Block Transceivers-OFDM and Beyond - Livro sobre Modulação Multicarrier

SYNTHESIS LECTURES ON COMMUNICATIONS Morgan Claypool Publishers w w w m o r g a n c l a y p o o l c o m Series Editor William Tranter Virginia Tech C M Morgan Claypool Publishers SYNTHESIS LECTURES ON COMMUNICATIONS About SYNTHESIs This volume is a printed version of a work that appears in the Synthesis Digital Library of Engineering and Computer Science Synthesis Lectures provide concise original presentations of important research and development topics published quickly in digital and print formats For more information visit wwwmorganclaypoolcom Series ISSN 19321244 William Tranter Series Editor ISBN 9781608458295 9 781608 458295 90000 DINIZ MARTINS LIMA BLOCK TRANSCEIVERS MORGAN CLAYPOOL Block TransceiversOFDM and Beyond Paulo SR Diniz Universidade Federal do Rio de Janeiro Wallace A Martins Universidade Federal do Rio de Janeiro and CEFETRJ Markus VS Lima Universidade Federal do Rio de Janeiro The demand for data traffic over mobile communication networks has substantially increased during the last decade As a result these mobile broadband devices spend the available spectrum fiercely requiring the search for new technologies In transmissions where the channel presents a frequencyselective behavior multicarrier modulation MCM schemes have proven to be more efficient in terms of spectral usage than conventional modulations and spread spectrum techniques The orthogonal frequencydivision multiplexing OFDM is the most popular MCM method since it not only increases spectral efficiency but also yields simple transceivers All OFDMbased systems including the singlecarrier with frequencydivision equalization SCFD transmit redundancy in order to cope with the problem of interference among symbols This book presents OFDMinspired systems that are able to at most halve the amount of redundancy used by OFDM systems while keeping the computational complexity comparable Such systems herein called memoryless linear timeinvariant LTI transceivers with reduced redundancy require lowcomplexity arithmetical operations and fast algorithms In addition whenever the block transmitter and receiver have memory andor are linear timevarying LTVit is possible to reduce the redundancy in the transmission even further as also discussed in this book For the transceivers with memory it is possible to eliminate the redundancy at the cost of making the channel equalization more difficult Moreover when timevarying block transceivers are also employed then the amount of redundancy can be as low as a single symbol per block regardless of the size of the channel memory With the techniques presented in the book it is possible to address what lies beyond the use of OFDM related solutions in broadband transmissions Block Transceivers OFDM and Beyond Paulo SR Diniz Wallace A Martins Markus VS Lima SYNTHESIS LECTURES ON COMMUNICATIONS Morgan Claypool Publishers w w w m o r g a n c l a y p o o l c o m Series Editor William Tranter Virginia Tech C M Morgan Claypool Publishers SYNTHESIS LECTURES ON COMMUNICATIONS About SYNTHESIs This volume is a printed version of a work that appears in the Synthesis Digital Library of Engineering and Computer Science Synthesis Lectures provide concise original presentations of important research and development topics published quickly in digital and print formats For more information visit wwwmorganclaypoolcom Series ISSN 19321244 William Tranter Series Editor ISBN 9781608458295 9 781608 458295 90000 DINIZ MARTINS LIMA BLOCK TRANSCEIVERS MORGAN CLAYPOOL Block TransceiversOFDM and Beyond Paulo SR Diniz Universidade Federal do Rio de Janeiro Wallace A Martins Universidade Federal do Rio de Janeiro and CEFETRJ Markus VS Lima Universidade Federal do Rio de Janeiro The demand for data traffic over mobile communication networks has substantially increased during the last decade As a result these mobile broadband devices spend the available spectrum fiercely requiring the search for new technologies In transmissions where the channel presents a frequencyselective behavior multicarrier modulation MCM schemes have proven to be more efficient in terms of spectral usage than conventional modulations and spread spectrum techniques The orthogonal frequencydivision multiplexing OFDM is the most popular MCM method since it not only increases spectral efficiency but also yields simple transceivers All OFDMbased systems including the singlecarrier with frequencydivision equalization SCFD transmit redundancy in order to cope with the problem of interference among symbols This book presents OFDMinspired systems that are able to at most halve the amount of redundancy used by OFDM systems while keeping the computational complexity comparable Such systems herein called memoryless linear timeinvariant LTI transceivers with reduced redundancy require lowcomplexity arithmetical operations and fast algorithms In addition whenever the block transmitter and receiver have memory andor are linear timevarying LTVit is possible to reduce the redundancy in the transmission even further as also discussed in this book For the transceivers with memory it is possible to eliminate the redundancy at the cost of making the channel equalization more difficult Moreover when timevarying block transceivers are also employed then the amount of redundancy can be as low as a single symbol per block regardless of the size of the channel memory With the techniques presented in the book it is possible to address what lies beyond the use of OFDM related solutions in broadband transmissions Block Transceivers OFDM and Beyond Paulo SR Diniz Wallace A Martins Markus VS Lima SYNTHESIS LECTURES ON COMMUNICATIONS Morgan Claypool Publishers w w w m o r g a n c l a y p o o l c o m Series Editor William Tranter Virginia Tech C M Morgan Claypool Publishers SYNTHESIS LECTURES ON COMMUNICATIONS About SYNTHESIs This volume is a printed version of a work that appears in the Synthesis Digital Library of Engineering and Computer Science Synthesis Lectures provide concise original presentations of important research and development topics published quickly in digital and print formats For more information visit wwwmorganclaypoolcom Series ISSN 19321244 William Tranter Series Editor ISBN 9781608458295 9 781608 458295 90000 DINIZ MARTINS LIMA BLOCK TRANSCEIVERS MORGAN CLAYPOOL Block TransceiversOFDM and Beyond Paulo SR Diniz Universidade Federal do Rio de Janeiro Wallace A Martins Universidade Federal do Rio de Janeiro and CEFETRJ Markus VS Lima Universidade Federal do Rio de Janeiro The demand for data traffic over mobile communication networks has substantially increased during the last decade As a result these mobile broadband devices spend the available spectrum fiercely requiring the search for new technologies In transmissions where the channel presents a frequencyselective behavior multicarrier modulation MCM schemes have proven to be more efficient in terms of spectral usage than conventional modulations and spread spectrum techniques The orthogonal frequencydivision multiplexing OFDM is the most popular MCM method since it not only increases spectral efficiency but also yields simple transceivers All OFDMbased systems including the singlecarrier with frequencydivision equalization SCFD transmit redundancy in order to cope with the problem of interference among symbols This book presents OFDMinspired systems that are able to at most halve the amount of redundancy used by OFDM systems while keeping the computational complexity comparable Such systems herein called memoryless linear timeinvariant LTI transceivers with reduced redundancy require lowcomplexity arithmetical operations and fast algorithms In addition whenever the block transmitter and receiver have memory andor are linear timevarying LTVit is possible to reduce the redundancy in the transmission even further as also discussed in this book For the transceivers with memory it is possible to eliminate the redundancy at the cost of making the channel equalization more difficult Moreover when timevarying block transceivers are also employed then the amount of redundancy can be as low as a single symbol per block regardless of the size of the channel memory With the techniques presented in the book it is possible to address what lies beyond the use of OFDM related solutions in broadband transmissions Block Transceivers OFDM and Beyond Paulo SR Diniz Wallace A Martins Markus VS Lima Block Transceivers OFDM and Beyond No text detected No text detected Synthesis Lectures on Communications Editor William Tranter Virginia Tech Block Transceivers OFDM and Beyond Paulo SR Diniz Wallace A Martins and Markus VS Lima 2012 Basic Simulation Models of Phase Tracking Devices Using MATLAB William Tranter Ratchaneekorn Thamvichai and Tamal Bose 2010 Joint Source Channel Coding Using Arithmetic Codes Dongsheng Bi Michael W Hoffman and Khalid Sayood 2009 Fundamentals of Spread Spectrum Modulation Rodger E Ziemer 2007 Code Division Multiple AccessCDMA R Michael Buehrer 2006 Game Theory for Wireless Engineers Allen B MacKenzie and Luiz A DaSilva 2006 Copyright 2012 by Morgan Claypool All rights reserved No part of this publication may be reproduced stored in a retrieval system or transmitted in any form or by any meanselectronic mechanical photocopy recording or any other except for brief quotations in printed reviews without the prior permission of the publisher Block Transceivers OFDM and Beyond Paulo SR Diniz Wallace A Martins and Markus VS Lima wwwmorganclaypoolcom ISBN 9781608458295 paperback ISBN 9781608458301 ebook DOI 102200S00424ED1V01Y201206COM007 A Publication in the Morgan Claypool Publishers series SYNTHESIS LECTURES ON COMMUNICATIONS Lecture 7 Series Editor William Tranter Virginia Tech Series ISSN Synthesis Lectures on Communications Print 19321244 Electronic 19321708 Block Transceivers OFDM and Beyond Paulo SR Diniz Universidade Federal do Rio de Janeiro Wallace A Martins Universidade Federal do Rio de Janeiro and CEFETRJ Markus VS Lima Universidade Federal do Rio de Janeiro SYNTHESIS LECTURES ON COMMUNICATIONS 7 C M cLaypool Morgan publishers ABSTRACT The demand for data traffic over mobile communication networks has substantially increased during the last decade As a result these mobile broadband devices spend the available spectrum fiercely requiring the search for new technologies In transmissions where the channel presents a frequency selective behavior multicarrier modulation MCM schemes have proven to be more efficient in terms of spectral usage than conventional modulations and spread spectrum techniques TheorthogonalfrequencydivisionmultiplexingOFDMisthemostpopularMCMmethod since it not only increases spectral efficiency but also yields simple transceivers All OFDMbased systems including the singlecarrier with frequencydivision equalization SCFD transmit re dundancy in order to cope with the problem of interference among symbols This book presents OFDMinspired systems that are able to at most halve the amount of redundancy used by OFDM systems while keeping the computational complexity comparable Such systems herein called mem oryless linear timeinvariant LTI transceivers with reduced redundancy require lowcomplexity arithmetical operations and fast algorithms In addition whenever the block transmitter and re ceiver have memory andor are linear timevarying LTV it is possible to reduce the redundancy in the transmission even further as also discussed in this book For the transceivers with memory it is possible to eliminate the redundancy at the cost of making the channel equalization more difficult Moreover when timevarying block transceivers are also employed then the amount of redundancy can be as low as a single symbol per block regardless of the size of the channel memory With the techniques presented in the book it is possible to address what lies