consequence is that for purely frequency modulated signals that is signals whose amplitude is constant the instantaneous bandwidth is zero Bt aWit 0 for st Ae0t if A constant 137 Physically what this means is that at each instant of time there is only one fre quency the derivative of the phase This is in conformity with our intuition that for a purely frequency modulated signal the instantaneous frequency is known pre cisely Notice also that for this case the global bandwidth is B2 fdt w 2 dt 138 For purely frequency modulated signals the spread in frequencies comes from the change in the instantaneous frequency only Constant Bandwidth We now ask for what signals is the instantaneous bandwidth constant Solving for IAt AtI p we immediately have At a ept 139 Therefore signals whose amplitudes are decaying exponentials have a constant in stantaneous bandwidth For these signals als the global bandwidth is B2 p2 J Wp t w 2 dt 1310 Decay Rate Model and Formant Bandwidth11491 For a decaying exponential the broadness is proportional to the decay constant This fact can be used to obtain a sense of the broadness of an arbitrary signal at time t by fitting an exponential at that time SFt a ept coswt coo 1311 Sec 2 Instantaneous Bandwidth 181 where SFt signifies the fitted curve around time t The adjustable numbers to obtain the best fit are the three constants a p cpo The fitting is done locally that is we take a small piece of the function at that time and do the best possible fit by obtaining the constants a p and W o Since we are trying to fit locally we must decide on the interval of time around the time of interest In the case of speech 5 milliseconds is typically used Once the curve is fitted the broadness at that time is then given by p For different times we have different sets of constants This idea has been utilized in speech where the ps thus obtained are called for mant bandwidth However speech is a multicomponent signal and the fitting is done by a sum of decaying exponentials In this way we obtain the spread of each component at each time We now want to show the relation between this method of estimating a local bandwidth and Eq 132 For simplicity we consider one component Suppose the signal we are trying to fit is of the form st At coscpt 1312 Expanding the amplitude and phase separately in a Taylor series about some time t to we have st I Ato Atot to cos oto cP tot to 1313 Similarly we expand the presumed fitted curve SF aePto coswttowo a 1ptto coswttocpo 1314 In the limit of t to we see that the best fit is when a Ato ap Ato 1315 which gives Ip AAtoto 1316 We see that this method of obtaining the spread in frequencies approximates the instantaneous bandwidth defined by Eq 132 Poletti Formulation An interesting formulation of these concepts was devised by PolettiE41 who defined a new signal by Qt at logst Al t jWt 1317 This signal is called the dynamical signal The relation between the dynamical sig nal and the real signal is st 3t st Now the instantaneous frequency and instantaneous bandwidth are simply given by wti U t IM OM Bt Re Qt 1318 182 Chap 13 Further Developments where Im and Re stand for the imaginary and real parts respectively Group Delay and Its Spread All the concepts and results developed for instan taneous frequency can be applied to group delay and its spread The analogy is both physical and mathematical We can think of group delay as the average time for a particular frequency It is given by the negative of the derivative of the spec tral phase In complete mathematical analogy with our consideration of the spread about the instantaneous frequency we have the spread about the group delay Qt1w Bw 1319 where Bw is the spectral amplitude 133 MULTICOMPONENT SIGNALS We have frequently alluded to multicomponent signals and emphasized that a mul ticomponent signal is not just the sum of signals We are now in a position to un derstand their nature and in particular to address the question of when a signal is multicomponent The concept of instantaneous frequency and instantaneous bandwidth developed in the previous section will be central to the explanation The origin of the concept of multicomponent signals arose with the observation that sometimes there are well delineated regions in the timefrequency plane Per haps the earliest observation of this was in the study of speech The delineated regions are called formants It is important to note that we get components with almost any distribution although the particular details may be different Before we address the case of multicomponent signals let us characterize a mono component signal Generally speaking a monocomponent signal will look like a single mountain ridge as illustrated in Fig 131 At each time the ridge is charac terized by the peak If it is well localized the peak is the instantaneous frequency The width of the ridge is the conditional standard deviation actually about twice that which is the instantaneous bandwidth A typical multicomponent signal is illustrated in Fig 132 It consists of two or more ridges each characterized by its own instantaneous frequency and instanta neous bandwidth Why do we have two ridges instead of one Because the width of each one is small in comparison to the ridge separation Therefore if we have a signal of the form 3t sit 32t A1t dW1t A2t e2t 1320 we will have a multicomponent signal if the instantaneous bandwidths of each part are small in comparison to the separation between the ridges But the separation between the ridges is given by the difference in instantaneous frequency and hence the condition for a multicomponent signal is that Alt I I A2t I C 1 P2 t P1 t Sec 3 Multicomponent Signals 12 02 175 Bowhead Whale 183 st At eivt Fig 131 The character At r1 istics of a monocomponent At f signal pt 225 275 325 175 225 275 325 FREQUENCY Hz FREQUENCY Hz Southern Right Whale 0L 100 150 200 50 100 150 200 FREQUENCY Hz FREQUENCY Hz Fig 132 A multicomponent signal is characterized by the narrowness of both parts in comparison to their separation Global or Local There is a tendency to say that a signal is or is not multicomponent The preceding discussion shows that the condition is a local one it applies for one given time Of course it is possible that it will be the case for all time but that does not have to be so Hence a signal may be multicomponent at some times and monocomponent at others Spectrum of a Multicomponent Signal Generally the spectrum gives no indica tion as to whether a signal is mono or multicomponent This is reasonable since the spectrum is just the projection of the timefrequency plot on the frequency axis An example is given in Fig 133 a where the spectrum does not give any indica tion that we have a multicomponent signal However there are situations where the spectrum may indicate components and this is illustrated in Fig 133 b The reason is the components are limited to mutually exclusive bands for all time Multicomponent in Time The above discussion has used situations where the components have a narrow spread in the frequency direction Of course we can have a situation where components are narrow in the time direction or a combina 184 0 4 0 0 10 10 14 RU hS 2 to En M Chap 13 Further Developments 024621012U181820 FII M04cY e Fig 133 Two examples of multicomponent signals and their energy density spec tra Generally the spectrum cannot be used to determine whether a signal is multi component as in a Sometimes it can as in b lion of the two The spread along the time dimension of each part must be small relative to the separation The separation is the difference in the group delay and so the condition for a multicomponent signal in time is Biw B w 102w V511 wI 1322 There are many situations where the components may not be well delineated and this will depend on the amplitudes and phases As with real mountain ridges there are many hazy situations in which separate ridges cannot be distinguished 134 SPATIAL SPATIALFREQUENCY DISTRIBUTIONS The intuitive concept of frequency is how often something repeats Thus far we have emphasized repetition in time although we can have any other variable and ask for the repetition For example an ordinary two dimensional picture is an ex ample of variation in density as a function of position in space Now fix on a partic ular direction The frequency is then the number of ups and downs in the density per unit distance Therefore instead of timefrequency we can sensibly speak of positionpositionfrequency or spatialspatialfrequency distributions Everything we have previously done is immediately applicable to this case with the proper transliteration Signals are functions of space rather than time Also we can have two dimensional situations and have frequencies in each direction the distribu tions will then be four dimensional Just as timefrequency analysis gives us the local behavior in frequency spatial spatialanalysis gives us the local behavior of spatial variations The advantage and use of this type of distribution have been developed by a number of people and excellent discussions regarding them can be found in the articles by Jacobson and 2731 and in the review article by Cristobal Gonzalo and Bescos1611 Sec 5 Delta Function Distribution for FM Signals 185 135 DELTA FUNCTION DISTRIBUTION FOR FM SIGNALS We have seen in Section 132 that for constant amplitude signals the instantaneous bandwidth is zero Therefore for signals with constant amplitude one would like the joint timefrequency distribution to be totally concentrated along the instanta neous frequency Pt w 6w cpt for st ewt 1323 Given a distribution we know how to obtain the kernel It is Eq 919 Applying that formula we have 09 T f ejetjrWt dt f eet3l wtT2wtT2i dt 1324 This kernel insures that we get Eq 1323 for signals that are purely frequency mod ulated Note that it is a functional of the signal Example 131 Quadratic Phase Take Wt 3t2 2 wot then tpt 2 T cpt 2T tT woT and since cp t 3t wo we have 1325 f eetjrwt dt f ejetirtwo dt 1 1326 which is the kernel of the Wigner distribution Example 132 Cubic phase We take wt 27t3 and use Eq 1324 to obtain f ejetjryt2 dt 00 T f jetj7trr912 dt 1327 Marginals The condition for the frequency marginal is 09 0 1 Taking r 0 in Eq 1324 we see that the time marginal is satisfied Now consider the condition for the frequency marginals f ei t dt 00 0 f ealwt2Twt2r1 dt 1328 186 Chap 13 Further Developments which shows that the marginal cannot be satisfied exactly for this kernel However it is approximately satisfied To see that expand Wtr2 and cptr2 in a Taylor series cpt 2r cpt 2r r cpt 1329 and O r f eiwt dt f eirwt dt 1 1330 Oe7P0 dt f eiwt2Twtz1Tl dt f e7P0 These results are clearly unsatisfactory and argue that there should be another for mulation of the general class that takes this difficulty into account in an exact way However that problem is unsolved as of this writing Concentration for Bilinear Distributions In general we can not have a bilinear dis tribution satisfying the marginals that is a delta function along the instantaneous frequency The reason is that the delta function is positive and we know that we cannot have positive bilinear distribution satisfying the marginals The seeming exception of the Wigner distribution for the signal given by Eq 843 is readily ex plained because for that signal the Wigner distribution can be expressed in a form which is a not bilinear but of higher order 1280 1411 The fact that the bilinear distribu tions may go negative presents a problem in the definition of concentration since for example standard deviations may go native Using the square of a distribu tion as a measure of concentration janssen 2761 has considered this problem and shown that for distributions characterized by kernels of the form 09 r ell the Wigner distribution a 0 is generally the most concentrated However the question for a general kernel remains open 136 GABOR REPRESENTATION AND TIMEFREQUENCY DISTRIBUTIONS When a one dimensional signal is expanded in a complete set of functions the co efficients give an indication of the relative weight of the particular expansion func tion For example in the Fourier case expanding a signal in the complex exponen tial the coefficients that is the Fourier spectrum give an indication of the intensity or relative dominance of a frequency for the particular signal being expanded Gabor210 conceived of the possibility of expanding a one dimensional signal in terms of two dimensional timefrequency functions The timefrequency plane is discretized into a lattice where the coordinates are ti nT wi Trail oc n m oo 1331 and where T and Il are the time and frequency lattice intervals Gabor proposed that an arbitrary signal be expanded in the form st Cnm hnm t hnmt ht rnT ejnstt n m oo oo 1332 nm Sec 6 Gabor Representation and TimeFrequency Distributions 187 where ht is a one dimensional function as yet to be specified If such an expansion can be done the coefficients squared would give an indication of intensity at the time frequency point ti wi Gabor suggested that an appropriate function for h is the Gaussian because it is the function that is most compact in the sense of the timebandwidth product ht a7r4 eat22 1333 On the issue of whether such an expansion is possible in principle it has been shown to be possible when TO 1 However the expansion coefficients are not unique and for that reason a number of different approaches have been developed for their calculation We shall not delve into their calculation here because it is not germane to our main point Suffice it to say that they can be obtained for an arbi trary signal Qian and Morrisi411 showed that this type of expansion can be effectively used to understand the cross terms in the Wigner distribution and in addition offers an easy way for systematically adding them to the self terms The method was further developed by and Qian and Chen 470 The approach can be applied to any of the bilinear distributions Notice that what appears in all bilinear distributions is the factor s u I rsu 1T which when expanded using Eq 1332 is 2 2 8u 2T8u 2T Cn m C hn m u 2T hn mu 2T 1334 nm nm We find the generalized distribution to be Ct W E E Cnm Cnm Cnmnmt W 1335 nm nm where Cnmnm 47r2 JffhimU 2T hnmu 2T 007 ej8jrwjAu du d7 d9 1336 Let us now specialize to the Wigner distribution in which case the Cnmnm can be done analytically since the hnms are Gaussians If we break up the summation as W t w E I Cnm I2Wnmnmt w E Cnm Cnm Wnrnmt w nm n m fin rn 1337 then the first summation is manifestly positive since the hnm is a Gaussian If all the terms of the second summation are added thenwe get the exact Wigner distri bution An effective way to add the terms of the second summation is to first add nearest neighbor terms then add second nearest neighbors and so on The motive is to systematically add the relative importance of the cross terms 188 Chap 13 Further Developments 137 EXPANSION IN SPECTROGRAMS From a calculation point of view the spectrogram is particularly easy to compute since the only calculation involved is a single Fourier transform of the windowed signal Furthermore Fourier transforms are very efficiently calculated by way of the fast Fourier transform technique We now show that an arbitrary real time frequency distribution may be expressed as an infinite sum of spectrograms This decomposition offers an efficient method for the calculation ofa timefrequency dis tribution because for many cases the timefrequency distribution can be approxi mated with a finite number of such terms From a computational point of view there may still be a considerable saving This method was developed into an effec tive calculation scheme by Cunningham and Williams We write here Eq 912 one of the ways to express the general timefrequency distribution Ct w 2 Jf rt 2x x x x eixx s x sx dx dx 1338 Let us momentarily assume that we can express r in the following form 00 rt a x x x x unx t unx t 1339 n1 n where Ans and un are constants and functions as yet to be determined Substituting this expansion in Eq 1338 we have 27r Yn n1 1 27r J 8x unx t a3 dx 2 1340 which is a sum of spectrograms with windows un All this depends on whether indeed we can decompose r as indicated by Eq 1339 which we now consider Hermitian Functions If we have a two dimensional function Kz z that satisfies the Hermitian property Kz z K z z then solving the integral equation 1341 uz A J Kz z uz dz 1342 results in eigenvalues and eigenfunctions Ans and uns which form a complete set For this to be possible the kernel must be square integrable Such functions are called HilbertSchmidt kernels The kernel K is then expressible in terms of the eigenfunctions and eigenvalues 00 Kz z unz u z 1343 n1 Sec 8 Spectrogram in Terms of Other Distributions 189 Now to specialize to our case Let z x t and z x t and define Kz z r z z z z 1344 As a function of z and z r is Hermitian if the distribution is real That is Kz z K z z if ruT r u T 1345 Hence the solution of r uz aJ r1z z z z uz dz 1346 will result in eigenvalues and eigenfunctions so that r may be expanded as indi cated by Eq 1339 In practice we usually have a signal in a discretized form and the eigenvalue problem is formulated in terms of a matrix equation for which there are standard routines for solving for the eigenvalues and eigenfunctions This has to be done only once Once the eigenvalues and eigenvectors have been obtained for a specific kernel they may be used for any signal The windows are not the usual windows used in calculating spectrograms but that is of no consequence since the main mo tive is for the purpose of numerical computation Complex Distributions If r does not satisfy the Hermitian property then it is pos sible to expand it in the following form 00 r 1 unxtvx t n1 an 1347 where now the uns and vns are different complete sets Such a decomposition is achieved by a method known as singular value decomposition which is discussed in Section 139 Substituting into Eq 1338 we have Ct w 1 00 1 l 3x Vnx t ajx dx I 1348 27r f 8x unx t ax dx L 27rf which is a sum consisting of products of shorttime Fourier transforms This ap proach was developed by OHair and Suteri411i 138 SPECTROGRAM IN TERMS OF OTHER DISTRIBUTIONS The concept of expanding a spectrogram in terms of other distributions was con sidered by Whiteiml and Amini19i who made a singular value decomposition of the kernel We restrict ourselves to real distributions which insures the kernel is Her mitian White showed how the spectrogram can be expanded in terms of modified 190 Chap 13 Further Developments Wigner distributions We shall consider the general case where we expand any dis tribution in terms of modified distributions Recall from Section 97 that any two distributions are related by CitW Jf 912ttwwC2twdtdw 1349 with 9121w 12 if 010r e 9t7rw dB dT 47r 020r 1350 If the distributions are real then 912 t w is Hermitian and 912 t w F 1 7 t rlw n1 en where the ens and i s are obtained by solving the integral equation 1351 On t En J 912tW17nw dw 1352 Substituting this decomposition into Eq 1349 we have 1 C2n t w 1353 00 C1t W E n1 En where C2n t W Jf rrnt t 17 W w C2t w dt dw 1354 This shows that an arbitrary distribution can be expanded as a sum of modified distributions C2nt w The modification indicated by Eq 1354 is that each term in the sum is the distribution smoothed independently in the time and frequency directions 139 SINGULAR VALUE DECOMPOSITION OF DISTRIBUTIONS If we have a function of two variables Ct w it is possible to expand it in terms of a sum of product functions 00 1 Ct w unt V W 1355 n1 Qn where the uns and vn are complete sets obtained by solving the coupled integral equations unt Qn f Ct w vnw dw i vnw vnf CW t unt dt 1356 Sec 10 Synthesis 191 The solution of these equations results in common eigenvalues but two complete sets of functions the us and the vs This decomposition of a two dimensional function is called a singular value decomposition Note that for the case Ct w C w t we have Hermiticity in which case we can use the standard approach dis cussed in Section 139 Since in our case Ct w will be a joint timefrequency dis tribution it is in general not Hermitian In the general case we need in principle an infinite number of terms to represent the distribution exactly However in many situations the first four or five terms are sufficient to achieve a very high accuracy The concept of decomposing a distribu tion in this manner was first done by Marinovich and Eichmanni3631 for the Wigner distribution There are two basic reasons for seeking such a decomposition First suppose the signal is in a noisy environment If we decompose the distribution and keep only the first few terms we will reduce the noise significantly because the signal is well represented by the first few terms but the noise is typically spread out over all the terms Therefore by truncating the series after four or five terms we retain most of the signal but lose most of the noise The second reason is for the purpose of classification The basic idea is that the as contain unique characterizations of the timefrequency structure of a distribu tion and may be used for classification Suppose for example we have ten signals and do a singular value decomposition of each one This will result in ten sets of singular values and eigenfunctions all different from each other Suppose we keep just the first five terms For the eigenvalues we will have ten sets of five numbers Now suppose we have an unknown signal one of the ten but in a noisy environ ment and want to classify it We do a singular value decomposition and compare the first five singular values to our sets Since we are comparing numbers the com parison is easy and fast Any reasonable measure of comparison can be used such as the Euclidean distance The closest distance is used to classify the signal 1310 SYNTHESIS Suppose we want to design a signal that has a timefrequency structure of our choosing A way to accomplish this is to form the timefrequency distribution and then calculate the signal that generates it If the timefrequency distribution we have constructed is a legitimate one that is a representable one then the sim ple inversion formula Eq 956 yields the signal However in general the time frequency distribution we construct will not be a representable one For example suppose we want a Wigner distribution to have certain time frequency structure In general we will not be able to construct a representable Wigner distribution be cause we will not be able to properly construct the cross terms We are simply not that capable What is done is to seek a signal that reproduces the given time frequency distribution as dose as possible for example in the least squares sense Obtaining a signal from a distribution is the synthesis problem It was first con sidered by BoudreauxBartels and Parks93 9si for the Wigner distribution For the 192 Chap 13 Further Developments ChoiWilliams distribution where the cross terms are less of a problem Jeong and Williams have devised an effective synthesis scheme P13 Another important application of the synthesis problem arises in the following circumstance Suppose we have a timefrequency distribution of some signal that is multicomponent and we want the signal that produces only one of the compo nents What we can do is to erase everything in the timefrequency plane but the component of interest Having erased parts of a legitimate distribution the result ing distribution will not usually be a representable one because for example we may have erased cross terms which are needed to have a representable distribu tion Nonetheless we want the signal that generates that component The synthe sis problem will again have to be done in terms of the best signal that generates a distribution close to the one we have at hand A further reason for the synthesis problem that is if we have a signal with noise and calculate the timefrequency distribution the noise will typically be spread throughout the timefrequency plane If we literally erase the noise but keep the signal the result will be the signal and noise only around the signal in time and fre quency Finding the signal this produces this curtailed distribution will hopefully result in a signal with less noise than the original 1311 RANDOM SIGNALS The main consideration in this book has been deterministic signals To apply the ideas developed to random signals one ensemble averages Ensemble averaging means averaging over all possible realizations of the signal each one having a cer tain probability of occurrence For deterministic signals we have used averaging for many purposes such as in the definition of global and local averages These two uses of averages must be differentiated We shall use an overline to denote en semble averaging Very often both averages are taken For example if we have a random signal then t means that we are calculating the average time for a partic ular signal of the ensemble and then averaging over all possible signals Sometimes the order of averaging matters and sometimes not but we will not address this issue here We define a random timefrequency distribution Ct w by ensemble averag ing the deterministic joint distribution over all possible signals Using the general timefrequency distribution Eq 91 we have rr eietiTWieu du dr d8 r 00 Ct 1 N 13 57 w s u r s u 7r 4 2 2 2 If we assume that hekernel is independent of the signal then r du d7 d6 r 09 t u 2 u fff 13 58 w s r s 2 r du dr dO 2 1 69 Jff R 13 59 u r u 7 4 2 Sec 12 Numerical Computation 193 where Rt t is the autocorrelation function of the random process Rt t st st 1360 If we want to work with the form given by Eq 99 then Ct w 2 ff rt u rRu Zr u 2T du dr 1361 If in addition we assume that the random process is stationary the autocorrelation function is then a function of the difference of the times Rt t Rt t 1362 in which case we have Ct w Ru 2r u 2r Rr 1363 Ct w 2 Jf rt u rRr ajT du dT 1364 For the Wigner distribution Martin13761 and Martin and Flandrin3781 have devel oped the main ideas and Martin has coined the phrase WignerVille spectrum to indicate that we have ensemble averaged the Wigner distribution White devel oped a comprehensive theory for the general case and devised specific methods for obtaining the important parameters of a random process and the errors involved in estimating the parameters Amins1201 early work on spectrum estimators is closely related to the concept of kernal choice for the random case Considerable work has been done in this area recently by Amin1261 Chaparro ElJaroudi and Kayhan11101 Riedel and Sidorenko14811 and Sayeed and JonesI5011 who have addressed the fun damental problems and specific applications Kernel Choice Most of the ideas regarding the choice of kernel carry over to the random case For example if we have a widesense stationary process we expect the marginal in frequency to be white noise The condition for that as Poschl45 has shown is that 00 0 1 which is the condition to obtain the frequency marginal for the deterministic case The issue of which is the best kernel to use for estimating the properties of the random process has been formulated by Amin1261 who has shown that the random case necessitates unique constraints on the kernel He has developed a comprehensive theory of kernel choice 1312 NUMERICAL COMPUTATION The calculation of timefrequency distributions is fairly straightforward However as is usual with numerical methods there are many tricks and procedures that have been devised and that are best obtained from the original sources We just give a broad outline of the basic ideas and discuss some of the fundamental issues that 194 Chap 13 Further Developments have arisen We assume that we have discrete samples and that the signal is suffi ciently bandlimited so that the Nyquist sampling theorem applies For a bandlim ited signal the signal can be reconstructed from discrete sampled values if the sam pling is done at a sampling rate w 2w where is the highest frequency in the signal All distributions have at least one Fourier transform to be done Historically there have been a number of methods proposed to calculate integrals involving sines and cosines for example the method of Fillon However the advent of the Fast Fourier Transform has over shadowed these methods because indeed it is fast The simplest distribution is the spectrogram Eq 75 One chooses a specific time t calculates the modified signal sr hr t as a function of r then takes the Fourier transform with respect to T This is repeated for each time desired For the Wigner distribution st 2T 8t 2T is calculated as a function of r for a fixed time t and the Fourier transform taken That gives us the Wigner distribution of frequencies at time t The procedure is repeated for any other time Note that it appears because of the factor of one half we must have signal values at points in between the sampled values If we do have them then of course there is no problem But that requires that we over sample by a factor of two If we do not one can for example interpolate to get them Because of this it has been believed that to construct the Wigner distribution from discrete samples one must sample the signal at twice the Nyquist rate otherwise aliasing will occur That can not be the case since in principle having a sampled signal at the Nyquist rate allows us to construct the signal for any time Having constructed the signal for an arbitrary time one can then calculate the continuous Wigner distribution Therefore we see that in principle we should not have to over sample For the Wigner distribution Poletti11 and Nuttalll405 have shown that the Wigner distribution can be computed from a signal sampled at the Nyquist rate without interpolation of the sampled values or reconstitution of the continuous signal The same considerations apply to the calculation of any of other bilinear time frequency distributions The basic issue is how to define the discrete version of bilinear timefrequency distributions so that at the discrete time frequency points they have the same value as the continuous version and where the calculation can be performed directly from the discrete version of the signal sampled at the Nyquist rate In addition the discrete version should satisfy the same general properties of the continuous version that is the marginals and so forth This is a fundamen tal problem that has recently been solved by Morris and Wu11871 OHair and B W Suter1410 Jeong and Williams1286 and Cunningham and WilliamsJ1671 For the general bilinear case three Fourier transforms have to be performed but for many kernels one of the integrations can be done analytically resulting in the form given by Eq 99 This is the case for example with the ChoiWilliams distri bution For such distributions we have two integrals only one of which is a Fourier transform To calculate such a distribution the inner integral that is Eq 910 is ef fectively done by the rectangular rule Subsequently the Fourier transform is taken We also mention that decomposition of a timefrequency distribution as a sum of weighted spectrograms is a very effective calculational method This is described Sec 13 Signal Analysis and Quantum Mechanics 195 in Section 137 Also the computational approaches for calculating the positive dis tributions are described in Section 142 and the calculation method for obtaining optimum kernels is described in Section 115 1313 SIGNAL ANALYSIS AND QUANTUM MECHANICS Historically the origins of the mathematics and many of the ideas used in time frequency analysis originated and were guided by corresponding developments in quantum mechanics In fact the original papers of Gabor and Ville continu ously evoked the quantum analogy This parallel development will undoubtedly continue because there is a very strong mathematical similarity between quantum mechanics and signal analysis and results can be mathematically transposed How ever the transposition cannot be carried over necessarily to the interpretation of the ideas because the physical interpretations are drastically different The most fundamental difference between the two subjects is that quantum mechanics is an inherently probabilistic theory while signal analysis is deterministic We empha size that the fundamental idea of modern physics is that we can only predict prob abilities for observables such as position and velocity and that this is not a reflection of human ignorance but rather the way nature is The probabilities are predicted by solving Schrodingers equation of motion The reason for the mathematical similarity is that in quantum mechanics the fundamental quantity is the wave function In the position representation the wave function is bq where q is position In the momentum representation it is Op where p is the momentum The main point is that the two wave functions are Fourier transforms of each other 2rrfi Oq aill dq Pq J OP e gplh dp Op 1365 where h is Plancks constant divided by 2ir Moreover the probability distribution Pq for finding the particle at a certain position is the absolute square of the wave function and the probability of finding a particle with a certain momentum Pp is the absolute square of the momentum wave function Pq Iq I2 Pq I Op 12 1366 Therefore mathematically and mathematically only we can associate the signal with the wave function time with position and frequency with momentum The marginal conditions are formally the same although the variables are different and 1It should be kept in mind that in quantum mechanics and signal analysis there is another layer that is probabilistic In the case of quantum mechanics this comes about if we do not know the possible wave functions and assign probabilities for obtaining them Hence in calculating averages for such a situation we have a double probability average one due to the inherent probabilistic distribution where the absolute square of the wave function is