beyond the use of OFDMrelated solutions in broadband transmissions KEYWORDS block transceivers multicarrier modulation MCM orthogonal frequencydivision multiplexing OFDM reducedredundancy transceivers broadband digital commu nications vii Contents Preface xi Acknowledgments xv List of Abbreviations xvii List of Notations xxi 1 The Big Picture 1 11 Introduction 1 12 Digital Communications Systems 2 13 Orthogonal FrequencyDivision Multiplexing 3 131 Wired Systems 4 132 Wireless Systems and Networks 4 133 Basics of OFDM 7 14 Cellular Division 8 15 Multiple Access Methods 9 151 TDMA 9 152 FDMA 10 153 CDMA 10 154 OFDMA 13 16 Duplex Methods 16 161 TDD 16 162 FDD 17 17 Wireless Channels Fading and Modeling 17 171 Fading 17 172 Modeling 19 18 Block Transmission 20 19 Multicarrier Systems 22 110 OFDM as MIMO System 24 111 Multiple Antenna Configurations 26 112 Mitigating Interference and Noise 27 113 Concluding Remarks 28 viii 2 Transmultiplexers 29 21 Introduction 29 22 Multirate Signal Processing 30 23 FilterBank Transceivers 35 231 TimeDomain Representation 36 232 Polyphase Representation 36 24 Memoryless BlockBased Systems 44 241 CPOFDM 45 242 ZPOFDM 49 243 CPSCFD 51 244 ZPSCFD 51 245 ZPZJ Transceivers 51 25 Concluding Remarks 53 3 OFDM 55 31 Introduction 55 32 Analog OFDM 56 321 From TDM to FDM 56 322 Orthogonality Among Subcarriers 58 323 Orthogonality at Receiver The Role of Guard Interval 62 324 Spectral Efficiency PAPR CFO and IQ Imbalance 69 325 Implementation Sketch 71 33 DiscreteTime OFDM 73 331 Discretization of The OFDM Symbol 73 332 Discretization at Receiver The CPOFDM 76 333 DiscreteTime Multipath Channel 79 334 BlockBased Model 82 34 Other OFDMBased Systems 88 341 SCFD 88 342 ZPBased Schemes 90 343 Coded OFDM 93 344 DMT 94 35 Concluding Remarks 103 4 Memoryless LTI Transceivers with Reduced Redundancy 105 41 Introduction 105 42 ReducedRedundancy Systems The ZPZJ Model Revisited 107 ix 43 Structured Matrix Representations 114 431 DisplacementRank Approach 115 432 Toeplitz Vandermonde Cauchy and Bezoutian Matrices 118 433 Properties of Displacement Operators 124 44 DFTBased Representations of Bezoutian Matrices 128 441 Representations of Cauchy Matrices 129 442 Transformations of Bezoutian Matrices into Cauchy Matrices 132 443 Efficient Bezoutian Decompositions 133 45 ReducedRedundancy Systems 137 451 Complexity Comparisons 141 452 Examples 142 46 Concluding Remarks 147 5 FIR LTV Transceivers with Reduced Redundancy 149 51 Introduction 149 52 TimeVarying ReducedRedundancy Systems with Memory 151 521 FIR MIMO Matrices of LTI Transceivers 151 522 FIR MIMO Matrices of LTV Transceivers 156 53 Conditions for Achieving ZF Solutions 157 531 The ZF Constraint 157 532 Lower Bound on The Receiver Length 160 533 Lower Bound on The Amount of Redundancy 163 534 Achieving the Lower Bound of Redundancy 164 535 Role of The TimeVariance Property 168 54 Transceivers with No Redundancy 169 55 Examples 170 56 Concluding Remarks 174 Bibliography 175 Authors Biographies 183 More People at School Less Free Time for Play The figure shows how childrens time use varies by income group xi Preface The widespread use of mobile devices with high processing capabilitieslike smartphones and tablets as well as the increasing number of users and growing demand for higher data rates are some of the main reasons why data traffic over mobile communication networks has increased so much during the last decade As a result these mobile broadband devices spend the available spectrum fiercely requiring the search for new technologies From the physicallayer viewpoint the first step to address this problem is to choose a mod ulation scheme which is more adequate for the type of channel through which the signal wave propagates Indeed the spectral efficiency of communications systems can significantly increase by properly choosing the modulation scheme For example in broadband transmissions in which the channel presents a frequencyselective behavior multicarrier modulation MCM schemes have proven to be more efficient in terms of spectral usage than conventional modulations and spread spectrum techniquesThe multicarrier transmission illuminates the physical channel utilizing several nonoverlapping narrowband subchannels where each subchannel appears to be flat thus turning the equalization process simpler Among the existing MCM schemes the orthogonal frequencydivision multiplexing OFDM is the most notorious since it not only increases spectral efficiency but also yields simple transceivers OFDM is capable of eliminating the intersymbol interference ISI with very simple transmitter and receiver by performing lowcomplexity computations such as insertion and removal of a prefix and by using fast algorithms such as the fast Fourier transform It is worth noting that ISI is one of the most harmful effects in broadband transmissions Simple transceivers are attractive since they lead to lower latency and require less power consumption Therefore it is no surprise that OFDM has been adopted by many wired and wireless broadband communication technologies For instance the longterm evolution LTE is a wireless communication standard whose down link connection is based on OFDM whereas its uplink connection is based on the singlecarrier with frequencydivision equalization SCFD which is similar to OFDM in many aspects and is composed of the same building blocks The enhancements introduced by LTE physical layer are so significant that LTE achieves much higher data rates as compared to 3rd generation 3G systems and is already being considered a 4th generation 4G system At this moment given the desirable features of OFDM and SCFD schemes and their widespread use in both wired and wireless communication standards one could ask the following questions Is this the best we can do in terms of spectral efficiency When LTE spectrum gets overloaded what comes next Can we further improve these schemes This book tries to provide some directions to address these questions Both OFDM and SC FD systems transmit redundancy the prefix in order to cope with the problem of ISI Indeed a xii PREFACE portion of each data block is reserved for the prefix whose size must be larger than the channel memory for OFDM and SCFD systems Thus spectral efficiency may be increased if less redun dancy could be used This book presents OFDMinspired systems that are able to at most halve the amount of redundancy used by OFDM systems while keeping the computational complexity comparable Such systems herein called memoryless linear timeinvariant LTI transceivers with reduced redundancyrequire lowcomplexity arithmetical operations and fast algorithmsIn addition whenever the block transmitter and receiver have memory andor are linear timevarying LTV it is possible to reduce the redundancy in the transmission even further as discussed in the last chapter of this book For the transceivers with memory it is possible to eliminate the redundancy at the cost of making the channel equalization a more difficult task Moreover when timevarying block transceivers are also employed then the amount of redundancy can be as low as a single symbol per block regardless of the size of the channel memory An example is the codedivision multiple access CDMA system with long spreading codes which is always able to achieve ISI elimination as long as the system is not at full capacity ie at least one spreadingdespreading code is unused TheapproachfollowedinthisbookistopresentbothOFDMandSCFDsystemsasparticular cases of the socalled transmultiplexer TMUX In fact these two systems belong to the category of memoryless TMUXes Special attention is given to OFDM and its desirable properties since OFDM is being employed in many standards Then the TMUX is used as the main framework to derive both LTI and LTV transceivers with reduced redundancy With the techniques presented in the book it is possible to address what lies beyond the use of OFDMrelated solutions in broadband transmissions In summary this book presents solutions to reduce the redundancy in transmission aiming at increasing data throughput However it is worth mentioning that reducing redundancy might increase mean square error MSE and biterror rate BER as well as turn the design of the block transceiver more challengingThe optimum solutions are environmentdependent and its proper sensing leads to much more efficient spectral usage ORGANIZATION OF THE BOOK Chapter 1 aims at providing a big picture of digital and wireless communications This chapter differs from the others in the sense that the material is presented in a pictorial manner avoiding the mathematics whenever possibleThe reasons for this choice are i to provide a quick overview of the field without wasting time explaining concepts that are not central to this book most of which are thoroughly explained in digital communication and wireless communication textbooks and ii to concentrate on ideas rather than mathematicsTherefore at the end of this chapter the reader should have recollected topics such as digital modulation channel encoder cellular systems multiple access methods frequencyselective channels multicarrier modulation schemes and OFDM Chapter 2 briefly presents multirate signal processing fundamentals that are of major impor tance to fully understand theTMUX frameworkwhich is employed throughout the rest of the book In this chapter it is shown that TMUXes are general structures that can be used to representmodel several communications systems In particular OFDM and SCFD systems can be interpreted as PREFACE xiii memoryless TMUXes whose implementations are based on memoryless blockbased transceivers This chapter also introduces some initial results related to what is beyond OFDMbased systems Chapter 3 introduces OFDM from its original analog conception to its actual discretetime practical usage The chapter starts with the analog OFDM exploring the role played by the guard interval in maintaining the orthogonality among OFDM subcarriers and explaining the choices for some parameters such as the OFDM symbol duration and distance between adjacent subcarriers Then the discretetime OFDM is studied in connection with its analog version In addition many other topics related to OFDM are coveredsuch as the coded OFDM COFDMissues of OFDM transmissions like the peaktoaverage power ratio PAPR discrete multitone DMT systems and optimal power allocation Chapter 4 presents multicarrier and singlecarrier memoryless LTI transceivers that use a reduced amount of redundancy as compared to OFDM and SCFD systems That is this chap ter describes how transceivers with reduced redundancy can be implemented employing superfast algorithms based on the concepts of structured matrix representationsThus part of this chapter de scribes structured matrices and the displacement theory that allows the derivation of these superfast algorithms The focus of Chapter 5 is on the fundamental limits of some parameters related to LTV transceivers In particular we consider the memory of the multipleinput multipleoutput MIMO receiver matrix and the number of transmitted redundant elements which are inherent to finite impulse response FIR LTV transceivers satisfying the zeroforcing ZF constraint In addition it is shown that ZF equalizers cannot be achieved when no redundancy is used and as alternative pure MMSEbased solutions are presented PREREQUISITES We attempted to make this book as self contained as possible Although basic knowledge of wireless communications digital transmission and multirate signal processing is highly desirable it is not necessary since the first two chapters revisit the main concepts which are used throughout the book Thus the main prerequisites to follow this book are digital communications basic concepts of stochastic processesinvolving expected values means and variances of random variablesand linear algebrainvolving operations with vectors and matrices ranks determinants null and range spaces More advanced concepts such as structured matrices and displacement theory are explained Paulo SR Diniz Wallace A Martins and Markus VS Lima Rio de Janeiro June 2012 Getting to know the school day 1 What kind of information do the Mountain Stream Public School details give you about the school 2 Do you think this school day is a typical day for a school student Whywhy not 3 If you could change something about the school day what would you change Why xv Acknowledgments The authors are grateful to Joel Claypool for kindly pushing us to finish this project They are also thankful to Professors MLR de Campos EAB da Silva LWP Biscainho and SL Netto of UFRJand ProfessorTNFerreira of UFF for their incentive and always being available to answer our questionsThey would like to thank Professors R Sampaio Neto of PUCRJ and VH Nascimento of USP who influenced some parts of this book Wallace thanks his colleagues at the Federal Center for Technological Education Celso Suckow da Fonseca CEFETRJUnEDNI in particular at the Department of Control and Automation Industrial Engineering We also would like to thank our families for their patience and support during this challenging process of writing a book Paulo would like to thank his parents his wife Mariza and his daughters Paula and Luiza for illuminating his life Wallace thanks his fiancee Claudia and his parents Renê and Perpétua Markus thanks his parents Luiz Álvaro and Aracy and his girlfriend Bruna Paulo SR Diniz Wallace A Martins Markus VS Lima Rio de Janeiro Brazil June 2012 Vienna International School School Day Planning Times 900915 Arrival 915925 Assembly 925945 Break 9501030 Lesson 1 10301110 Lesson 2 11151155 Lesson 3 12001240 Lunch 1240120 Lesson 4 125205 Lesson 5 205210 Break 215230 Register and Notices 230315 Activity PEMusicLibrary 315 Home xvii List of Abbreviations 2G 2nd Generation 3G 3rd Generation 3GPP 3rd Generation Partnership Project 4G 4th Generation ADSL Asymmetric Digital Subscriber Line BER Bit Error Rate CDMA CodeDivision Multiple Access CFO CarrierFrequency Offset CPOFDM CyclicPrefix Orthogonal FrequencyDivision Multiplexing CPSCFD CyclicPrefix SingleCarrier with FrequencyDivision equalization CSI ChannelState Information DAB Digital Audio Broadcasting DFT Discrete Fourier Transform DHT Discrete Hartley Transform DMT Discrete MultiTone DSCDMA Direct Sequence CDMA DSP Digital Signal Processing ETSI European Telecommunications Standards Institute FDD FrequencyDivision Duplex FDM FrequencyDivision Multiplexing FDMA FrequencyDivision Multiple Access FFT Fast Fourier Transform FHCDMA FrequencyHopping CDMA FIR Finite Impulse Response GSM Global System for Mobile communications IBI InterBlock Interference ICI InterCarrier Interference IDFT Inverse Discrete Fourier Transform IEEE Institute of Electrical and Electronics Engineers IIR Infinite Impulse Response ISI InterSymbol Interference LAN Local Area Network LTE Long Term Evolution LTI Linear TimeInvariant xviii LIST OF ABBREVIATIONS MA Multiple Access MAN Metropolitan Area Network Mbps Megabits per second MCMRBT MultiCarrier MinimumRedundancy Block Transceiver MCRRBT MultiCarrier ReducedRedundancy Block Transceiver MIMO MultipleInput MultipleOutput MISO MultipleInput SingleOutput MMSE Minimum Mean Square Error MSC Mobile Switching Center MSE Mean Square Error MUI MultiUser Interference OFDM Orthogonal FrequencyDivision Multiplexing OFDMA Orthogonal FrequencyDivision Multiple Access OLA OverlapAndAdd PAM PulseAmplitude Modulation PAN Personal Area Network PAPR PeaktoAverage Power Ratio PSD Power Spectrum Density PSK PhaseShift Keying QAM Quadrature Amplitude Modulation QPSK Quadrature PSK SC Single Carrier SCFD SingleCarrier with FrequencyDomain equalization SCFDMA SingleCarrier FrequencyDivision Multiple Access SCMRBT SingleCarrier MinimumRedundancy Block Transceiver SCRRBT SingleCarrier ReducedRedundancy Block Transceiver SGSN Serving GPRS Support Node SIMO SingleInput MultipleOutput SISO SingleInput SingleOutput SNR SignaltoNoise Ratio SVD Singular Value Decomposition VDSL Very highspeed Digital Subscriber Line TDD TimeDivision Duplex TDM TimeDivision Multiplexing TDMA TimeDivision Multiple Access TMUX Transmultiplexer UMTS Universal Mobile Telecommunications System LIST OF ABBREVIATIONS xix WAN Wide Area Network WiFi Wireless Fidelity WiMAX Worldwide interoperability for Microwave ACCess WLAN Wireless Local Area Network WPAN Wireless Personal Area Network WSS WideSense Stationary xDSL highspeed Digital Subscriber Line ZF ZeroForcing ZP ZeroPadding ZPOFDM ZeroPadding OFDM ZPOFDMOLA ZPOFDM OverLapandAdd ZPSCFD ZeroPadding SCFD ZPSCFDOLA ZPSCFD OverLapandAdd ZPZJ ZeroPadding ZeroJamming the WiMAX worldwide interoperability for microwave access is usually considered an example of MAN but it can also be seen as a WAN system Many wireless network standards use OFDM in the air interface Two examples are the LTE longterm evolution and WiMAX standards LTE was designed to fully replace the 3rd generation 3G networks for mobile communications The WiMAX although originally conceived to provide wireless broadband services to homes has been upgraded to be employed by mobile phones as access method in recent years competing with LTE xxi List of Notations Scalars Lowercase letters eg x Vectors Lowercase boldface letters eg x Matrices Uppercase boldface letters eg X Definition N Set of natural numbers which is defined as N 1 2 3 Z Set of integer numbers R Set of real numbers C Set of complex numbers t Realvalued variable representing continuous time n Integer number representing discrete time j Imaginary unit j2 1 ω Angular frequency δt Dirac impulse δn Kronecker delta WM Unitary DFT matrix of size M M IM Identity matrix of size M M em Canonical vector eg e0 1 0 0T T Transpose of matrix H Hermitian conjugate transpose of matrix 0MN M N matrix with all entries equal to 0 E Expected value of Z Ztransform applied to Z1 Inverse Ztransform applied to tr Trace of matrix rank Rank of matrix diag Diagonal matrix whose entries in its diagonal are ker Kernel Null space of matrix R Range Column space of matrix F Fourier transform of N Decimation operator by N N Interpolation operator by N ml Entry of matrix in the mth row and lth column DFT Discrete Fourier transform of sequence A WiMAX The main target of IEEE 80216 standard known as WiMAX is to deliver wireless highspeed Internet access over longer distances than the ones supported by the IEEE 80211 standard commonly known as WiFi wireless fidelity Indeed WiMAX provides a MAN with wireless broadband service in an area of 50 km about 30 miles of radius The WiMAX is guided by an association called WiMAX Forum and its data rates can reach up to 40 Mbps for low mobility access and up to 15 Mbps for mobile access The capabilities and coverage area of WiMAX systems especially due to the amendment e of IEEE 80216 standard IEEE 80216e also called Mobile WiMAX increased in such a way that the IEEE 80220 standard was put to hibernation WiMAX employs an adaptive modulation scheme as illustrated in Figure 13 in which the digital modulator is adjusted according to the signaltonoise ratio SNR Through lowSNR channels usually consisting of channels where the user is far from the base station a sparser modulation scheme the quadrature phaseshift keying QPSK is used On the other hand users near the base station are likely to have highSNR channels and therefore transmissions with higherorder modulation schemes such as the 16 or 64quadrature amplitude modulation 1 C H A P T E R 1 The Big Picture 11 INTRODUCTION In communications the ultimate goal is the transport of as much information as possible through a propagation medium Currently most transmissions are performed by propagating electromagnetic energy through the air or wired channels The majority of our current communications systems transmit digital data to benefit from the widely available digital technology which has become relatively cheap The digital technology is also reliable amenable to error detection and correction and reproducible All these features reduce the cost of transmission and make available new services to the end users The wired channels involve a physical connection between fixed communication terminals usually consisting of guided electromagnetic channels such as twistedpair wirelines and coaxial cables In addition optical fibers are becoming increasingly popular due to their channel bandwidth Indeed the bandwidth of optical fiber is in general some orders of magnitude larger than in coaxial cablesEven though there is a trend for replacing wireline by opticalfiber channelsthe low costs and improvements in modem designs have extended the lifetime of many wireline connections which were already deployed In wireless communications the channel is the medium through which the electromagnetic energy propagates Examples of such mediums are the air and the water In this book the focus is on wireless communications through the air which we will refer to only as wireless communications1 In such communications systems the electromagnetic energy is radiated to the propagation medium via an antenna Unlike wired transmissions wireless transmissions require the use of radio spectrum which in turn should be carefully managed by government regulators The current trend of increasing the demand for radio transmissions shows no sign of settling The amount of wireless data services is more than doubling each year leading to spectrum shortage as a sure event in the years to come As a consequence all efforts to increase the efficiency of spectrum usage are highly justifiable at this point In response this book addresses the issue of how to increase the spectral efficiency of radio links by properly designing the transceivers especially for multicarrier systems which include the popular orthogonal frequencydivision multiplexing OFDM as a special case This chapter starts with a brief description of digital communications systems in Section 12 Section 13 motivates the study of OFDM by giving examples of several communications standards that use it with emphasis to the WiMAX and LTE systems The next two sections contain key 1Communications through the water are usually referred to as underwater communications 2 1 THE BIG PICTURE concepts of mobile and multiuser communications Indeed Section 14 introduces the cellular division paradigm while Section 15 briefly describes the most used multiple access schemes and also explains how multiple access can be performed in OFDMbased systems leading to the so called OFDMA orthogonal frequencydivision multiple access After that Section 16 describes duplex methodsThenSection 17 introduces the main problems present in wireless communications systems emphasizing the multipath effect In Section 18 a central topic of this book namely block transceivers is introduced In this section the use of a guard time between blocks is exemplified as a naive solution to avoid interblock interference Section 19 presents the fundamental idea of multicarrier systemswhich is the division of the channel spectrum in narrowband and approximately flat fading subchannels that allows simple equalization schemes at the receiver as it will be discussed in the following chaptersSection 110 shows for the first time the simpleyet powerfulmathematical model for representing OFDM as a MIMO multipleinput multipleoutput system The target is to illustrate that the low complexity of the OFDM is due to parameter decoupling which is achieved by judicious design of both the precoder transmitter and postcoder receiver Finally in Sections 111 and 112 we briefly discuss multiple antenna systems and the interference issue respectively 12 DIGITAL COMMUNICATIONS SYSTEMS Using a simplified buildingblock representation the main elements of a digital communication system are depicted in Figure 11 see 4 7 