the probability and the other an ensemble average over the possible wave functions That aspect of the subject is called quantum statistical mechanics In signal analysis we start with a deterministic theory and if the possible signals are probabilistically determined random signal then we have to ensemble average over the possible signals 196 Chap 13 Further Developments the interpretation is certainly different In quantum mechanics the marginals are probability densities in signal analysis they are deterministic intensities In Table 131 we list some of the physical quantities in each field and the correspondence be tween them We also point out that the operator method we describe in this book is fundamental in quantum mechanics We now come to the issue of interpretation and address some of the fundamen tal distinctions One must be particularly cautious in transposing ideas because blind transpositions can lead to preposterous results In quantum mechanics phys ical quantities are represented by operators and it is the fundamental tenet that what can be measured for an observable are the eigenvalues of the operator That is the basis for the quantization of physical quantities From a classical point of view this produces bizarre results which are nonetheless true and have been veri fied experimentally For a dramatic example of the difference consider the sum of two continuous quantities In quantum mechanics each quantity can be continuous and yet the sum is not necessarily continuous Specifically consider the position q and momentum p which are continuous in quantum mechanics because the eigenvalues of the po sition and momentum operators are continuous Now consider the physical quan tity made up of position and momentum q2 p2 appropriately dimensioned It is never continuous under any circumstances for any particle It is always quantized that is it can have only certain values The reason is that the eigenvalues of the op erator q2 p2 are discrete If we were to make a blind analogy the corresponding statement in signal analysis would be that time and frequency are continuous but that t2 w2 appropriately dimensioned is never so and is always discrete That would be a ludicrous statement to make in signal analysis We now address the issue of the uncertainty principle We have already dis cussed some aspects of its interpretation in quantum mechanics and signal analysis in Chapter 3 The term uncertainty was coined in quantum mechanics where it properly connotes the fact that quantum mechanics is an inherently probabilistic theory In quantum mechanics the standard deviations involve the measurement of physical observables and are probabilistic statements However in nonproba bilistic contexts the uncertainty principle should be thought of as expressing the fact that a function and its Fourier transform cannot both be made arbitrarily nar row It has nothing to do with uncertainty as used in quantum mechanics There is another important difference In Chapter 2 we discussed that observ able signals are real and that the energy density spectrum of real signals does not properly indicate the physical situation For example for real signals the average frequency is zero which is not a reflection of the physical situation This led to the desire to define a complex signal In quantum mechanics wave functions are in herently complex although they may be real If a wave function is real then the average momentum is zero That is perfectly all right and does reflect the physical situation It means that we have equal probability for the particle traveling to the right and the particle traveling to the left Therefore there is no need to introduce the equivalent of an analytic signal Momentum unlike frequency can be negative or positive Sec 13 Signal Analysis and Quantum Mechanics 197 Table 131 The relationship between quantum mechanics and signal analysis The formal mathematical correspondence is position momentum time frequency The wave function in quantum mechanics depends on time but this has no formal correspondence in signal analysis Quantum Mechanics inherently probabilistic Position Momentum Time Wave function Momentum wave function 4P t 1 ftq t aj9Pn dq 27ri Probability of position at time t Probability of momentum Expected value of position Expected value of momentum Standard deviation of position Standard deviation of momentum Uncertainty principle Current Iwq t 12 10p t 12 q f gIV qt12dq p fpl0PtI2 dp aq q2 q2 P P2p2 aq ap h Z d phase of Oq dq Time Frequency Signal Spectrum Energy density Signal Analysis deterministic t w No correspondence 8t Sw 1 9t ait dt 27r 18t12 Energy density spectrum ISw12 Mean time t f t Ietl2 dt Mean frequency w f w I Sw I2 dw Duration T t2t2 Bandwidth B w2w2 Time bandwidth relation BT z Instantaneous frequency dt phase of st Chapter 14 Positive Distributions Satisfying the Marginals 141 INTRODUCTION Wigner showed that manifestly positive bilinear distributions satisfying the time and frequency marginals do not exist The spectrogram for example is man ifestly positive but does not satisfy the marginals while the Wigner distribution satisfies the marginals but is not manifestly positive Historically there has been a sense expressed in the literature that manifestly positive distributions satisfying the marginals cannot exist But they do They are obtained by a simple procedure which ensures that the marginals are satisfied These distributions are of course not bilinear in the signal 142 POSITIVE DISTRIBUTIONSI27 121 Take Pt w I Sw I2 I st I2 fu v 141 where u v are functions of t and w t ut f I st 12 dt vw J 1 Sw 12 dw 142 00 and where St is any positive function satisfying i If0 S2u v du 1 143 198 Sec 2 Positive Distributions 199 It is sufficient to define Qu v only for 0 u v 1 Note that u and v are the cumulative margins that is they are the sum of the densities up to given time and frequency values Marginals First note that dv J Sw I2 du 144 Now integrate with respect to w f Pt w dw I st 2f I Sw 12 u v dw 145 Ist 12 J1 1u v dv 146 0 I st 12 147 Similarly for the frequency marginal Positivity The distributions given by Eq 141 are positive as long as 12 is positive for the range 0 u v 1 cl may or may not be a functional of the marginals or signal but to generate all possible positive joint distributions we have to consider it a functional of the signal 12311 Examples of cls satisfying the above conditions are easily made up for example iluv 1 nui1 1mvm1 1 148 where n m are any integers Relationship Between Kernels We have introduced the form given by Eq 141 because it makes the manifestly positive aspect dear It is possible to obtain this form using the general lass developed in Chapter 9 There exists a relationship between the kernels11281 but generally it is easier to work directly with Eq 141 The Wigner Distribution for a Chirp In Section 86 we pointed out that for one signal and one signal only the Wigner distribution was manifestly positive That case is given by Eq 843 The reason it is positive is the it is not really bilinear It can be obtained from Eq 141 Weak and Strong Finite Support For the distributions given by Eq 141 the factor I Sw12 1 st 12 appears Therefore if the signal is zero at a certain time or the spectrum is zero at a frequency the distribution will likewise bezero assuming that fl is not infinity at those values Therefore these positive distributions satisfy the strong finite support property and hence also weak finite support Uncertainty Principle Since these distributions satisfy the marginals they satisfy the uncertainty principle 200 Chap 14 Positive Distributions Satisfying the Marginals Scale Invariance We now show following Loughlinlm4i that these distributions are scale invariant The distribution of the scaled signal sat sat is Pact w Sacw I2 3act 2 9uact vecw I Swa 12 1 sat 2 Quec t v w 1410 But t t at uact J I S1 I2 dt a J 1 sat 12 dt f I st I2 dt t1 at 00 1411 and similarly vaw vwa Therefore we have that P3t w I Swa I2 I sat 2 nuat vwa 1412 Plat wa 1413 149 Time and Frequency Shifts For a signal that is shifted by a time to and frequency wo the distribution is Pah t w I Sahw 12 I Saht I2 cZuaht vahw 1414 I Sw wo I2 I st to I2 1zuaht vahw 1415 But u I st I2 dt ut to sht f I Saht 12 dt f I st to I2 dt tto t t 00 00 00 14 16 and also vahw vw wo which shows that Pahtw Pt tow wo 1417 Conditional Distributions The conditional distributions are PtI Pt w IS t I2 S2 v 14 18 w u I Sw I2 Pwt Pt w I Sw 121u v 14 19 1 3t 12 Sec 3 The Method of Loughlin Pitton and Atlas 201 Consequences of Positivity Our experience in everyday life is of course with pos itive densities However our experience in timefrequency analysis has dealt with either positive distributions that do not satisfy the marginals or nonmanifestly positive ones that do satisfy the marginals We now examine the nature of time frequency distributions that are both manifestly positive and satisfy the marginals First let us note that if the marginal is zero at a certain point in time then the joint distribution must be zero at that time for all values of frequency the strong finite support property is satisfied This is so because to get the time marginal we add nonnegative pieces which cannot add up to zero unless the pieces are zero themselves This is no different than saying that if there are no people who are six feet tall then there are no people at that height for any weight Similarly if the spectrum is zero for a certain value of frequency the joint distribution will be zero for all times at that frequency Now consider the issue of instantaneous frequency and positive distributions Can we obtain the derivative of the phase as the conditional average wt st y w Pt w dw t 2 1 Jo 1420 The answer is no in general but yes sometimes The general answer of no comes about by the following argument of Claasen and Mecklenbrauker11201 Suppose we have an analytic signal and therefore the spectrum is zero for negative frequencies By the arguments given above the joint distribution will also be zero for negative frequencies If the distribution is positive the conditional average will be positive for an analytic signal This is a satisfying result However we know from Section 27 that there are analytic signals that produce an instantaneous frequency defined as the derivative of the phase which may go negative Hence for those signals Eq 1420 cannot be satisfied But those are precisely the signals for which the derivative of the phase as a representation of the concept of instantaneous fre quency apparently makes no sense Therefore a meaningful question is whether for cases where the derivative of the phase meets our expectations of instantaneous frequency positive distributions can satisfy Eq 1420 No general results regard ing this question are known 143 THE METHOD OF LOUGHLIN PITTON AND ATLAS Construction of positive distributions has been achieved by Loughlin Pitton and Atlas I They have formulated the problem in the following way With our present knowledge we cannot fix the joint distribution There are too many Sls The prob lem of finding a function for conditions that do not fix the function is a long standing one in many fields One approach is the method of maximum entropy The idea is that we find all the functions that satisfy the conditions and then choose among them the one that maximizes the entropy the reason being that the maximum en tropy solution is the one that is unbiased In practice this is achieved by taking a 202 Chap 14 Positive Distributions Satisfying the Marginals guess at the joint distribution POrdefining the cross entropy by A JJ Pt w log Pt dt dw 1421 Potw and maximizing this expression with the constraints of the marginals positivity and other possible constraints one may want to impose In Figs 141 143 we give a number of examples Fonollosa and Nikias2021 have added the additional constraint that the distribution along a given axis be a specified function In particular if we have a two dimensional distribution and want to obtain the density of the variable u at bw where a b are the direction cosines specifying the axis then the density of u is given by Eq 497 Pu ffou at bw Pt w dt dw 1422 J Pt u atb dt 1423 In our case we do not know Pt w Fonollosa and Nikias used the Wigner distri bution to obtain Pu and impose the constraint for a finite number of axis a co I IStI A ISWW w Fig 141 Positive joint time St 12 frequency of a chirp a and the sum of two chirps b and correspond ing marginals Notice that the time frequency distribution in b oscil t lates and is not simply the sum of the time frequency distributions of I i S rw 2 two chirps This must be so if the marginals are to be satisfied For a iI r1 fuller discussion of this issue refer to Section 110 Courtesy of P Lough w lin J Pitton and L Atlas Sec 3 The Method of Loughlin Pitton and Atlas 203 B A I I FMVINERM a myoil 1111111 Illt a b rmtr c d d Fig 142 For A the signal is a sequence of decaying exponentials shown in a The frequency is constant In b and c are the wide band and narrow band spectrograms The manifestly positive distribution that satisfies the marginals is shown in d B is the same as A except that the frequency is continually increasing linearly Courtesy of P Loughlin J Pitton and L Atlas r 1 M M yw I 1 13 b OTr Fig 143 The signal is the sound of a grinding machine The spectrogram is a and the positive distribution is b The spectrograms gives no in dication of the fine detail The pos itive timefrequency distribution re solves the lines and moreover shows that some of the lines have periodic amplitude modulation Courtesy of P Loughlin J Pitton L Atlas and G Bernard Data courtesy of Boeing Commerical Airplane Group 0 Time rnsec 50 Chapter 15 The Representation of Signals 151 INTRODUCTION In the previous chapters we developed the theory of joint representations for time and frequency We now extend the methods to other physical variables For a phys ical quantity we want to obtain its density average value spread and other results as was done for the case of frequency This is accomplished by expanding the signal in the representation of the physical quantity in the same sense that the Fourier rep resentation is the appropriate representation for the physical quantity frequency The basic idea of representing a signal in another representation is to write the sig nal as a linear combination of other functions the expansion functions They are obtained by solving the eigenvalue problem for an operator that represents or is associated with the physical quantity of interest For example the complex sinu soids are the eigenfunctions of the frequency operator and the frequency operator is associated with or corresponds to the physical quantity frequency We will use a to signify the physical quantity we are studying and script capital A for the associated operator 152 ORTHOGONAL EXPANSION OF SIGNALS A signal is expanded in the form st JFauatda 151 where ua t are the basis functions and Fa are the expansion coefficients or the transform of the signal As we will prove momentarily Fa is given by Fa fstuatdt 152 204 Sec 2 Orthogonal Expansion of Signals 205 The basis functions ua t are always functions of two variables in this case time and a The as are the numerical values of the physical quantity and they may be continuous discrete or both Furthermore the range of as may be infinite or limited The integration in Eq 152 implies a specific region of integration For example for frequency the range is all possible values while for scale we will see that the range is from zero to infinity In this section we assume that the variable a is continuous and at the end of this section we address the discrete case The functioh Fa gives us an indication of how important a particular value of a is for the signal at hand If Fa is relatively large only in a particular region we can then say that the signal is concentrated at those values of a The Expansion Functions Operators and the Eigenvalue Problem Where do we get the expansion functions the us that presumably are the natural functions for the physical quantity we are interested in Also how do we determine the possible numerical values the as that the physical quantity may attain Both are obtained by solving the eigenvalue problem for the operator that corresponds to the physical quantity Operators are commands that change functions For example multiplica tion squaring differentiation integration and combinations of these are operators Generally the eigenvalue problem is written as A ua t a ua t 153 In the eigenvalue problem the operator is given and one seeks to find those func tions the us that when operated upon give back the same function multiplied by a number in this case a Generally there are an infinite number of such functions each paired with an eigenvalue a The ua ts are called the eigenfunctions and the as are the eigenvalues Solution of the eigenvalue problem means solving for both the u s and as For example the