22 26 47 57 65 66 75 for further details Firstly typical transmitted input signals like data speech audio image and video are com pressed at the source encoder By exploring the redundancy2 of the input signal the source encoder is capable of representing it in a more compact form ie using less bits Depending on the input signal nature the compression process can be either lossless which means that the original signal can be exactly recovered from the compressed signalor lossywhich does not allow exactly restoration of the input signal but yields a much higher compression rate Lossy compression schemes use perceptive criteria to discard information that is not perceived by the end users For instance an audio signal at the receiver end should keep the perceptive quality as close as possible to the original input signal The next building block is the channel encoder whose primary task is to protect the compressed input information against the physicalchannel impairments The channel encoder allows for error detection and correction by adding some bits to the compressed signal This building block acts as a wrap to encase and secure the information to be transported 38 90 Once our package of information is ready it must be represented in a proper format to cross the channelThis task is performed by the digital modulator which is responsible for mapping bits into waveforms These waveforms are mathematically represented by complex numbers and are usually called symbols After crossing the channel the received waveform symbol is usually a distorted 2 Here the term redundancy means predictability which is a common characteristic of natural signals and is related to the concept of entropy 14 In the rest of the book the term redundancy is related to the prefix or suffix used in block transmissions a topic that will be covered in detail in the following chapters 13 ORTHOGONAL FREQUENCYDIVISION MULTIPLEXING 3 Source Decoder Source Transmitter Signal Input Channel Decoder Demodulator Digital Modulator Digital Channel Encoder Encoder Channel Output Signal Receiver Figure 11 Simplified representation of digital communication systems version of the transmitted waveform since the latter suffers attenuation and other wave propagation effects and interferences caused by environmental noise and by other signals being transmitted through the same channel At the receiver end all the strategies utilized at the transmitter to improve the channel usage are undoneThe received waveform is converted to bits by the digital demodulator the channel decoder is responsible for correcting some bits that were erroneously detected and then the source decoder undoes the compression process which is not a perfect reversion process in cases of using lossy compression schemes as explained before generating an output signal as close as possible to the input signal The next section addresses the OFDM a transmission scheme that enables efficient trans mission of symbols waveforms through the channel 13 ORTHOGONAL FREQUENCYDIVISION MULTIPLEXING Orthogonal frequencydivision multiplexing OFDM is a transmission technique that is currently used in a number of practical systems such as Digital radio broadcasting or digital audio broadcasting DAB Wireless local area network WLAN Wireless broadband links Highspeed digital subscriber line xDSL 4th generation 4G cellular communications Digital broadcasting TV 4 1 THE BIG PICTURE Because of its great importance and widespread use Chapter 3 is dedicated to explain OFDM and some of its variants By now it is important to notice that OFDM is applicable to both wired and wireless links as exemplified in the previous list of applications 131 WIRED SYSTEMS An important player in wired communications is the xDSL a general term for all broadband access technologies based on digital subscriber line The xDSL systems provide customers with high data rates using the already existing copper pairs inherited from the fixed telephony The main xDSL systems are the following Asymmetric DSL ADSL the asymmetric term means that upstream iefrom costumer to network and downstream ie from network to costumer data rates are different Nowadays evolved versions of the ADSL such as the ADSL2M can achieve upstream rates up to 35 Mbps megabits per second and downstream rates up to 24 Mbps Very highspeed DSL VDSL the first VDSL systems known as VDSL1 provided data rates higher than the ADSL The drawback of VDSL1 is that its data rate decreases too fast as the distance from the subscriber premises to the network increases limiting its usage to short local loops Nowadays VDSL2 systems can achieve upstream rates up to 10 Mbps and downstream rates up to 50 Mbpsand perform quite similar to the ADSL2 when transmitting over long distances unlike VDSL1 Both ADSL and VDSL use discrete multitone DMT which is essentially a sophistication of OFDM 132 WIRELESS SYSTEMS AND NETWORKS A set of complementary wireless standards are availableThese standards are mainly divided accord ing to their coverage area and their main classes are3 Personal area network PAN Local area network LAN Metropolitan area network MAN Wide area network WAN Figure 12 depicts these main classes according to their coverage areaand also shows examples of wireless standards Note that some technologies may appear in more than one class of wireless networks since they may receive upgrades that extend their original coverage area For instance 3 Some of these classes also exist for wired systems Indeed PAN and LAN can be used for both wired and wireless systems In cases when one wants to refer just to the wireless part of these systems the terms wireless PAN WPAN and wireless LAN WLAN can be used Figure 12 Wireless networks and corresponding technologies WAN IEEE 80220 MobileFi ETSI GSM 2G 3GPP UMTS 3G LTEAdvanced 4G MAN IEEE 80216 WiMAX ETSI HIPERMAN LAN IEEE 80211 WiFi ETSI HIPERLAN PAN IEEE 80215 Bluetooth 6 1 THE BIG PICTURE Medium SNR QPSK High SNR Low SNR 16QAM 64QAM Figure 13 Adaptive modulation in WiMAX QAMare allowedIn additionthe multiple access scheme used in WiMAX is the orthogonal frequency division multiple access OFDMA Other goals of WiMAX are Broadband on demand the deployment of LAN hotspots can be accelerated by MAN especially at locations not served by xDSL or other cablebased technologies and the phone companies cannot provide broadband services at short notice Cellular operator backhaul these operators usually use wired and leased connections from thirdparty service providers and they could be replaced by MAN Residential broadband in many areas it is difficult to provide wired broadband services Wireless services to rural and scarcely populated areas Wirelesseverywhereasthenumberofhotspotsincreasesthedemandforwirelessservices in areas not covered by LAN increases as well B LTE LTE is a standard for wireless data communication that is capable of overcoming some limi tations of GSMUMTS global system for mobile communicationsuniversal mobile telecom munications system standards It was developed by the 3rd Generation Partnership Project 3GPP and its first version was established in December 2008 when 3GPP Release 8 was frozen The main motivations for LTE are the user demands for higher data rates and quality of service the necessity of optimizing the packetswitched system and the demands for cost and complexity reduction Figure 14 Basics of OFDM overlapping but not interfering subcarriers Similar to the WiMAX LTE uses OFDMA in the downlink connection However in the uplink connection LTE opted for using the singlecarrier frequencydivision multiple access SCFDMA In addition LTE has very low latency and can operate on different bandwidths 14 3 5 10 15 and 20 MHz 3GPP Releases are continuously provided such as the most recent version called Release 11 It is worth mentioning that from Release 10 and beyond the LTE is usually called LTEAdvanced since it became IMTAdvanced compliant Some of the key features of IMTAdvanced compliant standards are Worldwide functionality and roaming Compatibility of services Interworking with other radio access systems Enhanced peak data rates to support advanced services and applications 100 Mbps for high and 1 Gbps for low mobility 133 BASICS OF OFDM In the basic application of OFDM a data stream is divided into blocks and each entry of the data block modulates a subcarrier with overlapping but not interfering frequency spectrum As illustrated in Figure 14 the OFDM subcarriers are overlapped in frequency and the distance between the mainlobes of adjacent subcarriers is equal to f Hertz Once a bandwidth is defined for provision of a certain service it can be occupied by a set of OFDM subcarriers The summation of bandlimited 8 1 THE BIG PICTURE Figure 15 Coverage area cellular division and frequency reuse OFDM subcarriers should occupy most of the available bandwidth so that the amount of symbols transmitted is maximized In the next section we introduce the concept of cellular division which was crucial to the success widespread use of mobile communications 14 CELLULAR DIVISION The mobile communication capitalizes on the significant attenuation a transmitted signal faces in the wireless channelThis allows the wirelessservice providers to reuse the available frequency range at distinct locations distant enough from each other During wireless service deployment the service provider defines a coverage region that consists of cell clusters Within these clusters each cell is assigned with a different channel group Figure 15 illustrates a coverage area in which each cell is represented by a hexagon and each collection of hexagons including all colors is a cluster Hexagons of the same color represent cells sharing the same radio resources ie transmitting on the same frequencies Each cluster utilizes the whole radio resources that can be reused by all other clusters In actual wireless networks there are equipments such as the mobile switching center MSC for GSM systems and the serving GPRS support node SGSN for UMTSwhich dynamically distribute the radio resources among the users These equipments can also manage an exchange of wireless channels among cells and among clusters 15 MULTIPLE ACCESS METHODS 9 When the mobile moves toward the frontier of two cells the signal connection with the original cell becomes weak Before it reaches the minimum level of acceptance in quality a handoff to the neighboring cell should be made without interruption Note that the cellular division enables different users to transmit simultaneously provided they are in different cells But if the users are in the same cell then a multiple access scheme must be used in order to allow simultaneous transmission for these users 15 MULTIPLE ACCESS METHODS Multiple access MA methods enable multiple users to transmit over the same channel Indeed the choice of the MA method determines how the radio resources sometimes called channel resources are shared among usersThe fundamental idea underlying MA methods is the concept of separability among users which means that the signals transmitted by different users although sharing the same transmission medium should be completely separable at the receiver Mathematically this implies that there must exist some domain eg time frequency space or code in which the waveforms corresponding to different users are orthogonal to each other The appropriate choice of the access method is key to achieve high datarate transmissions thus increasing the capacity of the networks to provide multiple services to several users The MA methods can be classified according to the domain in which the users are separable The main MA methods under use are Timedivision multiple access TDMA Frequencydivision multiple access FDMA Codedivision multiple access CDMA Orthogonal frequencydivision multiple access OFDMA These MA methods are briefly described in the following subsections Note however that they are not mutually exclusive Indeed the most successful 2G system the GSM is a hybrid of TDMA and FDMA In GSM each 200 kHz of channel bandwidth is shared among 8 users using TDMA 151 TDMA In TDMA the separability among users occurs in the time domain ie each user receives a time slot or a set of them for his transmission Figure 16 illustrates two users sharing the same medium using a TDMA schemeThe duration of each slot is T seconds and each user receives two of them During the slots that User 1 is transmitting User 2 is in silence and vice versaThe signal that arrives at the receiver assuming the channel does not introduce any distortion to the transmitted signals is the superposition of the signals sent by the two users Therefore in TDMA the transmitters must be tightly synchronized in order to avoid interference among them It is clear that the receiver must be synchronized with the transmitters as well since it must know how to properly chop the received signal in order to isolate the signals sent by the different users Figure 16 Example of TDMA involving two users 152 FDMA Unlike TDMA in FDMA users may transmit all the time The separability among them occurs in the frequency domain ie for each user it is assigned a different frequency band Figure 17 depicts two users employing an FDMA scheme to share the channel resources Both users transmit over the channel using a bandwidth of f Hz but the central frequency of User 1 is f1 whereas it is f2 for User 2 Clearly f1 and f2 must be distant enough from each other in order to avoid interference between the two users At the receiver side the signals belonging to the two users can be separated through bandpass filters that dramatically attenuate all frequencies except the ones that fall within the desired band As the number and types of services available to users increase the fixed channel assignment inherent to FDMA and TDMA becomes less efficient As a result more spectrum would be required with such fixed assignment Even if some kind of flexible channel assignment is incorporated in the current FDMA and TDMA schemes there is always a fixed upper bound in the number of users that can be served 153 CDMA The CDMA sometimes called spreadspectrum appears in two distinct forms the direct sequence CDMA DSCDMA and the frequencyhopping CDMA FHCDMA In DSCDMA the spectrum of the baseband signal is spread to occupy a wider bandwidth as depicted in Figure 18 In this figure the power spectrum density PSD is represented as a function of the frequency Before transmitting the waveform conveying information this waveform 15 MULTIPLE ACCESS METHODS 11 User 1 User 2 Δf Δf f f Δf Δf f Received Signal Without Distortion f2 f1 f1 f2 f1 f2 Figure 17 Example of FDMA involving two users is modified in order to spread it over a wider frequency band At the receiver the reverse process is performed and the spreadspectrum waveform is converted back to the original waveform from which the information can be extracted Frequency Frequency PSD PSD Spreading Despreading Original Signal Spread Original Signal Figure 18 Spreading the spectrum in CDMA One of the main benefits from using spreadspectrum is reducing narrowband interference For instance consider Figure 19 in which the spread original signal suffers interference due to two narrowband interferers In this case the despreading process performed at the receiver de spreads the original signal and also spreads the interferers as depicted in Figure 110 Consequently a significant portion of the signal power corresponding to the interferers is spread over frequency bands different from the ones used by the original signal thus reducing the inband interference Another main advantage of CDMA is that it allows users to share all the available bandwidth simultaneously Hence since the signal transmitted by each user is wideband CDMA must resolve such type of wideband interference in order to guarantee the separability among users Indeed CDMA introduces a new domain called code domain in which multiple users can be fully separated 12 1 THE BIG PICTURE Frequency PSD Spread Original Signal Narrowband Interferers Figure 19 Narrowband interference in CDMA Signal Original Frequency PSD Frequency PSD Narrowband Spread Interferers Figure 110 Mitigating narrowband interference in CDMA despreading process Figure 111 illustrates a DSCDMA passbandtransmission scheme In this case the mth user m N intends to transmit a sequence of symbols bmnTb in which Tb denotes the symbol duration and the sequence of symbols is indexed by n ZBefore transmittingthe symbols bmnTb are multiplied by a code sequence cmkTc where k Z represents the sequence index and Tc correspondstothedurationofeachelementofthecodesequenceInadditioncmkTcisaverysimple sequence whose elements are 1 or 1 The main difference between the sequences bmnTb and cmkTc is that the sampling rate of the latter is much higherieTc TbHencethe signal resulting from the multiplication of bmnTb by cmkTc is a spreadspectrum signal whose bandwidth is increased by a factor equal to TbTc also known as spreading factor or processing gain Finally this resulting signal modulates the carrier cos ωct where ωc is the carrier frequency resulting in a passband signal ready to be transmittedThe key point that guarantees the separability among users is that these code sequences are unique ie a different code sequence is assigned to each user and they are also orthogonal to each other4 4In fact due to some transmission issues such as synchronization orthogonal code sequences are sometimes exchanged by or used together with pseudonoise sequences 15 MULTIPLE ACCESS METHODS 13 cos ωct bm nTb cm kTc Figure 111 DSCDMA passbandtransmission scheme Note that whileTDMA and FDMA have a maximum number of simultaneous usersCDMA does not have such a constraint Nevertheless in CDMA if the number of users increases the service smoothly degrades In cellular systems CDMA allows soft handover between neighboring cells Close to the frontier where the signal is weaker the user communicates with two base stations simultaneously so that the diversity helps compensating for signal degradation at cell edges In TDMA and FDMA neighboring cells must use different frequencies because they control interference based on spatial attenuation of the signals frequency reuse In CDMA all cells use the same frequency range elim inating the necessity for frequencyuse planning whereas TDMA and FDMA may use adaptive frequency reallocation In addition CDMA requires strict power control and base station synchro nization and allows intercell interference to be suppressed at the receiver Another type of spreadspectrum technique is the FHCDMA in which each user employs a different frequency band within a given time frame During transmission the user frequency band hops to different bands according to a prescribed hopping pattern code The receiver hops synchronously with the transmitter with the knowledge of the codeThe hopping can be slow where hopping occurs at the symbol rate or fast where more than one hop occurs during symbol duration The latter case is more difficult to implement As a rule wideband MA schemes can operate in the frequency range of existing narrowband services and allow flexibility in the number of users and services provided to each userThey also allow improved interference rejectionoriginated from multiusermultipathand narrowband interferences On the other hand wideband MA systems require more advanced technology for implementation Figure 112 summarizes how the radio resources are assigned to one user forTDMA FDMA and CDMA schemesAs we have already discussedthe separability among the multiple users occurs in one of the following domains time frequency or code domains Therefore in TDMA the user receives a time interval for his transmissionwhich may use the whole channel bandwidthIn FDMA a frequency band is dedicated for the users transmission In CDMA a single user may transmit all the time and using the entire channel bandwidth but with a unique code 154 OFDMA In some standards sets of OFDM subchannels can be assigned to distinct users leading to an MA scheme known as orthogonal frequencydivision multiple access OFDMA In the simplest case 14 1 THE BIG PICTURE Time Slot Frequency Code a TDMA Code Time Slot Frequency b FDMA Code Time Frequency Slot c CDMA Figure 112 Channel sharing for a TDMA b FDMA c CDMA multiple access can be implemented in a TDMA format where at a given time slot a specific user is allowed to employ all subchannels for his transmission as illustrated in Figure 113 As depicted in Figure 114 it is also possible to assign distinct frequency bands for different users provided the users know in which subchannels bands they can transmit at a given time slot 15 MULTIPLE ACCESS METHODS 15 User 3 User 2 User 1 Frequency Time Figure 113 OFDMA system TDMA case User 3 User 2 User 1 Frequency Time Figure 114 OFDMA system FDMA case The most efficient way to assign the subchannels to multiple users is through random assign ment which guarantees that all users enjoy approximately the same quality of service Figure 115 illustrates an OFDMA scheme with random assignment of subchannels As depicted in this fig ure the subchannels used by each user may change at each time slot and therefore this kind of OFDMA scheme avoids users to get stuck in lowquality subchannels Note however that the OFDMA schemes that are usually employed in standards are the ones depicted in Figures 113 and 114 This is justified by some issues and limitations concerning the OFDMA with random assignment For example some multipleantenna transmission schemes especially those transmis sions with diversity cannot be employed in OFDMA with random assignment since they usually require the transmission over adjacent subchannels 16 1 THE BIG PICTURE User 3 User 2 User 1 Frequency Time Figure 115 OFDMA system random assignment In this section we have seen how different users can transmit over the same medium But what if the users want to be capable of both transmitting and receiving data In this case duplex methods must be used 16 DUPLEX METHODS In cellular systemsduplexmethods are used to separate the signal sent by the mobile to the base station called uplink connection from the signal sent by the base station to the mobile station known as downlink connection In fact in any bidirectional communication system the duplex method has to be specified in order to determine how these connections can coexist without interfering with each other5 There are mainly two duplex methods the timedivision duplex TDD and the frequency division duplex FDD which are briefly explained in the following 161 TDD As illustrated in Figure 116 TDD schemes assign different time intervals to uplink and downlink connections Both of these connections can use the whole channel bandwidth during their trans missions It is common practice to separate the uplink and the downlink connections in TDD by a time interval known as guard time which avoids interference between these two connections that might be caused by propagation effects such as multipath which are addressed in the next section 5That is bidirectional communications systems also have uplink and downlink connections like cellular systems The difference can be on the elements that are at the ends of these connections which can be computers and Internet service providers instead of mobile stations and base stations 162 FDD Figure 117 illustrates the FDD method in which different frequency bands are assigned to the uplink and downlink connections 17 WIRELESS CHANNELS FADING AND MODELING The signals waveforms transmitted by wireless communications systems are ordinary electromagnetic waves and as such they suffer from wavepropagation effects A detailed description of wireless channel effects over the transmitted signal as well as channel modeling can be found in 67 80 