operator d dx operating on ell returns a ex hence eax is an eigenfunction with eigenvalue a For this case there are an infinite number of eigenfunctions because we can take any number for a Linear Hermitian Operators A linear operator is one that satisfies AfgAfAg 154 For example the operation of differentiation is linear because the derivative of the sum is the sum of the derivatives However the operation of squaring is not because the square of the sum is not the sum of the squares An operator is Hermitian or self adjoint if for any pair of functions f t and gt fgt A f t dt 11t Agt dt 155 To prove that an operator is Hermitian one has to show that Eq 155 does indeed hold for any pair of functions 206 Chap 15 The Representation of Signals Importance of Hermiticity The importance of Hermiticity is threefold First Her miticity of the operator guarantees that the eigenfunctions are complete and or thogonal This means that the eigenfunctions satisfy f u a t ua t dt 6a a 156 f u a t ua t da 6t t 157 It is these properties of the expansion functions that allow us to transform between st and Fa as given by Eqs 151 and 152 In particular to see how to obtain Fa multiply Eq 151 by u a t and integrate with respect to time f stuatdt Jf Fa ua t if a t da dt 158 f Fa 8a a da 159 Fa 1510 which proves the inverse relation Eq 152 Second if the operator is Hermitian the eigenvalues are guaranteed to be real This is important because in nature measurable quantities are real Hence if the eigenvalues are to be measurable numerical values the operator should be Hermi tian to assure that this is so That is not to say that nonHermitian operators are not important but simply that they do not represent measurable physical quantities For example the translation operator that we studied in Section 14 is not Hermi tian but is very useful Third if an operator is Hermitian we can manipulate it in advantageous ways We have already seen this in Section 14 and we will see it over and over The proof that Hermitian operators do produce real eigenvalues and complete and orthogo nal eigenfunctions can be found in any book on mathematical methods Range and Values of a It is important to understand that the numerical values and range of the eigenvalues are obtained by solving the eigenvalue problem It is not imposed or assumed If the solution of the eigenvalue problem predicts certain values for the physical parameter a and we experimentally obtain a value that does not correspond to one of those values then there are only three possibilities the experiment is wrong we have made an error in solving the eigenvalue problem or we do not have the correct operator for that physical quantity Possible and Actual Values of the as One must be very dear about the following distinction The solution of the eigenvalue problem gives us the possible values that the physical quantity may attain that is the as However for a particular signal the actual values attainable are given by the function Fa which may or Sec 2 Orthogonal Expansion of Signals 207 may not range over all possible values of a For example any frequency is possible but for a particular signal only certain frequencies may exist Note that solving the eigenvalue problem has nothing to do with the signal at hand The eigenvalue problem is solved once and for all but Fa must be recalculated for each different signaL Normalization For linear operators a constant times an eigenfunction is also a solution to the eigenvalue problem To fix the constant the eigenfunctions are nor malized so that Eqs 156 and 157 are satisfied In that case we say that we have normalized to a delta function The Usage of spectrum The set of eigenvalues obtained by solving the eigen value problem is often called the spectrum and the terms continuous and discrete spectrum are used to communicate whether the d s are continuous or discrete The use of the word spectrum for the general case is unfortunate because spectrum is associated with frequency which is only one particular case But even more un fortunate the word spectrum is often used to denote the Fourier transform Sw However the particular usage is usually clear from the context Example 151 Frequency From Chapter 1 we know that the frequency operator is First let us prove that it is Hermitian For any two functions f and g we have by integration by parts that f 9t dt ftdt f9 0 00 f f t d 9t dt 1512 dt 1513 f f t 4t and hence the frequency operator is Hermitian The eigenvalue problem is W uw t w uw t 1514 and the solutions are uwt ce 1515 Notice that all real values of w are possible Therefore we say that the range of fre quencies are the real numbers from oo to oo To obtain the normalization consider f u w t uw t dt cf et e t dt 2ac26w w 1516 208 Chap 15 The Representation of Signals Since we want to normalize to a delta function we must take c2 121r and hence the normalized frequency eigenfunctions are 1517 Example 152 Time The time operator in the time representation is t and the eigenvalue problem is t ut t tut t 1518 where t are the eigenvalues This is the equation that led Dirac to invent the Dirac delta function The solutions are ut t 6t t 1519 where t can be any number Therefore the eigenvalues are continuous The eigen functions are complete and orthogonal Discrete Case If the eigenvalues are discrete then the notation a is used to denote them where n is an integer index By convention whenever possible the eigen values are arranged in order of increasing magnitude The corresponding eigen functions are denoted by u t and the eigenvalue problem is written as Aunt a u t 1520 We emphasize that before we have solved the eigenvalue problem we do not know whether the solutions will be discrete or continuous or both As in the continuous case the eigenfunctions are orthogonal and complete J unt unt dt 6n 1521 E unt unt 6t t 1522 n A signal can be expanded as where the coefficients cn are 8t E Cn unt n 1523 c J unt st dt 1524 The proof of Eq 1524 is the same as for the continuous case which we gave in the previous section Sec 3 Operator Algebra 209 Signals in different representations Terminology One can uniquely go back and forth between st and Fa Nothing is lost Therefore we say that st is the signal in the time representation and that Fa is the signal in the a representation Similarly for the discrete case we say that the set of cs is the signal in the u1 representation 153 OPERATOR ALGEBRA We now give an elementary exposition of the basic operator methods we will subse quently use The two fundamental operators are the time and frequency operators and generally other operators will be functions of these The primary idea is that for a physical quantity we will associate an operator Very often we will be dealing with two or more physical quantities and thus with two or more operators If we have two operators A and B then the operator AB means to operate first with B and then with A Generally the order of opera tion is not interchangeable as for example putting on socks and shoes The order matters If the order doesnt matter as for example putting on a hat and shoes then we say the operators commute To determine if two operators commute we operate with AB and BA on an arbitrary function to see whether the same answer is obtained Equivalently we can operate with AB BA to examine whether zero is obtained The operator AB BA is called the commutator of A and B and is denoted by A B A B1 AS BA Commutator of A and B 1525 As we will see the commutator of two quantities plays a fundamental role Some properties of the commutator are AB BA 1526 cAB c AB 1527 ABC AB AC 1528 It is also useful to define the anticommutator A B AB BA Anticommutator of A and B 1529 Note that A B and A B are respectively symmetric and antisymmetric with the interchange of the two operators Forming Hermitian Operators One often forms a new operator from Hermitian operators and it is important to know whether the new operator is Hermitian As suming that A and B are Hermitian the following are readily verified to be Hermi 210 tian operators cA An AB ABl ABlj Chap 15 The Representation of Signals are Hermitian if A and B are Hermitian and c is real 1530 Operator Equations and Fundamental Commutation Relation An operator equa tion means that when we operate with either side on an arbitrary function the same result is obtained For example the fundamental operator commutation rule be tween time and frequency TW WT j 1531 means that operating with the left hand side on an arbitrary function is the same as multiplying that function by j To prove this consider TW WT st t 1 d 1 d t l st 15 32 j dt j dt 1 tat tat 8 1533 ist 1534 which proves Eq 1531 Operator equations are manipulated almost like algebraic equations but careful attention must be paid to the fact that operators may not commute and therefore can not be switched around an expression Functions of an Operator By a function of an operator QA we mean by def inition that in the ordinary Taylor expansion of the function one substitutes the operator for the variable QA E cnAn if Qx 1535 n n When will QA thus defined be Hermitian We already know that An is Hermitian and therefore c An is Hermitian if cn is real But having the cns real means that the function Qx is real hence we conclude that QA is Hermitian if A is Hermitian and Qx is a real function 1536 A problem that often arises is the operation of QA on an arbitrary function Generally we just do it However a useful formula can be derived Consider first the action of QA on an eigenfunction of A Sec 3 Operator Algebra 211 QA ua t E cAn ua t E can ua t n n But the sum is recognized to be Qa and therefore we have 1537 QA ua t Qa ua t Now consider the operation of QA on an arbitrary function st 1538 QAst QA J Fa ua t da J FaQA ua t da and therefore using Eq 1538 1539 r QA st J Fa Qa ua t da The Inverse of an Operator The inverse of A A1 is defined so that 1540 AA AA1 T 1541 where T is the unit operator typically we leave out the unit operator because it is understood to be there where appropriate For example we write T W j instead of T W jZ The inverse of an operator that is the product of two operators is given by AB1 B1A1 1542 This is so because ABAB BAAB B1B Z 1543 The Adjoint of an Operator The adjoint is another operator denoted by At which forces the equality J gAf dt f f Atg dt 1544 If the adjoint of an operator happens to equal the operator itself then we see that Eq 1544 becomes the definition of a Hermitian operator Eq 155 and hence if A At then A is Hermitian self adjoint 1545 As can be seen from the definition of adjoint the adjoint of a constant is the complex conjugate of the constant At c if A c constant 1546 212 Chap 15 The Representation of Signals The adjoint of a product of operators is given by ABt Bt At 1547 Another important property of the adjoint is that A At A Atlj are Hermitian whether A is Hermitian or not 1548 An arbitrary possibly non Hermitian operator can be written in the following way A 2AAtijAAtj 1549 Notice that we have multiplied and divided by j in the second term The impor tance of this decomposition is twofold First it breaks up an operator into sym metric and antisymmetric parts Second it expresses an operator as a sum of a Hermitian operator plus j times a Hermitian operator This is the operator analog of writing a complex number in terms of its real and imaginary part Product of Two Hermitian Operators The product of two Hermitian operators is not necessarily Hermitian However using the above decomposition by letting A AB it can be written in the following useful way AB 2 AB ABt 2 AB ABt 1550 AB BA 1 AB BA 1551 ABI1AB1j 1552 This expresses AB in terms of its commutator and anticommutator Unitary Operators An operator u is said to be unitary if its adjoint is equal to its inverse Ut U1 Unitary 1 1553 The importance of unitary operators is that they preserve normalization when op erating on a function That is f t and U f t have the same normalization if U is unitary J f t 12 dt J U f t 12 dt 1554 To see this consider f I U f t I2 dt fU ftI U ft dt f f t Utu f t dt f I ft12 dt 1555 Sec 4 Averages 213 In general an operator of the form uejA 1556 will be unitary if A is Hermitian To see this first note that if A is Hermitian then Ut ejAt 1557 This can be proven straightforwardly t Ut q t 7n An j Atn ejAt n n n n To show that u is unitary if A is Hermitian consider 1558 UUt e1 aAt e3A ajA 2 1559 Therefore Ut U1 which is the definition of a unitary operator Unitary operators are not Hermitian but they do obey the simple manipulative rule J gUf dt fi Utg1 dt rf Ulg dt 1560 This is the case since the middlestep is the definition of adjoint and the right side follows because of Eq 1553 Example 153 Tlranslation Operator The translation operator defined by U eTl is unitary That follows because the frequency operator is Hermitian Also we can see that from the fact that eN f t f t r and by noting that f t and f t r have the same normalization for any r The inverse of the translation operator is U1 e2 W This follows from Eq 1553 or can be verified directly eTw eTwst est r st r T st 1561 154 AVERAGES If the density of a is taken to be I Fa I2 then the average value of a is a J a I Fa 12 da 1562 and more generally the average of any function ga is g f ga I Fa I2 da 1563 214 Chap 15 The Representation of Signals In Section 14 we saw that we can calculate frequency averages directly from the signal without calculating the Fourier transform This is a special case of the more general result whereby the average of g a can be calculated directly from the signal and the calculation of the transform Fa can be avoided This is a fundamental result of operator theory In particular g J ga I Fa I da J s t gA st dt 1564 To prove this consider J s t gA st dt ff F a u a t gAFa ua t dt dada 1565 Now J sgA ua t ga ua t we have s t gA st dt ff F a u a t gaFa ua t dt dada 1566 ffFalgaFa6a a dada 1567 f ga I Fa I2 da 1568 Averages of Hermitian Operators Are Real We use both A and a to signify the average of a a fstAstdt 1569 It is important to appreciate that the average defined by Eq 1569 will always be real provided the operator is Hemtitian The reason is that if the operator is Her mitian then Eq 1564 holds and the middle term is real if g is a real function Example 154 Frequency The frequency operator is W j d and therefore gw f I Sw I Zgw dw f s t g Q 1 st dt 1571 which is a relation we have repeatedly used in the previous chapters to simplify cal culations Sec 4 Averages 215 Averages of NonHermitian Operators If A is not Hermitian then its average calculated by way of Eq 1569 will be a complex number The real and imaginary parts can be explicitly written down by using Eq 1549 Taking averagevalues of both sides of that equation we have A1AAt zjAAtlj 1572 This is the average value of an operator in terms of its real and imaginary parts If A is Hermitian then the second term the imaginary part is zero A particularly important case is when the operator is the product of two Hermi tian operators Taking average values of Eq 1551 we have AB zABIjABlj 1573 which expresses the average value for the product of two operators in terms of the real and imaginary parts since both AB BA and A B j are real The Covariance In Section 18 we defined the covariance between time and fre quency as the average of t cpt where W is the phase of the signal We now define the covariance of any two quantities byil Covab 1ABBA A B 1574 2AB AB 1575 2A AB B 1576 The rational for this definition will be discussed in Section 1710 Example 155 TimeFrequency Covariance Using the commutation relation TW WT j we have TWWT2TWj2f stWsdtj2 ft4dt 8t 2 dt 2t t 1577 and therefore Covt tW t tw 1578 which is the same definition we used in Eq1124 We mentioned in Section 18 that f tP t 18t 12 dt f w 0w I Sw 12 dw 1579 A simple proof will now be given Instead of evaluating TW WT in the time representation we evaluate it in the spectral representation Using the same steps as above TWWT 2WTj 2f SwTSdwj r 2 J SwI2 dw 2w 0 JJ 1580 Since the two methods of calculation must give the same result we have Eq 1579 216 Chap 15 The Representation of Signals 155 THE UNCERTAINTY PRINCIPLE FOR ARBITRARY VARIABLES If we have two physical quantities then we will have an uncertainty principle be tween them if the operators do not commute Specifically the uncertainty principle for any two quantities represented by the operators A and B is CaCbZ1AB 1581 where Qa is the standard deviation defined in the usual way Qa A2 A 2 1582 r s t A A 2 st dt 1583 J 14 A st 12dt 1584 Similarly for B Furthermore the signal that minimizes the uncertainty product is obtained by solving BBst A A A st 1585 where A 28 1586 a Note that in general A may be complex since A B is not Hermitian As with the uncertainty principle for time and frequency Eqs 33 and 34 a more general result can be proven 0a vb 2 A B 2 4 Covab 1587 Proof of the Uncertainty Principle First some preliminaries For convenience de fine Ao