In this section we briefly explain some of the main problems introduced by wireless channels emphasizing the multipath fading in order to motivate the mathematical model used throughout this book These main impairments are path loss shadowing and multipath fading The problem of multiuser interference MUI associated with multiple access schemes is not addressed here 171 FADING The main effects on transmitted signals inherent to wireless channels can be summarized in one word fading Fading is a phenomenon concerning the timevariation of the channel strengths If these variations are due to transmissions over long distances ranging from hundreds to thousand meters then they are known as largescale fading whereas the term smallscale fading is used for channel variations due to transmissions over short distances of the order of the carrier wavelength Figure 116 Timedivision duplex TDD 18 1 THE BIG PICTURE Uplink Downlink ϕ1 ϕ2 ϕ3 t2 Time Frequency ϕ4 t1 Figure 117 Frequencydivision duplex FDD Some examples of wavepropagation effects that fall into the category of largescale fading are the path loss and the shadowing effects Path loss also known as path attenuation is the power reduction of an electromagnetic signal as it propagates through the medium ie longer distances between transmitter and receiver leads to lower power of the received signalassuming that the power of the transmitted signal does not vary The path loss Ploss is defined as Ploss Pt Pr 11 where Pt and Pr stand for the power of the electromagnetic signal at the transmitter and receiver respectively The path loss measured in decibels has a linear dependence on log10d where d is the distance that the signal travelsThe other effect mentioned above namely shadowing is a fading caused by the obstruction of the lineofsight between transmitter and receiver The shadowing generates signal variations that are usually modeled by lognormal distribution There are two main types of smallscale fading the frequencyselective and the flat fading If different frequency components of the electromagnetic signal are affected differently by the channel such effect corresponds to frequencyselective fading Otherwise in cases the variation induced on the signal by the channel does not depend onvary with the frequency of the electromagnetic wave then such effect falls into the category of flat fading The multipath fading is the most common type of smallscale fading that is present in mobile communications It originates from the transmittedsignal reflections in local buildings hills or structures around a few hundred wavelengths from the mobile Therefore the multipath fading can be understood as variations on the signal caused by interferences from attenuated and delayed versions of the same signal the reflections Figure 118 illustrates an example in which two reflected signals are interfering with the signal received from the lineofsight This figure depicts the three waveforms independently but it is clear that the received signal yt is a summation of the three signals Note also that in the time interval 0 Ts in which Ts stands for the duration of one symbol generated by the digital modulator the interference among the three signals will be constructive whereas some destructive interferences occur within other intervals such as Ts 2Ts Figure 118 Example of multipath fading An illustrative behavior of the path loss shadowing and multipath fading is depicted in Figure 119 in which PrPt is represented as a function of log10d 172 MODELING In rough environments such as urban areas frequencyselective fading occurs due to the reflections of the transmitted signal that arrive with distinct delays at the receiver multipath In these cases the propagation medium channel is usually modeled as a linear system with memory which can be characterized by its impulse responses Indeed since wireless channels are timevarying their impulse responses htτ can change along the time t Thus the signal arriving at the receiver at a time instant t ℝ can be written as yt₀ᵗ sτhtτdτ where htτ corresponds to the channel response to an impulse applied at the instant t and sτ represents the transmitted signal concatenation of waveforms representing symbols In addition it is common practice to consider that the memory of htτ is finite since due to path loss not all existing reflections will have enough power to be sensed at the receiver Figure 119 Example of power loss in wireless environment If one is interested only in the discretetime representation of the signals the signal arriving at the receiver at instant k ℤ can be written as ykᵢ₀ᵏ sihki where hki is the channel response to an impulse applied at instant k and si represents the transmitted signal as a sequence of symbols Note that in practical communications systems the sequence of symbols si are usually divided into blocks and then each block is transmitted through the channel as will be shown in the next section 18 BLOCK TRANSMISSION Block transmission schemes are employed in some modern digital communications systems Figure 120 illustrates the singleinput singleoutput SISO model of a block transmission In this figure each data block sn where n ℤ represents the block index is comprised of eight symbols coming from a digital modulator Each data block sn is processed and prepared for transmission at the SISO transmitter generating the block un that is propagated through the SISO channel If the SISO channel has memory as usually happens in wireless channels it will give rise to intersymbol interference ISI as well as interblock interference IBI at the receiver end In this case the currently received block yn and the previously received block yn1 corresponding to the transmitted blocks un and un1 respectively will have an overlap as depicted in Figure 120 Then the SISO receiver is responsible for eliminating both ISI and IBI and detecting the transmitted symbols yielding a data block ŝn that must be as close as possible to the originally transmitted block sn In the case of wideband transmission the ISI can be severe enough to make the SISO receiver very complex to implement Roughly speaking each transmitted symbol would spread over the time slots of the neighboring symbols turning their correct detection more challenging A naive but widely used solution to avoid interference among symbols of different blocks ie IBI is to allow a guard period between each block transmission In Figure 120 this solution corresponds to separate each block un by an amount of time that is sufficiently large to guarantee that overlaps do not occur between the received blocks yn n ℤ The drawback of such a solution is the reduction of the datatransmission rate This reduction can be significant in cases the blocks un are not much larger than the guard period A more general blocktransmission framework is illustrated in Figure 121 This MIMO model for block transmission encompasses many blocktransmission schemes including the SISO model multicarrier schemes and multiple antenna configurations In the MIMO model for block transmission a given data block s is modified by a MIMO Transmitter yielding the block u whose length is greater than the length of s This larger length is due to many different reasons such as the replication of previously transmitted blocks in case the transmitter has memory or just by the use of a guard period The MIMO channel model can be described through a MIMO transfer function possibly with memory generating ISI and IBI It is up to the MIMO receiver to process the received signal block y in order to generate a reliable estimate of the transmitted signal block ŝ 22 1 THE BIG PICTURE u s y ˆs Channel MIMO MIMO Transmitter Receiver MIMO Figure 121 MIMO model for block transmission The MIMO model might represent a wide range of signalprocessing tasks In this book the main type of MIMO processing is the multicarrier transmission which consists of transmitting each symbol in a block through a narrowband subcarrier The benefits from using this technique are the following Each subcarrier illuminates a narrow range of channel frequencies so that the equivalent subchannel appears to be flat This turns the equalization for each subcarrier much simpler Since it consists of a block transmission the time support for transmission of a symbol modu lated by a subcarrier is roughly multiplied by the number of symbols in the blockThat means there is much more time to decode the information conveyed by each subcarrier reducing or even avoiding ISI within a block If a guard period is inserted to avoid IBI the time overhead is relatively low as long as each block carries several symbolsThe guard time is a function of the length of the channel impulse response timedelay spread As we have seen if ISI and IBI are not tackled they can deteriorate the performance of communications systems We have also seen that IBI can be avoided just by using a simple guard period between the blocks In the following section we will introduce systems that tackle the ISI problem in a simple and efficient manner 19 MULTICARRIER SYSTEMS In currently deployed communications systems multicarrier transmissions seem to be the standard choice Multicarriermodulation methods play a key role in modern data transmissions to deal with channels with moderate to severe intersymbol interference Figure 122 Multicarrier system dividing the channel bandwidth into nonoverlapping flat subchannels The basic idea of multicarrier systems whose most popular implementation is the OFDM is the transport of information through a wideband channel by energizing it with several narrowband subcarriers simultaneously The success of this technique relies on the partition of the physical channel into nonoverlapping narrowband subchannels through a transmultiplexer as will be explained in the next chapter If the subchannels are narrow enough the associated channel response in each subchannelfrequency range appears to be flat thus avoiding the use of sophisticated equalizers Figure 122 illustrates the effect of splitting a wideband channel in flatter subchannels Figure 123 depicts a transmultiplexer implementing a 4band multicarrier system ie the channel is divided into 4 subchannels At the transmitter end a set of symbols represented by colors is prepared for transmission through distinct subcarriers where each subcarrier is represented by a finiteimpulse response FIR filter whose transfer function is denoted by Fiz i 0 1 2 3 At the receiver side there are related FIR filters Giz The FIR filters can be thought as narrowband filters with distinct central frequencies so that the symbols sent at different subcarriers travel through different subchannels thus not interfering with each other In addition Figure 123 assumes that perfect transmission reconstruction is possible ie the symbol transmitted at each subcarrier is perfectly recovered at the receiver 24 1 THE BIG PICTURE Channel Noise G1 z G2 z G3 z F2 z F1 z F0 z F3 z G0 z Figure 123 Example of a 4band multicarrier system In practical systemsperfect reconstruction is usually not achievable due to degradations caused by physicalchannel and noise effects as well as power limitations In this case the subchannel division allows whenever possible the exploitation of the signaltonoise ratio SNR in the distinct subbands by managing their data load in each subchannel Indeed if the transmitter has knowledge about the SNR at the channel output for each subcarrierthen some loading scheme could be applied as illustrated in Figure 124As can be observed in this figureat the subcarriers with higher SNR it is possible to transmit symbols belonging to higherorder modulations such as an 8PSK modulation whereas low SNR ones use lowerorder modulation schemes such as binaryPSK BPSK For very low SNR subcarriers it can be even decided not to transmit any symbol at all In a general setup some redundancy is required at the transmission in order to keep the equalization as simple as possible This is an important issue that will be addressed in the following chapters In addition several methods for jointly optimizing the transmitter and receiver of FIR MIMO systems can be employed to combat nearend crosstalk and additivenoise sources 110 OFDM AS MIMO SYSTEM In a noiseless environment an OFDM system can be described using the MIMO framework de picted in Figure 121 For this case the estimated signal vector ˆs can be described as a function of Figure 124 Example of a loading scheme applied to a 4band multicarrier system the inputsignal vector s as ŝ GHF s 14 where F represents the precoder matrix applied at the transmitter G represents the postcoder matrix applied at the receiver and H is the MIMO channel matrix In this simplified description all matrices are considered memoryless so that each inputsignal vector is processed independently In addition allowing the existence of additive noise v at the channel output and assuming that F modifies and inserts a prefix on s the block transmission can be modeled as ŝ GH Fs Gv 15 as represented in Figure 125 For channels with memory in which IBI exists the OFDM system adds some redundancy to the inputsignal vector in order to be able to eliminate the IBI at the receiver As will be explained later if the redundancy consists of a cyclic prefix and the transmitter and receiver matrices are based on discrete Fourier transform DFT the detection of the symbols at the receiver are decoupled from each other meaning that ISI within each block is also eliminated Figure 125 Parameter decoupling in OFDM using a MIMO model 111 MULTIPLE ANTENNA CONFIGURATIONS Although the main topic of this book is multicarrier systems in many current applications the MIMO formulation allows the incorporation of multiple antennas at the transmitter and receiver on top of the usual precoder and postcoder blocks inherent to these systems as shown in Figure 126 These multipleantenna building blocks introduce another degree of freedom the space that enables an efficient use of the radio resources For instance this new degree of freedom can be exploited to increase the system throughput by employing a spatial multiplexing scheme or to enhance a transmission ie decrease biterror rate by using a transmission with diversity scheme In a general multipleantenna setup we can consider the transmission of several blocks of data belonging to one or multiple users where all the preprocessing at the transmitter is incorporated in a single matrix building block Tx and transmitted through an array of antennas At the receiver there is also an array of antennas whose output signals feed a single post processing building block Rx that is responsible for separating and detecting each transmitted signal block as illustrated in Figure 127 There are several ways to compose the input and outputsignal vectors as well as the channel matrix in a digital communication setup In any case by properly stacking the transmitted and received information the representation given by Equation 15 is quite powerful and accommodates several transceiver configurations The capacity gains of multipleantenna systems with respect to the conventional singleantenna systems depend on number of antennas at the transmitter 112 MITIGATING INTERFERENCE AND NOISE 27 Rx ˆsM ˆs1 Tx s1 sN Figure 126 General setup of MIMO precoding with multiple antennas SpaceTime Coding Interleaving RF Modulation Prefilter Post Filter Demodulation RF Receiver Space Deinterleaving ˆs s Figure 127 General setup of multiple block MIMO precoding with multiple antennas number of antennas at the receiver number of paths in the channel 112 MITIGATING INTERFERENCE AND NOISE Practical communications systems must be able to deal with interference and noise in an efficient mannerAs we have already seenmobile communications suffer ISI and IBI due to multipath fading and they also suffer interference caused by other users sharing the same radio resources which is usually called multiuser interference MUI or cochannel interference In this contextthe signaltointerferenceplusnoise ratio SINR plays a key role in assessing the quality of a transmission A high SINR indicating a highquality transmission can be achieved by mitigating interference andor noisepromoting enhancements in the performance of the physical layer 28 1 THE BIG PICTURE There are several strategies and techniques to increase SINR such as designing equalizers for MIMO systems employing transmission with diversity in multipleantenna systems optimizing multicarrier systems using subspace methods to mitigate noise etc In the next chapters we will discuss how multicarrier systems can combat ISI IBI and noise 113 CONCLUDING REMARKS In this chapter many aspects of digital communications and transmissions were briefly introduced In the following chapters some of these aspects will be used or carefully revisited Among the material covered in this chapter block transmissions and multicarrier systems in which OFDM is the most notorious case are the central topics of this book Indeed the rest of this book is dedicated to thoroughly explain the fundamental ideas of block transmissions and multicarrier systems to present the OFDM and SCFD systems and to introduce block transceivers that are capable of increasing the system throughput by reducing the amount of redundancy necessary to remove IBI In addition we tried to motivate and expose in an intuitive manner the importance and necessity of such topics to current communications systems Therefore the approach followed here consists of presenting the material in a pictorial way leaving the mathematical details to the following chapters 29 C H A P T E R 2 Transmultiplexers 21 INTRODUCTION The proposal of new techniques for channel and source coding along with the development of integrated circuits and the use of digital signal processing DSP for communications have allowed the deployment of several communications systems to meet the demands for transmissions with high dataratesTypical DSP tools such as digital filtering are key to retrieving at the receiver end reliable estimates of signals associated with one or several users who share the same physical channel There are various classes of digital filters Those employed in communications systems can be either fixed or adaptive linear or nonlinear with finite impulse response FIR or with infinite impulse response IIR just to mention a few Among such classes of systems fixed linear and FIR filters are rather common in practice because of their simpler implementation good stability properties and lower costs as compared to other alternatives Nonetheless modern communications systems require more sophisticated techniques thus calling for more features than fixed linear and FIR filters can offer In this context multirate signal processing adds some degrees of freedom to the standard linear timeinvariant LTI signal process ing through the inclusion of decimators and interpolators These degrees of freedom are crucial to develop some interesting representations of communications systems based on filter banksespecially multicarrier transceivers A filter bank is a set of filters usually LTI FIR filters sharing the same inputoutput pair and internally employing decimators and interpolators Filterbank representations are widely used in source coding and spectral analysis In com munications the transmultiplexer TMUX configuration can be employed to represent multicarrier or singlecarrier transceivers and can be considered a system dual to the filterbank configuration in the sense that the signal processing which takes place at the input of a filter bank actually appears at the output of a TMUX and vice versa Indeed several practical systems can be modeled using TMUXes Unlike filter banks that usually require sharp frequencyselective subfilters practical multicar rier transceivers can be modeled as TMUXes which use shortlength subfilters with poor frequency selectivity In the majority of practical cases these transceivers are implemented as memoryless block based transceiversThe most commonly used memoryless blockbased transceivers are the orthogonal frequencydivision multiplexing OFDM and the singlecarrier with frequencydomain equaliza tion SCFD systems The main feature of OFDMbased transceivers is the elimination of intersymbol interfer ence ISI with low computational complexity ie using just a small amount of numerical opera 2 TRANSMULTIPLEXERS tions to undo the harmful effects induced by frequencyselective channels A competing alternative to OFDM is the SCFD transceiver which presents lower peaktoaverage power ratio PAPR and lower sensitivity to carrierfrequency offset CFO as explained in 63 87 In addition for frequencyselective channels the biterror rate BER of SCFD can be lower than for its OFDM counterpart particularly for the cases in which the channel has high attenuation at some subchannels In this chapter some key multirate signalprocessing tools are revised Section 22 aiming at their use in the modeling of communications systems Section 23 These tools will be particularly utilized to represent OFDM and SCFD systems as well as to introduce some initial results related to what is beyond OFDMbased systems namely the memoryless LTI blockbased transceivers using reduced redundancy Section 24 22 MULTIRATE SIGNAL PROCESSING In many signalprocessing applications it is quite common that signals with distinct sampling rates coexist In general multirate signalprocessing systems include as building blocks both the interpolator and the decimator The interpolation consists of increasing the sampling rate of a given signal whereas the decimation entails a samplingrate reduction of its input signal The loss of data inherent to decimation may give rise to aliasing in the decimated signal spectrum The interpolation by a factor N N consists of including N 1 zeros between each pair of adjacent samples generating a signal whose sampling rate is N times larger than the sampling rate of the original signal Indeed given a complexvalued signal sn in which the integer number n denotes the time index at the original sampling rate the interpolated signal sintk is given by sintk sn whenever k nN 0 otherwise 21 where the integer number k denotes the time index at the new sampling rate In the frequency domain the effect of interpolation can be described as see for example 17 Sintejω SejωN 22 in which Xejω Fxn nZ xnejωn 23 is the discretetime Fourier transform of the sequence xn with ω R denoting the frequency variable1 1It is assumed that the discretetime Fourier transform of the sequence xn exists ie the series in expression 23 is convergent for all realvalued scalar ω For instance an absolutely summable ie an 𝓁1signal xn is sufficient to guarantee the convergence of the series 22 MULTIRATE SIGNAL PROCESSING The decimation by a factor N consists of discarding N 1 samples from each nonoverlapping block containing N samples of the input signal The resulting signal has a sampling rate N times lower than the sampling rate of the original signal Indeed given the signal sn the decimated signal sdeck is defined by sdeck skN 24 for all integer number k In the frequency domain it is possible to show that the decimated signal is represented by see for example 17 Sdecejω 1N n𝓃S ejω2πnN 25 where 𝓃 0 1 N 1 Unlike the interpolation the decimation is a periodically timevarying operation Figure 21 Interpolation N 2 It is worth mentioning that a more appropriate nomenclature for the interpolation and decimation processes just described should be upsampling and downsampling reserving the nouns interpolation and decimation for the cases in which a filtering process is also present However it is rather common in the literature and in practice to use interchangeably the nomenclatures upsamplinginterpolation and