A A Bo B B 1588 and note that the operators A0 and Bo are Hermitian and that their mean is zero Furthermore it is easy to verify that AoBo AB 1589 and AoBo AB2AB2BA2AB 1590 Sec 5 The Uncertainty Principle for Arbitrary Variables Taking expectation values of both sides of Eq 1589 we have correspondingly AoBo AB and 217 1591 Ao Bo 2 COVab 1592 With these preliminaries we can now derive the uncertainty principle for arbi trary quantities by considering r 2 2 dt 1593 dt x J Bo 3t 1 Oa Q6 J Ao st 1 J A0 st 8ost dt 1594 2 I fstAoBostdt 1595 I A0 Bo I2 1596 where in going from step 1593 to step 1594 we have used the Schwarz inequal ity Eq 310 and in going from 1594 to 1595 we have used the Hermiticity property of the operator Therefore OoCb 2 I AOBO I2 Using Eq 1592 and 1591 we have AoBo 2AoBo2jAoBolj and hence 2 2 oa Qb 1597 1598 COVab 2 j A0 BO j 1599 ICOVab27A0B07I2 Cova6q IA0BoI2 Cov b 4 I A B I2 15100 15101 15102 which is Eq 1587 Since COVab is positive it can be dropped to obtain the more standard uncertainty principle Eq 1581 The minimum uncertainty signal is obtained when the two functions are pro portional Aos Bos and Cov 0 That is AAAsBBs 15103 218 Chap 15 The Representation of Signals A AB B 0 15104 The constant A is thus far arbitrary To fix it let us first write the above in terms of Ao and Bo AAos Bos 0 AoBoBoAo 15105 15106 Multiply the first equation by Ao and take average values then multiply by Bo and take average values to obtain aBoAo Bo Since As as and Bo ab we may write this as 15107 Aas AO Bo abA Bo Ao 15108 Adding and subtracting these two equations we have A as ab a 0 15109 as 012A Ao Bo A B 15110 Solving for A we obtain 2aa Note that A is purely imaginary Example 156 Time and Frequency The commutator of the time and frequency operator is j so at aWzlTJIaljI i 15111 15112 Chapter 16 Density of a Single Variable 161 INTRODUCTION In this chapter we develop the basic methods to study the density of a single vari able The object is to generalize the ideas and methods that have been developed for the cases of frequency and time to arbitrary variables 162 DENSITY OF A SINGLE VARIABLE In the Fourier case the density of frequency is taken to be I Sw 12 and we can argue by analogy that for an arbitrary variable it should be I Fa 12 where Fa is the a transform of the signal Eq 152 We are now in a position to derive this by way of the characteristic function method The word derive is in quotes because the general result for calculating averages Eq 1564 has built into it that I Fa 12 is the density The following derivation using the characteristic function operator method illustrates the simplicity and consistency of the method and will be the basis for obtaining joint densities of two variables If Pa is the density of a then the characteristic function is Ma ejaa f aaPa da and the distribution is obtained from the characteristic function by 161 Pa 2 J Ma aaa da 162 As we have pointed out in previous chapters the characteristic function is an aver 219 220 Chap 16 Density of a Single Variable age and therefore it can be calculated directly from the signal by way of Ma J 3t eaAst dt 163 Expanding the signal in terms of the transform we have e3aA J Fa ua t da 164 r Fa a A ua t da 165 J Fa ua t da 166 e3aA3t and therefore ffJ Ma F a u a t a Fa ua t da da dt 167 J F a eaa 6a aFa da da 168 f FaI2e3aada 169 Comparing Eq 169 with Eq 161 we have Pa I Fa 12 1610 Discrete Case For the characteristic function we have Ma J 3 t eASt dt 1611 JEECkUkt eyaACnuntdt 1612 n k e2aA f Ck ukt aa cn unt dt 1613 n k ck5knCn elaan 1614 n k E IcnI2eaa n 1615 Sec 2 Density of a Single Variable 221 which gives Pa f Ma aja da 1 1616 1 I C l2 jcka ajaa da 27r n 1617 E I cn I2 6an a 1618 n The only values which are not zero are the ans and their density is Pan I Cn I2 Example 161 Distribution of Frequency 1619 For frequency Mr eTW Jste3TW3t 1620 The operator e is the translation operator eTWst st r 1621 and therefore Mr J s t st r dt 1622 The distribution is given by Pw 2a I Mr a d7 1623 2 ff s t at r aT dr dt 1624 I Sw 12 1625 as expected Example 162 Distribution of Inverse Frequency Method 1 We define inverse frequency by 1626 where wo is an arbitrarily chosen reference frequency As we are making an ordinary transformation we can use the method developed in Section 47 Since Pw ISwI2 1627 222 Chap 16 Density of a Single Variable the distribution of r is according to Eq 486 Pr Pw I d wo ISwor I2 1628 Wo r2 Method 2 Alternatively we can define the inverse frequency operator R W 1629 and use the operator methods developed Solving the eigenvalue problem in the fre quency representation WO ur w rur w gives 1630 ur w 6w wor 6r wow 1631 Calling Gr the inverse frequency transform we have Gr J Sw u r w dw J Sw 6w wor dw Swor 1632 The distribution is Pr IGrI2 I Swor I2 1633 which is the same as Eq 17124 163 MEAN VALUES As we showed in the previous chapter the average of a can be calculated directly from the signal by way of a I s t A 9t dt 1634 We now obtain an interesting form analogous to the one we obtained for frequency Eq 190 Rewrite Eq 1634 as a J 1 I st 12 dt 1635 and break up Ass into its real and imaginary parts 1636 AsSIR Asl j1A to write r r 1 l 1 a J L j l J I st 12 dt 1637 e R 8 I Sec 4 Bandwidth 223 Since the mean value of a Hermitian operator has to be real we must have which leaves J stI2dtO 8 st l2 dt a fR Example 163 Spectral Average Taking A to be the frequency operator A W d we have W8 11 d 1 A t 8 a j dt e j At P t The real part is gyp t and using Eq 1639 we immediately have w 1638 1639 1640 1641 As with the average the standard deviation can also be written directly in terms of the signal Q2 a Sq fa a I Fa I2 da stA A 2st dt 1 I As Ast 2 dt As A 1642 1643 1644 2 dt 1645 The first two terms are real since A is real hence QQ f 1at 12 dt f k R A J 2 18t 12 dt 1646 Example 164 Bandwidth Equation From the previous example we have Ws At Ws P t 1647 8 At 8 I R 224 Chap 16 Density of a Single Variable 165 and therefore Ow2 J A2 A2tdt J U2 A2t dt 1648 which is the bandwidth equation for frequency ARBITRARY STARTING REPRESENTATION A JFbAFbdb 1650 a r F b A A 2 Fb db 1651 where now A has to be expressed in the variables of the b representation The identical derivation of Section 163 leads to a fFbI2db 1652 Fb J u b t st dt 1649 The average and bandwidth of a can be calculated using Fb as the signal instead of st In the above considerations we have been transforming between the time repre sentation and the a representation However there is no particular reason why we must start with the time representation We can work between any two represen tations say a and b Suppose we have a quantity b and its associated operator B and we solve for the eigenfunctions which we call ub t The signal in the b representation is then and Q2 f AF2 1 Fb I2db f f AJ Fb 12 db 1653 Example 165 Mean Time and Duration in Terms of the Spectrum We write the spectrum in terms of its amplitude and phase Sw Bwe to obtain Therefore TS 1 1 d 1 Bw S 3 S7Bw w 1654 TS Bw TS T S I Bw R and the mean time and duration expressed in terms of the spectrum are 1655 t fp1sI2 dw 1656 at f B2 1Bw12dw J 1Sw12dw Bw 1657 Chapter 17 Joint Representations for Arbitrary Variables 171 INTRODUCTION We now generalize the methods we developed for timefrequency representations to obtain joint representations for arbitrary quantities In generalizing we will use the timefrequency case for examples so that we may check by recovering the known results In the next chapter we specialize to the study of scale 172 MARGINALS For two quantities a and b represented by the operators A and B we seek joint distributions Pa b In the previous chapter we saw that the densities of a and b are given by I Fa 12 and I Fb 12 where Fa and Fb are the a and b transforms respectively The joint density should therefore satisfy the marginal conditions J Pa b db I Fa 12 171 J Pa b da I Fb 2 172 173 CHARACTERISTIC FUNCTION OPERATOR METHOD As for the timefrequency case we approach the problem of finding joint densities by the characteristic function method which was generalized for arbitrary operators by Scully and Cohen1505 The characteristic function is Ma Q eiaaAb ff e3craj13Pa bda db 173 225 226 Chap 17 Joint Representations for Arbitrary Variables and the distribution is obtained from Ma 3 by Fourier inversion Pa b 41r2ff Ma Q ejaajRb da dQ 174 Since the characteristic function is the average of jajOb we expect to calculate it by finding the average of a yet unspecified operator We shall call that operator the characteristic function operator and denote it by M a Accordingly we write Ma Q M a Q fstMa3 st dt 175 There are many possibilities for M Among them are Ma13 4 ejaAjQB e7aAejOB e3RB e3aA 176 177 178 or more involved combinations like Ma a ejaA2 B ejaA2 179 Since there are an infinite number of possible orderings there are an infinite num ber of distributions It is this ordering ambiguity that gives rise to the general class of timefrequency distributions and we will see that it also gives rise to a general class for arbitrary variables The method for obtaining a distribution is as follows Choose an ordering that is a characteristic function operator M a p calculate the characteristic function by way of Eq 175 and then obtain the distribution using ffMa 3 ejaaj da di Pa b 47r2 174 METHODS OF EVALUATION 1710 We now consider the evaluation of the characteristic function and particularly of the expectation value of ejaA3QB Although this is only one form it is an important one because very often other characteristic function operators may be expressed in terms of it The evaluation of ejcAjOB is a problem that appears in many fields and has a long historyis891 We discuss two methods for its simplification Method 1 Solve the eigenvalue probleml505 aAIi uAt 1 uAt 1711 Sec 4 Methods of Evaluation 227 Since a and6 are real numbers and A and B are Hermitian the quantity aA 08 is also Hermitian Therefore the solution to the eigenvalue problem gives rise to a complete set of eigenfunctions the us Any signal can hence be expressed in terms of them by st uA tF1 dA with the inverse transformation given by 1712 FA f u A t st dt Now consider 1713 aAiABSt ejaAiPB f uA tFA dA J eu tFA dA 1714 Substituting for FA we have eyaAjoB st f Gt t st dt where 1715 Gt t r e u A t uA t dA 1716 This method works quite well in many cases but has the disadvantage that the eigenvalue problem must be solved Example 171 Time and Frequency The eigenvalue problem OT rW uA t A uA t 1717 in the time representation is etjrdtuAt AuAt 1718 and the solutions normalized to a delta function are u1 t 1 ejatet22r 1719 27rr Calculating G of Eq 1716 Gt t 5r t t ejer2 ejet 1720 and substituting in Eq 1715 we have eeTirwst J Gt t st dt yOr2 e7etet r 1721 228 Chap 17 Joint Representations for Arbitrary Variables And so the characteristic function is MBT st ej6r2ejOt8tTdt 1722 J st 2Tejetst 2rdt 1723 which is the characteristic function of the Wigner distribution Method 2 The second method seeks to directly simplify the operator e3cAjsB This however is one of the most difficult problems in mathematical analysis and simplification has thus far been achieved only for special cases One such case is when both operators commute with their commutator IABA ABB 0 1724 in which case589 This relation holds for time and frequency since their commutator is a number and hence commutes both with the time and frequency operator We have already used this method in Chapter 10 Another special case is when AB c1 c2A 1727 e7cA705 ea0AS2 e3AB e7aA ea81AB12ejaAej3B 1725 1726 in which case 11131 where ejaAj9B e7µac1c9 e7aµAejB e7aA 1 jlic2µ jC2 11j9c2e 1729 1728 This case will arise when we study the scale operator These relations are proved in the appendix Note that for c2 0 Eq 1724 holds and we have A 0 jc2 j3 for c2 0 1730 and hence Eq 1728 becomes Eq 1725 Sec 5 General Class for Arbitrary Variables 229 175 GENERAL CLASS FOR ARBITRARY VARIABLES As with timefrequency case instead of dealing with all possible orderings we can use the kernel method to obtain all possible representations Choose a specific or dering to produce a specific characteristic function Ma 3 Using it form a new characteristic function by Mnewa Q 0a 0 Ma 0 1731 where 0 is the kernel The new distribution is Pnewa b 1 ff Mnewa 0 ejaau3b da d3 1732 fJ Oa 3 Ma 3 aajob da d0 1733 47r2 and it will satisfy the marginals of the original distribution if the kernel satisfies 00 Q 0a 0 1 1734 Equation 1733 is the general class for arbitrary variables It can be written in the triple integral form by using Eq 175 for M Pa b 47r2 fJMe2dad3 1735 s t Ma Q st ejajAb da d0 dt 1736 176 TRANSFORMATION BETWEEN DISTRIBUTIONS Suppose we have two distributions Pl and P2 with corresponding kernels 01 02 and corresponding characteristic functions M1 and M2 The respective character istic functions are Ml a 3 01a R Ma 3 1737 M2 a 0 02a 3 Ma 0 1738 which gives Ml a 0 01a 0 M2 a 3 1739 02 a 0 Now P b tea jQb d M 3 d 17 40 l a i a a a Q 47r2 ff 1a 0 M ja jpb d 3 d 1 17 41 2 a a a Q 2ff 47r 02a 0 230 Chap 17 Joint Representations for Arbitrary Variables and since we have M2a0 if ejaajOl P2 a b da db 1742 Pl a b 1 ffff 01 a Q ejaaa 30bb P2a b do d0 da db 1743 4 m2 a This can be written as with PIab JJga abbP2abdadb 1744 ga b 12 J ejaaj Ob 01 a A do df3 47r 02a3 177 LOCAL AUTOCORRELATION 1745 In studying timefrequency distributions we saw that a fruitful approach was to generalize the relationship between the power spectrum and the autocorrelation function by defining a local autocorrelation function A similar development holds for arbitrary variables Consider the variable b We write its density as Pb 1 f Rf3 ajOb d3 1746 where Rb is the characteristic function of b which we write as R instead of M as is the convention We generalize by taking the joint distribution of a 6 to be the local autocorrelation of b around some value a and write Pa b 2 f R0J ajb dO 1747 Comparing Eq 1747 with Eq 1733 we find Ra3 1 f Oa R Ma Q aa da 1748 In terms of the characteristic function operator this may be written as RaQ 2 If Oa 0 em Ma 0 9t ajaa da dt 1749 Sec 8 Instantaneous Values 231 178 INSTANTANEOUS VALUES We rewrite Eq 1646 here which expresses the bandwidth directly in terms of the signal as f As2 18t I2 dt f A 2 18t 12 dt 1750 8 I 8 R Comparing it to the general result that connects the global and local standard de viation Eq 471 as aalb Pb db fab a 2 Pb db 1751 we conclude that the instantaneous value at and its conditional standard devia tion aalt are given by l ata t As s J R 1752 2 a 2It As v In an eigenstate we expect the value of the physical quantity to be the fixed for all time For an eigenstate A ua t a ua t and therefore A uat a 17 53 uat Because the operator is Hermitian the eigenvalue a is real and Eq 1752 gives at a aaIt 0 1754 We see that in an eigenstate the instantaneous value is equal to the global average and the local standard deviation is zero for all time Relation Between Global and Instantaneous Quantities We expect the average value to be the average of the local value 1755 A f A t I st I2 dt fR This is seen to be the case since f stAst dt f As I St I2 dt 1756 f f As j I I st 12 dt 1757 L 8 R 3 IJ f As R I st I2 dt 1758 232 Chap 17 Joint Representations for Arbitrary Variables where the second term in Eq 1757 is zero because we know that the average value of a Hermitian operator is real 179 LOCAL VALUES FOR ARBITRARY VARIABLE PAIRS We now consider the more general question of the conditional value of a quantity for a fixed value of another arbitrary quantity We shall use the notation ab or ab for the conditional value of a for a given value of b In Section 164 we showed that the bandwidth of a can be written in the following way Fb Iz db 1759 Ora J c Fb 1z db f c A12 where Fb is the signal in the b representation Comparing with Eq 1751 gives ab a b AbFb Fb R 1760 z AbFb12 1761 aalb Fb r Again we expect that the global average be the average of the conditional value a A b i Fb 12 db 1R I Fb I2 db 1762 The proof is identical to that given in the previous section where b was time Example 172 Group Delay For a we take time and for b we take frequency Hence Fb is the spectrum Sw Writing the spectrum as Sw Bw etiw 1763 we have TS 11 d 1Bw S 17 64 w S d j Bw S Therefore tW Vw 1765 which is the group delay Its standard de 2 viation is according to Eq 1761 17 66 at1w Sec 10 The Covariance 233 1710 THE COVARIANCE In Chapter 1 we showed how the covariance can be defined for the case of time and frequency In Section 155 we gave an expression for the covariance for arbitrary variables but did not justify that definition We do so now Let us place ourselves in the b representation The joint moment is then bab since ab is the instantaneous value of a for a given b The covariance is COVab bab A B 1767 b CAbFb AB Fb rt 1768 We could have equally well put ourselves in the a representation in which case Covab aba A B 1769 a BaFa A B 1770 Fa R These two definitions are identical because b AbFb a BaFa Fb R Fa We prove this by showing first that the definition given in Section 155 for the co variance is the same as given above namely COVab 2ABBA AB 1772 Consider C a Fa 2 a Fa 2 a f F 1773 a 2JFABFJaFBtF1 1774 2ABBA 1775 If we start with the expression given by the left hand side of Eq 1771the identical derivation leads to the same result Hence we conclude that the two expressions for the covariance Eqs 1768 and 1770 are identical 234 Chap 17 Joint Representations for Arbitrary Variables 1711 GENERALIZATION OF THE SHORTTIME FOURIER TRANSFORM We now generalize the shorttime Fourier transform to arbitrary variables First con sider the case where the two variables are time and b and subsequently generalize to two arbitrary variables The ShortTune b Transform We focus in on a particular time by the usual proce dure of defining a modified signal according to stT sT hT t 1776 where t is the fixed time of interest r is the running time and ht is the windowing function Instead of taking the Fourier transform of this modified signal we take the b transform Ft b JsrhTtubTdT 1777 In analogy with the shorttime Fourier transform this shall be called the shorttime b transform The joint time b density is then Pt b I Ft b I2 fsrhr tubTdT 2 1778 The Shorta b Transform We first recall that the signals in the a and b representa tions are given by Fa J u a t at dt 1779 Fb J u b t st dt 1780 These two equations are the transformation from the time representation to the a and b representations However we can transform directly from the a to the b representation Fb JcxbaFada 1781 where ab a is called the transformation matrix It is given by aba J ubtuatdt 1782 which can be verified by substituting it into Eq 1781 Now we window the signal in the a representation to obtain a modified signal Fa Fa ha a 1783 Sec 12 Unitary Transformation 235 where a is the running variable and a is the fixed value of interest Taking the b transform of the modified signal we have Fab f ab a ha aFa da 1784 and it is reasonable to call this the shorta b transform The joint density is therefore r 2 Pa b I Fab I2 J ab a ha aFa da 1785 which is a joint distribution for the variables a and b analogous to the spectrogram 1712 UNITARY TRANSFORMATION Another method for obtaining joint distributions for variables other than time and frequency is to start with the general class of timefrequency distributions PtW 412 N dudadQ 1786 and for the signal st use a transformed signal The transformed signal is obtained by a unitary transformation so chosen to be in the representation of one of the variables a we are interested see section 153 The other variable b is as yet unspecified In the general timefrequency representation Eq 91 replace st by ust and replace the variables t w by a b Pa b 412 ffJ us u ejoTjAwusu 09 7 ejaajOl du da dQ 1787 This approach was developed by Baraniuk and Jones It can be directly verified that the marginals for this distribution are J Pa b db I Usa 12 1788 J Pa b da I Suab 12 1789 where Suab is the Fourier transform of the transformed signal Usa Let us sup pose our interest was to have a distribution where one of the marginals is I Us a I2 We are not free to choose the other it is forced and not at our disposal The ad vantage of the BaraniukJones procedure is that we do not have to calculate new distributions but can use the already calculated timefrequency distribution It can not be applied to two arbitrary variables because both variables are not under our control We now ask what are the two variables or operators that generate Eq 1787 for the joint distribution52 236 Chap 17 Joint Representations for Arbitrary Variables Similarity and Unitary Transformation To see the relation between the form given by Eq 1787 and the general procedure developed in the previous sections we first have to define the concept of a similarity and unitary transformation of operators If we have an operator and form a new operator A according to A U1 AU similarity transformation 1 1790 then that is called a similarity transformation If in addition the operator is unitary then U1 Ut in which case A Ut AU unitary transformation 1 1791 and the transformation is called a unitary transformation The reason for defining a unitary operator transformation in this way is that by doing so we obtain the im portant result that averages retain their form in both representations In particular if we call s the transformed signal s Uts U1s 1792 then a fst4stdt 1793 JU1st Ut AUU1st dt 1794 fstAstdt 1795 a 17 An important property of a unitary transformation is that the eigenvalues of operators remain the same Also Hermiticity is retained that is A is Hermitian if A is Another important property of a unitary transformation is that it leaves algebraic equations functionally the same In particular it leaves the commutator invariant If AB C 1797 then A B C 1798 To see this multiply Eq 1797 on the left by Ut and on the right by U UtABU UtCU C 1799 But Ut A B U Ut AB BAU 17100 UtAUU1B BUUAU 17101 UtAUU1BU UtBUUAU 17102 ABl 17103 Sec 12 Unitary Transformation 237 and hence Eq 1798 follows We are now in a position to see which two operators generate the joint distri bution Eq 1787 They are operators that are related to the time and frequency operator by a unitary transformationP4 A LitTU B Ut WU 17104 To prove this we calculate the characteristic function and distribution for A and B as per the general procedure given by Eqs 173 and 174 Ma 3 Oa Q J s t e3AAB st dt 17105 0a Q J s t eia62 eA eJ8 st dt 17106 0a Q f st eja32 ll1 eiaTU U1 e3WU st dt 17107 Oa p J Us t ea73 Ust dt 17108 where in going from step 17107 to 17108 we have used the fact that e3aA Ut eaT U e70g Ut e331V U 17109 which can be readily proven The distribution is the double Fourier transform of M which gives Eq 1787 Therefore for the case where the two new variables are connected to time and frequency by a similarity transformation the general class of distributions for a and b is obtained from the general class of timefrequency distributions by substituting the transformed signal for st This simplification can be used whenever applica ble Generalizationi52 This result can be further generalized Suppose we have two arbitrary variables and use the procedure we have developed to obtain the general class that is Eq 1710 which we repeat here Pa b 47r2 JJJ s t m a 3 stiaa0b dt da d3 17110 Now suppose that we have two new variables which are related to A B by way of A Ut Au B Ut BU 17111 Then we do not have to recalculate their general class It can be obtained from Eq 17110 by substituting the transformed signal for st 238 Chap 17 Joint Representations for Arbitrary Variables Pa b 4 2 JJJ l t s t Ma 011 st eoarpb dt da d3 17112 The proof is the same as in the timefrequency case This shows that variables con nected by a unitary transformation have distributions that are functionally identi cal 1713 INVERSE FREQUENCY It is sometimes of interest to obtain a distribution of time and inverse frequency Method 1 The straightforward way to obtain such a distribution is to use the gen eral class of timefrequency distributions and the method of Section 47 to transform distributions The distribution in time and inverse frequency Pt r is given by Pt r dt dr Pt w dt dw 17113 where Pt w is a timefrequency distribution Using r wow dr wow2 dw r dw 17114 WO we obtain Pt r r2 Pt wor 17115 where the P on the right hand side is the timefrequency distribution Any time frequency distribution can be used for Pt w If we use the Wigner distribution Pt r we Wt wor 17116 r 1 WO st 2re21 Ost zTdTr 17117 27r r2 J 1 r2 f Swor 82 ajte Swor 82 d9 17118 To obtain the general class of timeinverse frequency distributions we can use the general class of timefrequency distributions Eq 91 together with Eq 17115 Doing so yields Pt r 41 r2 fJJi9t207i914Orsu 2r su 2r dudr dO 17119 Sec 13 Inverse Frequency 239 In terms of the spectrum it is JfJ eietiTWO 9ra 00r S u z 9Su 2 9 du d7 dB Pt r 47r2 T2 17120 The marginals of these distributions are thermarginals of time and inverse frequency 1 Pt r dr I St 12 J Pt r dt r2 I Swor 12 17121 Method 2 Alternatively although in this case much harder we can use operator methods Here we do not need to since we are mating a transformation of ordinary functions nevertheless it is of some interest to do so Take M8 c J S W ej9TjRR Sw dw 17122 for the characteristic function We first address the simplification of eioTRRSw Using the method given by Eqs 17111716 we solve the eigenvalue problem OT uAw AuAw 17123 in the frequency representation fj6 d 4 J uAw U W 17124 The solution is uA w 1 e71J1WWORIn IWI1B 17125 21x9 which gives for JSuAwuAw1d he G of Eq 1716 WRIRWW059w w 17126 Gww Cj and therefore BT7RTtSw rS1 59 w w Sw dw 17127 Sw 0 exp I j SON In W w e 1 17128 Using this result we calculate r 1 MB rc JSSwSW9 exp I j wOrv In w w B J dw 17129 JSW0Sw9expjndw 1 1 W 2B 17130 240 Chap 17 Joint Representations for Arbitrary Variables The distribution is therefore i Pt r fffSw9 Sw 10 exp I j wOK In w i e I eyetj dO dw z which after considerable algebra leads to 17131 Pt r 1 wou 2 ejwoutrS wpU eu2 S wou au2 du 8xr3 f sink u2 2r sinhu2 2r sinh u2 17132 This distribution satisfies the marginals of time and inverse frequency Eqs 17121 1714 APPENDIX The simplification of eAB is generally difficult but for the case where A B QaA 17133 simplification is possible This encompasses the timefrequency and timescale case If one tries to expand eAB and attempts to use the commutation relations to rearrange each term a very significant mess ensues One of the well known tricks to simplify this type of expression is to introduce a parameter A and write an equation similar to the one we are trying to simplify In particular we write f eaB eaA eaAB 17134 The idea is to obtain a differential equation for f A and solve it The solution hopefully will be a simpler form than what we started out with Before proceeding we establish some operator relations Taking the commutator of A with A B we have AABAaA0 17135 Also B A B a B A a a 2 A 17136 There is a well known relationIM91 that is very useful for simplification of operator identities 2 V eASe AB1AB 2iAAB 3iAAAB 17137 1 For the sake of neatness we use here the notation A B 0 aA rather than A B cl c2A used in the text The a 0 used in this appendix should not be confused with their usage in the text Sec 14 Appendix 241 Using Eqs 17135 and 17136 we have eAABeA B AJ3 A aA 17138 eBAeB eaA Qe 1 17139 a Differentiating Eq 17134 with respect to A and using the relations just given we establish that dfd A3 aA f 17140 Solving we have f a expe 1a3 aAa k 17141 where k is a constant of integration which is found by imposing the requirement that f 0 1 This gives k aA 3 17 142 a2 and f A eµa9a aµaA 1 1 a aa a 17 143 s Hence AB AA aAB iAA aAd e e a e e µ 17144 or eaAB eµapa eµa4 eaB e1A Taking1 1 gives 17145 eAB eµOa e eB eA 1 a a 17146 Chapter 18 Scale 181 INTRODUCTION In this chapter we develop using the general methods of the previous chapters the basic properties of scale Scale is considered to be a physical attribute just like frequency The frequency transform the Fourier transform allows us to ascertain the frequency content of a signal Similarly we will develop the scale transform which allows the determination of the scale content The first objective is to ob tain the operator that represents scale The scale transform is obtained by solving the eigenvalue problem for the operator Having the scale operator and transform allows us to obtain average scale scale bandwidth instantaneous scale and other properties in complete analogy to frequency Scaling can be applied to any variable For concreteness we develop scale for time functions and subsequently consider the scaling of other variables such as frequency In the next chapter we develop joint representations of scale and time and frequency 182 THE SCALE AND COMPRESSION OPERATOR For the scale operator C we take the Hermitian operator C 2TWWT 181 That this operator gives results that conform to our intuitive sense of scaling will be seen as we develop its properties In the time representation C is given by 182 Sec 2 The Scale and Compression Operator 243 Using the commutation relation for time and frequency TW WT j it can be written in the following alternative forms CTW2jWT2j 183 Compression Operator In Chapter 1 we showed that from the frequency operator W we obtain the translation operator by forming 0rw ejTW st st 7r 184 We expect a similar result for the scale operator in that it will define a compression operator In fact st eal2 sea2t 185 That is eiac compresses functions with the factor ea2 If we take In a for a we have ejInaC 8t sat The significance of the factors ea2 or f is that they preserve normalization They entered in an automatic way because the operator ejc is unitary Equation 186 is basic and we give two proofs The first proof relies strictly on the algebraic properties of the operator the second depends on the eigenfunctions of scale and is presented after we obtain them To show Eq 186 consider first the action of C on tn 186 Ctn TW 2 j t j tdt 2j to jtntn1 2 jt 187 and hence Ctn j n 2tn 188 By repeated action we have Now Cktn jkn 2ktn 189 ejaCtn Grktn 00 q jkn 2ktn eQn12tn 1810 kO k O k l k To obtain the action of ejac on an arbitrary function st we expand the function in a power series 00 st eac E ante ea2 E an eantn ea2seat 1811 n0 n0 244 Chap 18 Scale which is the desired result Basic Commutation Relations Unlike the commutator for time and frequency the commutator of scale with time or frequency does not produce a number The com mutation relations are TCjT WCjW 1812 which can be proven directly by using the definition of C and T W j These relations are fundamental and determine the uncertainty principle for scale as we discuss in Section 189 Furthermore TTC 0 WWCI 0 1813 CTCIIT CWCIIW 1814 All these relations are important for the simplification of expressions involving the scale operator 183 THE SCALE EIGENFUNCTIONS To obtain the scale transform we solve the eigenvalue problem for the scale op erator We shall use c and yc t to indicate the eigenvalues and eigenfunctions respectively The eigenvalue problem is C yc t c yc t 1815 Explicitly using Eq 183 jtdyct 1 dt yc t c yc t 1816 The solutions normalized to a delta function are 1 ejctnt yc t 2r f t 0 1817 It is possible to obtain this solution only if time is positive and thus for scaling we must have an origin in time This is reasonable since to scale is to enlarge or reduce That means we must multiply time t by a number from zero to one for enlargement and a number bigger than one for reduction At no time do we multiply by a nega tive number Also the appearance of the logarithm is reasonable because it has the effect of putting on an equal footing the range of zero to one enlargement with the range of one to infinity reduction Sec 4 The Scale Transform 245 Completeness and Orthogonality Since the scale operator is Hermitian we expect that the eigenfunctions are complete and orthogonal J000 y c t yc t dt 6c c 1818 f y c t yc t do 6t t t t 0 1819 These relations are proven in the appendix Properties of the Scale Eigenfunctions In Table 181 we list the main algebraic properties of the eigenfunctions and contrast them to the frequency eigenfunctions e2Wt 1 uw t 2 1820 Notice that with respect to the time variable multiplicationdivision in the scale case corresponds to additionsubtraction in the frequency case Table 181 Properties of the scale kernel and frequency kernel Scale kernel yc t 7c tt 7c t 7c t 7c tt 7c t y c t yc c t 7c tYc t yc t j In t yc t et Yc t jc t12 7c t ejacyc t eo2yc tea Frequency kernel uw t uw t t uw t uw t uw t t uw t u w t uw w t uw t uw t uw t jt uw t st uw t jw uw t e20 uw t uw o t 184 THE SCALE TRANSFORM Any function can be expanded in terms of the scale eigenfunctions st fDcctdc 1 ejctnt 2n J Dc dc vt 1821 1822 246 and the inverse transformation is Chap 18 Scale r Dc J sty c t dt 1823 0 1 roo ajclnt st dt 1824 27r J VIt We call Dc the scale transform It can also be written as Dc 1 f st tjc12 dt f 1825 2Tr which shows that it is the Mellin transform of the signal with the complex argument jc 2 We know from the last chapter that the density of a quantity is the absolute square of its transform and therefore the density of scale is Pc I Dc 12 Intensity or density of scale 1826 Invariance of the Scale Transform to Compression The fundamental property of the Fourier transform Sw is that the energy density function of a translated func tion is identical to the energy density of the original function That is the Fourier transform of the signal st and the translated signal at st to differ only by a phase factor st to Stw etc Sw if st b Sc1 1827 Hence they have identical energy density spectra I StrW12 I Sw 12 1828 A similar idea holds for scale in that the scale energy density of a scaled function is equal to the scale energy density of the original function If we compare the scale transform of st and sa f sat they will differ only by a constant phase factor sat b Dcc Dc if st b Dc 1829 Therefore Dacc I2 Dc 1 2 1830 This a basic result which shows that the scaling of any time function leaves the energy density scale spectrum unchanged Alternative Proof of the Compression Operator Using the fact that the scale eigen functions satisfy e35c t erl2ryc at we have eic st eJc J Dcyc t dc J Dc e3 yc t dc 1831 J Dc e2 ryc et dc el2 8et 1832 Sec 4 The Scale Transform 247 which is identical to Eq 185 The Scale Transform of the Sum of Scaled Functions Let us first consider a signal composed of the sum of translated functions st Ck f t tk k1 1833 The Fourier transform of st is the sum of the Fourier transform of each term But the Fourier transform of f t tk is etMFw where F is the Fourier transform of f t Therefore we have n n E Fw ckeytk Sw k1 k1 We now obtain the analog for the sum of scaled signals 1834 n st E Ck ak f akt 1835 k1 The scale transform of each term is ck eic In all Ec where Ec is the scale transform of f t Therefore n n Dc L Ck a clnakEc Ec Ck ejclnak IC1 l 1 k1 If we consider the situation where the amplitudes are arbitrary n 3t Akf akt k1 then DC EC k C7clnak k1 1836 1837 1838 Relation with the Fourier Transform An interesting viewpoint of the scale trans form is obtained by considering a new function constructed by replacing time with the logarithm of time ft t slnt 1839 The factor 1f is inserted to preserve normalization Now consider the scaletrans form of ft ftt ajnt dt 1840 DI c 1 Jo f 248 1 roo ajclnt 27r J slnt t dt 1 27rf oo sr ajcr dT which is the Fourier transform of st Dtc Sc 1843 Inversely we can consider the scale transform to be the Fourier transform of the function set et2 If we define f t set et2 then the Fourier transform of f t is Fc 1 fm set et2 ai dt 27r 1 0o ajclnt st v dt 27r o Dc 185 SIGNALS WITH HIGH SCALE CONTENT 1844 1845 1846 1847 What kind of signals have high scale content that is a high concentration of scale around a particular value Signals with the highest scale content are the scale eigen functions because their transform is a delta function of scale As far as other func tions are concerned a way to develop our intuition is to use the crutch of Fourier transforms developed in the above section For a function st consider the func tion et2set and ask if that new function has a high frequency content If it does then the original function will have a high scale content because its scale transform will be functionally the same as the Fourier transform of et2set Example 181 The Highest and the Lowest Scale Content Signals The eigenfunctions have the highest scale content because c In t 1 d Dc 6c c if st 2a f 1848 What functions have the lowest or flattest scale content We approach this by asking what functions have the lowest frequency content They are functions that have a flat Sec 6 Scale Characteristic Function 249 spectrum Signals that give rise to flat spectra are impulses 6t to Therefore using our argument above the functions that have the lowest scale content are 1 ejelntp Dc 1849 2 a to for st 6Int Into 6lntto 6t to 1850 Xto VrZ where for the last step in Eq 1850 we have used Eq 1893 We see therefore that impulses have the lowest frequency and scale content 186 SCALE CHARACTERISTIC FUNCTION The characteristic function for the density of scale is MQ r e7 I Dc I2 dc 1851 or J 00 fo st ec 8t dt 1852 s t a2 s at dt 1853 MQ o se2t se2t dt 1854 0 As usual the distribution is given by MQ aJ do 1855 Pc 1 27r a J J s e2t se2t ajac dQ dt 1856 Malting the transformation x e2 t y e2 t 1857 Q Inyx t xy t do dt dx dy do dt dx yy 1858 we have 1 O ejclnyInx Pc 2I Jo f 8 x sy xy dxdy 1859 1 Dc 12 1860 250 Chap 18 Scale Total Energy Conservation The total energy E of a signal is obtained by inte grating over all time the energy density I st 12 It should also be obtainable by integrating the scale density over all scale That is indeed the case since 00 J I st 12 dt J I Dc 12 do 1861 0 This shows that the total energy is preserved in the transformation This is the analog to Parcevals theorem for time and frequency It can be proven by direct substitution although it follows from general considerations since the compression operator is unitary 187 MEAN SCALE AND BANDWIDTH There are three equivalent ways we can calculate the mean value for scale The first is by direct use of the scale density the second is to use the general formula for calculating the average value of any operator and the third is to use Eq 1758 c J 1862 s t C st dt 1863 j f7 CS R I st 12 dt 1864 All three formulas will lead to the same result but the last is the easiest to use Calculating C9s 1 t A 2 J 1865 we see that the real part is t cpt and therefore c J ao t cp t A2 t dt 1866 0 The Scale Bandwidth Similarly the scale bandwidth can also be calculated by using any one of the three alternative formulas UC fcc2IDc2dc 1867 J s t Cc2stdt J Ccst12dt 1868 0 Cs12 st 12 dt r 1 Cs C 2 1 st 12 dt 1869 S I J 8 R Sec 8 Instantaneous Scale 251 where the last equation is obtained from Eq 1646 Again the last equation is easiest since we have just calculated the real and imaginary parts of CsE Eq 1865 We immediately have 188 INSTANTANEOUS SCALE In Chapter 16 we showed that the instantaneous value of a quantity is given by the real part of Ass where A is an arbitrary operator Therefore instantaneous scale is tcpttwb C9 R 1871 That is instantaneous scale is time multiplied by instantaneous frequency The gen eral relation of a conditional mean to the global mean is that the global mean is the average of the conditional mean For scale we expect 00 c f ct I st I2 dt 1872 0 and that is the case as can be seen by Eq 1864 189 UNCERTAINTY PRINCIPLE FOR SCALE In Section 155 we showed that there is always an uncertainty principle for two operators if they do not commute Specializing to scale and time we have T tl 2ITCI 1873 From Eq 1812 we know that T C jT and therefore we have the following uncertainty principle for time and scale 1111 or 01 2 ZItI 1874 where t is the mean time Minimum Uncertainty Product Signal Using Eq 15103 the minimum uncer tainty product signal is td 2 c st Attst 1875 252 where TC t 2vt tat and the solution isE1511 8t k t0 where k is a normalizing constant and t 1 t2 Chap 18 Scale 1876 1877 a1 t 02 2 at 1 1878 1810 FREQUENCY AND OTHER SCALING In the preceding sections we have considered the scaling of time We can equally consider the scaling of frequency or the scaling of any other variable Let us first consider frequency scaling From a structural point of view it is clear that to obtain the frequency scaling operator CW we just replace t by w in Eq 182 w4 1879 Everything we have derived for time scaling can beimmediately transliterated for frequency by substituting w for t appropriately In particular the frequency scale eigenfunctions are 1 ejclil W 7W c w 27r V 1880 and the transformations between the frequency domain and the scale domain are Sw 1 JDc G9CI1W 27r do w 0 1881 D c Zr j oy Sw e v W dw 1882 We have used DW c to emphasize that we are defining a different transform the frequency scale transform For the frequency scaling representation only the posi tive half of the spectrum is considered which is equivalent to considering analytic signals For frequency functions we have e2c Sw eo2Sew ej Inoc Sw SQw 1883 For mean frequencyscale and bandwidth we have 00 c f w w B2 w duo 1884 0 Sec 11 Appendix 253 oc f tWW 22B2wdw fw wc2B2wdw 1885 where Bw and tiw are the spectral amplitude and phase Similarly instanta neous scale in the frequency domain is given by c wV w 1886 Other Domains For an arbitrary domain say the a domain the scale operator is Ca 2j aa 1887 and in general we have e3c Fa e12Fea e cc Fa V o Faa 1888 To convert the equations we have derived for time scaling to scaling in the a domain all we have to do is substitute a for t and sa for st in any of the formulas The above application for frequency was an example of this 1811 APPENDIX We prove the completeness relations Eq 1818 and 1819 Consider the left hand side of Eq 1818 00 1 00 ejdlnt ejclnt y c t yc t dt f 1889 0 27r o f f ao e3cc Int 1 dt 1 18 90 27r t 0 1 00 eccx dx f 1891 27r 6c c 1892 where in going from Eq 1889 to Eq 1891 we have made the transformation x In t To prove Eq 1819 we first establish the following delta function identity 6lnxa a 6x a with a positive To prove it consider 1893 fo M 6lnxa f xdx a f 00 6y f a el ebdy a f a 1894 254 Chap 18 Scale which is the same answer that would be obtained if we integrate the right hand side of Eq 1893 with f x Now consider J ryctyctdc 1 ejcut e3clnt 27r t V 1895 t bln t In t 1896 1 blntt 1897 t bt t 1898 6t t 1899 which proves Eq 1819 Chapter 19 Joint Scale Representations 191 INTRODUCTION In this chapter we obtain joint representations where scale is one of the variables and the other variable is time or frequency We also consider joint representations of the three variables time frequency and scale The general approach we use is the characteristic function operator method developed in Chapters 10 and 17 192 JOINT TIMESCALE REPRESENTATIONS We use Pt c for the timescale distribution and M9 a for the corresponding characteristic function As usual they are connected by M9 Q ff SBtiac Pt c dt dc etoc 191 Pt c 4 ff M9 a ejetj dO do 192 Marginals The energy densities of time and scale are I st 12 and I Dc 12 respec tively Therefore we would ideally like Pc J Pt c dt Dc 12 193 Pt J Pt c dc 1 st12 194 However as in the timefrequency case there are joint representations that do not satisfy the marginals exactly but are nonetheless useful 255 256 Chap 19 Joint Scale Representations Joint Characteristic Function For the characteristic function operator we use the notation M8 a T C The average of the characteristic function operator is the characteristic function M9 a 10 s t M9 Q T C st dt 195 0 As we discussed in Chapter 17 there are many possible orderings In Table 191 we list some of the possibilities and the corresponding characteristic functions and joint distributions In the appendix we work out the first two choices in detail because they utilize and illustrate the main mathematical techniques Marinovich1179 1 Bertrand and Bertrandl8 711 and Altes112 introduced the concept of joint scale repre sentations They obtained by other methods the first two representations of Table 191 Further development of the theory has been made by Rioul4Ml Posch14541 Ri oul and Flandrin14901 Papandreou Hlawatsch and BoudreauxBartels1421 42a Bara niuk and Jones 149511 Shenoy and Parks151o 5111 and Cohen11411511 It is straightforward to verify by direct integration that the timescale distri butions of Table 191 satisfy the marginals of time and scale Eqs 193 and 194 However the easiest way to ascertain that is by noting that each of those character istic functions satisfy the characteristic functions of the marginals That is MOOTC ejOT Mo o C e2c 196 Of course checking the characteristic function does not check whether the algebra to obtain the distribution was correctly done 193 GENERAL CLASS OF TIMESCALE DISTRIBUTIONS To obtain the general class of joint timescale representations we use the method developed in Section 175 We start with any characteristic function and define a new one by Mnew9 a W a M6 a 197 where 0 is the kernel function whose role is identical to that in the timefrequency case The general class of timescale functions is then Pt c 47r2 ff 09 a M9 a ejetioc d9 da 198 Suppose we choose the characteristic function given by ordering one of Table 191 Then Pt c 412 N Seo2u eietjCcieu 46O a se2u dO du da 199 Any other starting characteristic function would serve equally well Sec 3 General Class of TimeScale Representations Table 191 Timescale representations 257 Ordering C aracteristic FvnctionMB a Dlstri tionP t c 1 ejC2ej8TejC2 h O e f et 7 2 Jae2teacaetT2tdc o s e2t8e2t dt 2 ejOTjC f 00 e2jOtenh2 ejc 21r 2 sinha2 a E 2tae2t dt a eO2 at se2 at da 2 sinha2 i2 3 expjet cosha2 1 J 1 e1 C 2a cosho2 J0 s e2t8e2t dt s eO2 cosha2 ee2 cosha2 da 4 ejCe70T j00 expj9e2t IJ stry c t D c o 8 eO2tae2t 5 ejaTejC TO expj9e2t a tc tDc 0 a e2tae2t 6 2 1141151 1 4 5 Real part a tyc tDc Relation Between Representations Suppose we have two timescale distributions Pl and P2 with corresponding kernels 1 and 02 Their characteristic functions are M18 C 01010 Me 0 M200 0200M00 1910 Hence Mje Q 010 a M29o 1911 020 0 By taking the Fourier transform of both sides we obtain Pit c ffgt t c cP2t c dt do 1912 with if gt c 412 JJ eetjc 2010 d8 du 1913 258 Chap 19 Joint Scale Representations Local autocorrelation method The general timescale distribution can be written in the form Pt c 1 JRoeidu 27r 1914 with RtCT 27r if 00 a MO Q eiet dO 1915 We shall call Rt the generalized local scale autocorrelation function in analogy with the local autocorrelation function for frequency Eq 97 Using Eq 199 for the general class Rt v is explicitly given by Rta 27r J1 e t 8 Q se2u sea2Udu dO 1916 194 JOINT FREQUENCY SCALE REPRESENTATIONS Because the scale operator has the identical form in any domain we can immediately write down joint distributions of frequency and frequency scaling Using MT Q and Pw c for the frequencyscale characteristic function and distribution respec tively and C for the frequency scale operator we calculate the characteristic func tion by MT c MT aWC 1917 which is achieved by way of 00 Mr v J S w M T a W C Sw dw 1918 0 This is identical in structure to the timefrequency case Therefore all we have to do is substitute T for 8 and S for s However it must be understood that now c stands for scaling in the frequency domain that is scaling of frequency functions The marginals are J Pw c dw I Dc l2 1919 I Pw c do I Sw I2 1920 where D4 c is the transform defined by Eq 1824 195 JOINT REPRESENTATION OF TIME FREQUENCY AND SCALE We now consider joint representations of the three variables time frequency and scale By scale we mean timescaling although with a small modification the proce dure can be used to obtain distributions of time frequency and frequencyscaling Sec 5 Joint Representations of Time Frequency and Scale 259 We use the notation Pt w c to signify the joint distribution and M9 T v to sig nify the characteristic function In Chapter 4 we discussed multivariate joint dis tributions but used the two dimensional case for the sake of clarity We are now dealing with a three dimensional situation in which case the characteristic function and distribution are related by M9 T a fff Pt w c ejOtj7W30 C dt dw dc 1921 d6 dv dr Pt fff M8 19 22 T a w c e The marginals we have to satisfy are f Pt w c do Pt w 1923 00 J Pt w c dt Pw c 1924 0 J Pt w c du Pt c 1925 where the right hand sides are the two dimensional densities of the corresponding variables Characteristic Function Many orderings are possible but it is sufficient to consider one ordering and write all other possibilities by using the kernel method We take the ordering M 0 T a e3aw2 eaoW2 The characteristic function is r000 MOT a s t ejarW2 e8Tjrw eiow2 st dt which evaluates to 1926 1927 east se2t T se2t T dt 1928 MBTQ f From Eq 1922 the distribution is Pt w c ffse012t 2T eaTWa se2t 2T dT dv 1929 To obtain the general class of timefrequencyscale distributions we define a gen eral characteristic function by Mnew9 T Q 00r a M9 T a 1930 260 Chap 19 Joint Scale Representations where 09 T a is a kernel of three variables Therefore Mnew9 T a 09 T Q f ejet se2t 2r sel2t 2r dt 1931 and the general lass is 87r3f Mnew B T Q ejetjrwjc dO da dT 1932 5 Ta 3e4u 2T se2u 2T ffff 00 X ejetujrwjc dO da dT du 1933 The marginals of this general class are the general lasses of the marginals For ex ample consider integrating out scale Integrating over c gives a 27r6a and there fore Pt W fPtwcdc 1934 1 JfJ jetujrw 47x2 49 T 0 s u 2r su z r a dB dr du 1935 This is precisely the general class of timefrequency distributions Eq 91 with the general kernel B Or 0 196 APPENDIX We derive the distributions corresponding to orderings 1 and 2 of Table 191 Ordering 1 MB a eiC2 ejOTe3C2 We have M8 a e3C2 ej9TeC2 r st ejC2 ejOTejc2st dt r st ejC2 ejet a4se12t dt J st expjOe2t a2set dt 1936 1937 1938 1939 or M9 a fsel2tei8tse2tdt 1940 Sec 6 Appendix 261 The distribution is a eyetjc dB dQ Pt ff M9 1941 C 4i2 12t eietse0 2e eietjc d6 do dt ffJ 1942 47r2 e 2 JJ se2t6t tsel2t aj dv dt 1943 Hence 2 se2t do Pt C J sel2t e 1944 Ordering 2 MO a ejOTc Since the scale operator satisfies TC jT 1945 we can use Eq 1728 to obtain ejOTAC ei0 T ejc ejeT 1946 with 77 1 11Qe 1947 01 The characteristic function is therefore r M6 a I s t ej9Ticst dt J st e7efT eic eiOTst dt 1948 Now we can follow essentially the same steps as in ordering 1 to obtain M9 a J s e2t e2jet dt 1949 In one of the steps leading to Eq 1949 use is 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TimeFrequency and TimeScale Analysis pp 417420 1992 625 B Zhang and S Sato A timefrequency distribution of Cohens class with a compound kernel and its application to speech signal processing IEEE Trans Sig Proc vol 42 pp 54641994 Bibliography 289 626 Y Zhao L E Atlas and R J Marks The use of coneshaped kernels for generalized time frequency representations of nonstationary signals IEEE Trans Acoust Speech Signal Processing vol 38 pp 108410911990 627 C Zhen S E Widmalm and W J Williams New timefrequency analyses of EMG and TMJ sound signals Proc IEEE Int Conf Engineering in Medicine and Biology pp 7417421989 628 C S Zheng W J Williams and J C Sackellares RID TimeFrequency Analysis of Median Fil ters and Lowpass Filters in Reducing EMG Artifacts in EEG Recording IEEE Intl Conference on Engineering in Medicine and Biology vol 1 pp 3503511993 629 Y M Zhu F Peyrin and R Goutte Equivalence between twodimensional analytic and real signal Wigner distributions IEEE Trans Acoust Speech Signal Processing vol 37 pp 163116341989 630 Y M Zhu F Peyrin and R Goutte The use of a two dimensional Hilbert transform for Wigner analysis of twodimensional real signals Signal Processing vol 19 pp 2052201990 631 Y M Zhu Y Gao and R Goutte An Investigation of the Wigner Model for Texture Discrimina tion Proc IEEESP Int Sympos TimeFrequency and rimeScale Analysis pp 5395421992 632 T P Zielenski New detection method based on the crossterms mechanism of the WignerV lle transform Signal Processing V Theories and Applications Eds L Torres E Masgrau and M A Lagunas Elsevier vol 1 pp 1571601990 Index A Ackroyd M H 45 Adjoint see operator Altes R A 256 Ambiguity function 96108116117125 137142166 generalized 137 Amin M G 189193 Amplitude definition of 3 29 3539 determination of 3 3539 modulation 3182138110121126 203 see also instantaneous frequency an alytic signal Analytic signal 2842130196 201 and the complex signal 2728 calculation of 3134 energy of 31 factorization theorem 3435 Hilbert transform and 3031 imaginary part of 30 interpretation of 3536 of a complex signal 31 of an analytic signal 32 of derivative 32 of the convolution of two signals 33 of the product of two signals 3436 of the sum of two signals 34 paradoxes 40 quadrature approximation 3639 Anticommutator 44 209 212 Antinormal ordering see correspondence rules Atlas L E 78 138 144 162 166 172175 201203 Autocorrelation function 23193 local 137 230 258 Average value 4 5 820 5456 and analytic signal 2829 conditional 6364 of an operator 213215 of frequency see frequency of Hermitian operators 214 of nonHermitian operators 215 of scale see scale of time see time B Bandlimited 3336 40 87 115 129 144 194 Bandwidth equation 1517179223224 Bandwidth 8194652 AM and FM contribution to 1719 expressed in terms of signal param eters 1517 instantaneous see instantaneous band width Baraniuk R G 166167 235256 BaraniukJones optimal kernel design 166 167 Barry D T135171 Basis functions 204205 Bedrosian E 35 Bernard G 78 203 Bertrand J 256 Bertrand P 256 Bescos J 184 Bilinearity 116140 Born M 44174 BornJordan distribution 113 138 158 160174 BornJordan rule see correspondence rule BoudreauxBartels G F 165191 256 Bunsen R W 2 291 292 C Calculation tricks 815 Car door slams 77 Cartwright N 132 CauchyRiemann conditions 30 Chaparro L F 193 Characteristic function operator and moment method 158161 and scale 256 257 260 261 averages157 evaluation 154156 for arbitrary variables 225229 for general class 156157 for time and frequency 152161 for time and scale 256 Characteristic function 5669 83 91 96 113116118 for spectrogram 96 for the frequency density 2223 for the general class 137 for the time density 2223 for Wigner distribution 116 one dimensional 5659 relation to moments 58 two dimensional 5965 when is a function a 58 Chen D 187 Choi H I 162 ChoiWilliams distribution 138 144 147 164165 168172 ChoiWilliams method 168172 Claasen TACM 114137 201 Cohen L 138153159 Cole N M 135 Commutator for arbitrary operators 209 210 for time and frequency 211 involving scale 244 Completeness 244253 Complex distributions 189 Complex energy spectrum 174176 Complex signal 32743 and real signals 124 reasons for 28 29 representation 27 see also analytic signal Compression operator 242243 250 Conditional average 64 134 Index timefrequency 8485 Conditional density 6364 timefrequency 8485 Conditional standard deviation 646585 90103134178179 Condon E U 44 Cone kernel see ZhaoAtlasMarks distri bution Correlation coefficient see covariance Correspondence rules and characteristic function 152156 and distribution 152157 antinormal 158 BornJordan 153159 general Cohen 153159 normal 153158 symmetrization153158 Weyl153158 Covariance 63 and characteristic function 63 and correlation coefficient 63 and uncertainty principle 4648216 218 for arbitrary quantities 216217 233 for time and frequency 2023 215 in terms of the spectrum 21 of a real signal 22 of a real spectrum 22 of a signal 2022 Cristobal G 184 Cross terms 104 116 125131 141 163 166169172175187192 Cross distribution 139 Cross Wigner distribution 124 129 Cumulative distribution function 54 Cunningham G S 162 168 188 194 D Darwin C G 44 Decay rate model 180 and formant bandwidth 180 Delta function distribution 185186 Density function 54 83 92 96 246 of a new variable 68 218 219 of a single variable 219224 of instantaneous frequency 41 one dimensional 5356 Index relation to characteristic function 56 59 relation to moments 58 two dimensional 5965 Detection 129 Diethorn E J 165 Dispersion 681 Duration 45 810 and uncertainty principle 455290 91 in terms of the spectrum 1920 Durationbandwidth theorem see uncer tainty principle E Eigenfunctions and eigenvalues 204209 discrete 208 for scale 244246 frequency 207208 normalization of 206207 problem 189 204208 222 227 242 244 time 208 El Jaroudi A 193 Energy 34 total 4 7 83 118142164 250 Energy density 34 78 2325 823 Energy density spectrum 7B 2325 82 83 Ensemble averaging 192 Expansion functions 204206 Expansion in spectrograms 187189 F Factorization theorem 34 Fault analysis 78 Finite support 8687 for general class 144 for spectrogram 98 for Wigner distribution 115116 strong 8687 weak 87 Flandrin E 129 193 256 Fonollosa J R 202 Fourier J B J 12 7 Fourier analysis 6 Fourier expansion 69 293 Fourier transform 6 and uncertainty principle 47 Fraunhofer J 2 Frequency and analytic signal 2729 and uncertainty principle 4649 average 8121517 average square 9 11 17 103 description of signals 67 eigenfunctions 207208 instantaneous see instantaneous fre quency negative 2829 3236 operator 1115 representation 1 67 G Gabor D 28 45187195 Gabor representation 186 and joint timefrequency represen tations 186188 General class 113151 156157 229230 256260 alternate forms 137 bilinear transformation 139 bilinearity 140 characteristic function 137 covariance 146147 finite support 144 for scale 256260 for time and frequency 113151 Fourier transform of kernel 139 functional dependence of kernel 140 global averages 146147 group delay 149 instantaneous frequency 147 kernel method 140141 local autocorrelation function 137 local averages 147149 marginals 141 positive bilinear distributions 151 positive spread 148 properties related to kernel 141146 reality 142 relation to spectrogram 150 scaling invariance 143 second conditional moment of frequency 148 294 strong finite support 144 sum of two signals 139 time and frequency shifts 142 transformation between distributions 149150 types of kernels 140 uncertainty principle 142 weak finite support 144 General class for arbitrary variables 228 230 characteristic function 225 covariance 233 instantaneous values 231232 local autocorrelation 230 marginals 225 relation between global and instan taneous values 231 transformation between distributions 229 unitary transformation 235238 Global and local averages relation between 6465 see also averages Gonzalo C 184 Group delay 19 102 105 119 134 149 168182184 232 H Heart sound 80171 Heat equation 2 Heisenberg W 44 Hermitian functions 188 Hermitian operator 1214 204215 averages 213214 eigenvalue problem 205206 forming of 209210 function of 210211 importance of 206 product of two 212215 range of eigenvalues 206 reality of eigenvalues 214 Hilbert transform 3031 Hlawatsch E 256 Hook R 80 Index I Instantaneous bandwidth 40101149178 182185 and multicomponent signals 178182 Instantaneous frequency 1522273039 43102110119124178182 and analytic signal 2730 distribution of 4143 general class and 147 spectrogram and 100119 spread of 42 Wigner distribution and 119 Instantaneous power 3 4 82 Instantaneous scale 251 Interference terms see cross terms Inverse frequency 222 238240 Inversion for general class 145146 for spectrogram 108 for Wigner distribution 127129 J Jacobson L D 184 Janssen A J E M 137 151 184 Jeong J 162 168 171 174 192 194 Jones D L 166 167 235 237 256 K Kayhan A S 193 Kernel design 162167 for product kernels 165 for reduced interference 162165 optimal 166167 projections onto convex sets 166 Kernel method 136151 Kernel and covariance 146 and finite support 144 and instantaneous frequency 147148 and marginals 141 and reality 142 and scale invariance 143 and total energy 142 Index and uncertainty principle 142 basic properties related to 141146 design 162167 determination of 141 for spectrogram 141 Fourier transform of 139 functional dependence of 140 product 115 relationship between kernels 149 types 140 Khinchin A 58 69 Kirchhoff G R 2 Kirkwood J G 113138 L Lagrange F L 2 Laplace P S 2 Lerner R M 45 Local and global averages 6364 relation between 64 65 Loughlin P J 78 144 162 172174 200 203 M Margenau H 113 Margenau Hill distribution 113136138 174177 Marginal 5964 and timefrequency distributions 83 for arbitrary variable 219 225 frequency 8283 inverse frequency 238239 scale 246 255 time 8283 Marinovich N M 191 256 Mark WD114 Marks II R J 138 172 173 162 166 Martin W 193 Mecklenbrauker WFG114137201 Mellin transform 246 Minimum uncertainty product signals for arbitrary variables 217218 for time and frequency 4849 for time and scale 251252 Modified signal 9495 Moment operator method 158161 Moments 5563 295 relation to characteristic function 56 5760 relation to distribution 58 Morris J M 187194 Moyal formula 128129 Moyal J E 113116129 Multicomponent signals 73116126127 and instantaneous bandwidth 182184 in time 183184 spectrum of 183 N Negative densities 69 Negative frequencies 284041 Nikias C L 202 Noise 114115130131191193 Normal ordering see correspondence rules Notation in brief xv Numerical computation 193194 Nuttall A H 35130133194 0 OHair J R 189194 Oh S 166173 Operator adjoint 211 algebra 209 eigenfunctions of 205 eigenvalues of 205214 commutator for arbitrary operators 209210 for time and frequency 211 involving scale 244 frequency 1115 functions of 210 Hermitian 1214 204215 averages 213214 eigenvalue problem 205206 forming of 209210 function of 210211 importance of 206 product of two 212 215 range of eigenvalues 206 reality of eigenvalues 214 inverse 211 linear 211 296 scale 228 242244 253 258 261 self adjoint see Hrmitian spectrum of 207 time 1314 translation 14 23 161 206 213 221 243 unit 211 unitary 212213 236 Optimal kernel design 166167 Optimal window 50 94101108150 Ordering rules see correspondence rules Orthogonal expansion of signals 204209 P Page C H 138175 Page distribution 138 175177 Papandreou A 165177 Parcevals theorem 7 Parks T W 191 256 Phase definition of 3 29 3539 determination of 3 3539 modulation 31821 see also instantaneous frequency an alytic signal Pielemeier W J 79 Pitton J W 78144162166172174201 203 Poletti formulation 181 Poletti M 181 194 Posch T E 138193 256 Positive distributions 198203 consequences of positivity 201 marginals 199 method of Loughlin Pitton and At las 201203 scale invariance 200 time and frequency shifts 200 uncertainty principle 199 weak and strong finite support 199 Poyntings theorem 3 Probability 45 53 59 92 192 195197 Projections onto convex sets 166 Q Qian S 187 Index Quadrature model 3639 and analytic signal 3639 energy criteria 37 point by point comparison 38 Quantum mechanics 4445 92113136 and signal analysis 195197 R Random signal 2325192195 Rayleigh J W Strutt 7 Rayleighs theorem 7 Reduced interference distributions 161 167 kernel design for product kernels 165 Representability 127128 Representation of signals 204218 Riedel K S 193 Rihaczek A W113136174176 Rioul 0 256 Robertson H F 44 Rohrbaugh R 78 Rosenthal C 72 75 76 Running spectrum 175 S Sayeed A M 193 Scale transform 245246 of the sum of scaled functions 247 Scale 242261 average 250251 bandwidth 250251 characteristic function 249250 compression operator 242244 eigenfunctions 244245 frequency scaling 252 general lass joint 256 instantaneous 251 joint characteristic function 256 joint frequency scale 258 joint timescale 255261 joint timescalefrequency represen tation 258260 operator 228 242244 253 258 261 other scaling 252 sum of scaled functions 247 transform 242 245248 uncertainty principle 251 252 Index Scaled and translated densities 67 Scaling invariance 86 143 Schrodinger E 44 Schwarz inequality 44 4748143 217 Sclabassi R J 136 Scully M 0 225 Shenoy R 0 256 Shorta b transform 234 Shorttime Fourier transform 50 51 75 9295 98103 107 108 234 see also spectrogram Signal analysis 1 and quantum mechanics 195197 Signals classification 25 26 frequency description of 25 time description of 25 Similarity transformation 236 Singular value decomposition of distribution 190191 of kernel 189 Skolnik M I 45 Spatialspatialfrequency distributions 184 Spectra 12 why do they change 8081 Spectral amplitude 7 analysis 1267 decomposition 1267 nonadditivity of properties 2326 phase 7 properties 2326 Spectrogram 93112 average of frequency functions 97 98 average of time functions 9798 characteristic function 96 covariance 99 expansion in instantaneous frequency 109110 expansion in 187189 finite support 98 global quantities 99 group delay 102103 in terms of other distributions 189 190 instantaneous bandwidth 101 instantaneous frequency 100 inversion 108 297 local averages 100 local square frequency 101 localization trade off 98 marginals 97 modified signal 9495 narrowband and wideband 95 narrowing and broadening the win dow 101102 optimal window 110 relation to other distributions total energy 9697 window function 509495101108 150 Spectroscope 2 Spectrum 2 68 1314 Speech 33 71 75 81 181 182 Standard deviation 4 5 8 4449 6165 8791 101103 conditional 64 see also instantaneous bandwidth Sun M 135 Suter B W 189 194 Symmetrization rule see correspondence rules Synthesis 191 192 T Terminology 25 54 60 92 209 Tune average of 451314 4748 in terms of spectrum 1920 duration 45 810 and uncertainty principle 4552 9091 in terms of the spectrum 1920 eigenfunctions 208 operator 1314 Tunebandwidth product see uncertainty principle Tune description of signals 25 Timefrequency analysis 2326284465 7081 83 184 195 201 Tunefrequency distributions basic problem 9192 characteristic function 83 finite support 86 frequency shift invariance 85 fundamental ideas 8292 298 general class 136151 global average 84 linear scaling 86 local average 84 marginals 83 need for 7081 terminology 92 time shift invariance 85 total energy 83 uncertainty principle and 8791152 157186193194201226230 235 238 Time representation 1 208 224 227 234 242 Transient 25 Translation operator 14 23 161 206 213 221 243 Tyack P L 79171 U Uncertainty principle 4452 8791 216 218 and covariance 46 216 and general class 142 and joint distributions 8791 and positive distributions 199 and the marginals 87 and the Wigner distribution 134 for arbitrary variables 216218 for the short timeFourier transform 5052 for time and frequency 4452 for time and scale 244 251 in quantum mechanics 4445 195 197 minimum uncertainty product signals for arbitrary quantities 217218 for time and frequency 4849 for time and scale 251252 proof of 4748 what does proof depend upon 46 Unitary transformation 235237 V Vainshtein L A 30 38 Vakman D E 30 38 41 Index Ville J 113 116 195 w Wakefield G H 79 Wechsler H 184 Weyl H 44 Weyl ordering see correspondence rules White L B 189193 Widmalm S E 172 WienerKhinchin theorem Wigner E P113 Wigner distribution 113135 characteristic function 116 comparison to spectrogram 133135 conditional spread 119120 covariance 119 cross terms 126130 duration 119 for the sum of two signals 124127 global averages 118119 group delay 119 instantaneous frequency 119 inversion 127128 local average 119120 marginals 117 mean frequency 119 mean time 119 modified 132133 Moyal formula 128 multicomponent signals 126130 noise 130 nonpositivity 116 positivity 132 pseudo 130131 range of 114115 range of the cross terms 129 reality117 relation to spectrogram 129130 representability127129 smoothed 13131 time and frequency shift invariance 118 uncertainty principle 119 uniqueness 127 Williams W J 162 165 168 188 194 kernel design method 165169172 Window function 509495101108f 150 Index 299 entanglement and symmetry with sig Wu DS 194 nal 9899 narrowing and broadening 101102 Z optimal window 110 Zaparovanny Y 139 Wood J C135171 Zhao Y 162172173 Woodward P M 116 ZhaoAtlasMarks distribution 172173