downsamplingdecimation We will follow this practice but the reader will be able to identify easily when a filtering process takes place or not Figures 21 and 22 depict the respective effects of interpolation and decimation by a factor N 2 in both time and frequency domains These signals are only for illustration purposes and they do not represent true timefrequency pairs By examining Figures 21 and 22 it is possible to 2 TRANSMULTIPLEXERS verify that in order to avoid aliasing due to decimation and to eliminate the spectrum repetition due to interpolation a digital filtering operation is required before the decimation and after the interpolation The decimation filter narrows the spectrum of the input signal in order to avoid that aliasing corrupts the spectrum of the resulting decimated signal For a lowpass real signal for instance we have to maintain the input signal information only at the low frequencies within the range πN πN so that the spectrum at this range is not corrupted after decimation The interpolation filter smooths the interpolated signal sintk eliminating abrupt transitions between nonzero and zero samples which is the source of the spectrum repetitions also known as spectral images The central frequencies of the spectrum repetitions are located at 2πN n with n 𝓃 Figure 23 illustrates how the decimation and interpolation operations are implemented in practice Figure 22 Decimation N 2 Figure 23 General interpolation and decimation operations in time domain Example 21 Decimation Interpolation Let hn be a signal defined as hn 2n whenever n 0 1 7 0 otherwise 26 Determine Hintz Hz3 and Hdecz Hz3 for all nonzero complex number z in which Hz Zxn is the Ztransform of the sequence xn In addition the notations N and N denote the interpolation and decimation by N applied to respectively Solution We know that HznZxn zn 12z14z28z316z432z564z6128z7 for all z0 The interpolation by a factor of 3 is equivalent to insert 2 zerovalued samples between adjacent samples of xn Hence we have Hintz10z10z22z30z40z54z60z60z88 z9 0z100z1116z120z130z1432z150z160z17 64z180z190z20128z21 12z34z68z916z1232z1564z18128z21 Hz3 for all z 0 The decimation by a factor of 3 will generate a discretetime signal hdeckh3k23k8k if k 012 or hdeck0 otherwise Hence we have Hdecz18z164z2 for all z0 In multirate systems there are very useful manners to manipulate the interpolation and decimation building blocks We are particularly interested in ways to commute the decimation and interpolation operations with linear timeinvariant filters Some forms of commuting are based on the socalled noble identities interpolated filter followed by the downsampling These operations can be mathematically described as FzSzN UzFzN SzN Gz YzN SzGzN YzN Analysis Bank Synthesis Bank g0k f0k g1k f1k gM1k fM1k Figure 25 Analysis and synthesis filter banks in time domain A widespread application of multirate systems is the filterbank design A filter bank consists of a set of filters with the same input signal or a set of filters whose outputs are added to form the overall output signal as depicted in Figure 25 The set of M N filters represented by the family of impulse responses gmkmM in which M01M1 is the socalled analysis filter bank whereas the set of filters represented by the family of impulse responses fmkmM is the synthesis filter bank It is possible to verify that the analysis filter bank divides the input signal in subbands generating narrowband signals which can be further decimated The subband signals can be employed for analyses and manipulations according to the particular application For reconstruction the subband signals are interpolated and combined by the synthesis filter bank Transmultiplexers also known as filterbank transceivers are considered systems dual to the filterbank configurations since the roles of analysis and synthesis filter banks are interchanged in transmultiplexers Indeed the inputs of a transmultiplexer are first combined by the synthesis bank and after some further processing stages the outputs are obtained as a result from the analysis bank as shown in Figure 26 It is worth mentioning that this section is based on 17 81 which contain a thorough treatment of this subject 23 FILTERBANK TRANSCEIVERS 35 hk yk xk uk vk g1k gM1k g0k f0k f1k fM1k sM1n ˆsM1n ˆs1n ˆs0n s1n s0n N N N N N N Figure 26 TMUX system in time domain 23 FILTERBANK TRANSCEIVERS Consider the transceiver model described in Figure 26 where a communication system is modeled as a multipleinput multiple output MIMO system The data samples of each sequence smn belong to a particular digital constellation C C such as PAM QAM or PSK2 The sequence smn represents the mth transceiver input where m M and n Z represents the time index The corresponding transceiver output is denoted as ˆsmn C which should be a reliable estimate of smn δ where δ N represents the delay introduced by the overall transmissionreception process A communication system can be properly designed by carefully choosing the set of causal transmitter filters with impulse responses represented by fmkmM and the set of causal receiver filters represented by gmkmM These filters operate at a sampling rate N times larger than the sampling rate of the sequences smn Note that the index n represents the sample index at the input and output of the transceiver whereas k Z is employed to represent the sample index of the subfilters and of the internal signals between the interpolators and decimators The transmitter and receiver subfilters are time invariant in our discussions The input signals smn for each m M are processed by the transmitter and receiver sub filters aiming at reducing the channel distortion so that the output signals ˆsmn may represent good estimates of the corresponding transmitted signals The usual objective in a communication system is to produce estimates of smn δ achieving low BER andor maximizing the data throughput The channel model can be represented by an FIR filter of order L N whose impulse response is hk CThe FIR transfer function accounts for the frequencyselective behavior of the physical 2Pulseamplitude modulation quadratureamplitude modulation or phaseshift keying respectively channel The additive noise vk C accounts for the thermal noise from the environment and possibly for the multiuser interference MUI 231 TIMEDOMAIN REPRESENTATION According to Figure 26 the channel input signal is given by u k imZMsmifm kiN 212 The relation between input and output of the channel is described as yk jZhjukjvk 213 At the receiver end the signal yk is processed in order to generate estimates of the transmitted data according to ŝm n lZgm l y nNl 214 By employing Equations 212 213 and 214 it is possible to describe the relation between the input signal sm n and its estimate ŝm n as given by ŝm n ijlmZ3Mgm l hj sm i fm nNljiNlZgm l vnNl 215 The description above is not the easiest one to analyze the system and draw conclusions For example it is possible to employ a timedomain approach using matrix description as described in 70 72 Another approach is to apply polyphase decomposition in a Zdomain formulation as described as follows 232 POLYPHASE REPRESENTATION As long as the interpolation and decimation factors are equal to N it is convenient to describe the transmitter and receiver filters by their polyphase decompositions of order N according to the Fmz riangleq mathcalZfmk sumk in mathbbZ fmk zk sumj in mathbbZ left fmjN zjN fmjN1 zjN1 cdots fmjNN1 zjNN1 right sumj in mathbbZ sumi in mathbbN fmjN i zjNi sumi in mathbbN sumj in mathbbZ fmjN i zjN zi sumi in mathbbN zi sumj in mathbbZ fmjN i zjN underbrace riangleq Fim zN sumi in mathbbN zi FimzN 216 and Gmz riangleq mathcalZgmk sumk in mathbbZ gmk zk sumj in mathbbZ left gmjN zjN gmjN 1 zjN1 cdots gmjN N 1 zjNN1 right sumj in mathbbZ sumi in mathbbN gmjN i zjNi sumi in mathbbN sumj in mathbbZ gmjN i zjN zi sumi in mathbbN zi sumj in mathbbZ gmjN i zjN underbrace riangleq GmizN sumi in mathbbN zi GmizN 217 where m in mathcalM and Fmz and Gmz are the mathcalZtransforms of fmk and gmk respectively The transfer functions Fimz are the TypeI polyphase components of order N associated with Fmz whereas the transfer functions Gmiz are the TypeII polyphase components of order N associated with Gmz Example 22 Polyphase Decomposition Let us consider the signal hn defined in Example 21 Determine the TypeI polyphase decomposition of order 3 associated with the transfer function Hz Solution Consider that i in 012 Thus we have Hiz sumj in mathbbZ h3j i zj 218 yielding H0z 1 8 z1 64 z2 219 H1z 2 16 z1 128 z2 220 H2z 4 32 z1 221 Observe that Hz H0z3 z1 H1z3 z2 H2z3 left1 8 z3 64 z6right z1 left2 16 z3 128 z6right z2 left4 32 z3right 1 2 z1 4 z2 8 z3 16 z4 32 z5 64 z6 128 z7 222 Comparing with the solution of Example 21 the reader should also notice that Hdecz H0z left H0z3 right3uparrow This is a useful property that will be further exploited By using a matrix approach we can rewrite Equations 216 and 217 as follows leftF0z cdots FM1zright left1 quad z1 cdots zN1 rightunderbracedTz leftbeginarrayccc F00zN cdots F0M1zN vdots ddots vdots FN10zN cdots FN1M1zN endarrayrightunderbracemathbfFzN mathbfdTz mathbfFzN 223 leftbeginarrayc G0z vdots GM1z endarrayright leftbeginarrayccc G00zN cdots G0N1zN vdots ddots vdots GM10zN cdots GM1N1zN endarrayrightunderbracemathbfGzN leftbeginarrayc 1 vdots zN1 endarrayrightunderbracemathbfdz1 mathbfGzN mathbfdz1 224 Figure 27 Polyphase representation of TMUX systems Now by defining Smz riangleq mathcalZsmn Uz riangleq mathcalZuk Xz riangleq mathcalZxk Vz riangleq mathcalZvk Yz riangleq mathcalZyk and hatSmz riangleq mathcalZhatsmn then one can write Uz mathbfdTz mathbfFzN underbraceleft beginarrayc S0zN vdots SM1zN endarray right riangleq mathbfsz 225 Xz Hz Uz 226 Yz Xz Vz 227 underbraceleft beginarrayc hatS0z vdots hatSM1z endarray right riangleq hatmathbfsz left mathbfGzN mathbfdz1 Yz rightdownarrow N 228 Figure 27 illustrates the transceiver model utilizing the polyphase decompositions of the transmitter and receiver subfilters By employing the noble identities described in Section 22 it is possible to transform the transceiver of Figure 27 into the equivalent transceiver of Figure 28 The highlighted area of Figure 28 that includes delays forward delays decimators interpolators and the SISO channel model can be represented by a pseudocirculant matrix mathbfHz of dimension 2 TRANSMULTIPLEXERS Vz Figure 28 Equivalent representation of TMUX systems employing polyphase decompositions N N given by Hz H0z z1HN1z z1HN2z z1H1z H1z H0z z1HN1z z1H2z HN1z HN2z HN3z H0z 229 in which Hz iN HizNzi and Hiz jZ hjN izj 230 0jNiL Indeed given the indexes m and l within the set N the m lth element of the matrix Hz denoted as Hzml represents the transfer function from the lth input element of the highlighted area shown in Figure 28 to the mth output element of this area Hence by assuming that vk 0 for all integer number k if Ulz is the lth input at the transmitter end of the highlighted area in Figure 28 and Ymz is the mth output of this area at the receiver end then Ymz zlHzUlzNzmN zmlHzUlzNN Ulz z mlHzN 231 23 FILTERBANK TRANSCEIVERS in which we have applied the noble identity described in Equation 211 and we also have considered that the only nonzero input of the highlighted area in Figure 28 is Ulz Therefore based on Equation 231 and on the first type of polyphase representation of the channeltransfer function we can write Hzml YmzUlz z mlHzN zml iN Hiz NziN iN Hiz NzmliN H0z Nzml H1zNzml1 HN1zNzmlN1N 232 We know that the decimation operation retains the first coefficient out of N coefficients within a block starting from the 0th element In the Zdomain this means that the decimation operation keeps only the coefficients which multiply a power of zN Thus the jth coefficient of the decimated signal corresponds to the jNth coefficient of the signal before the decimation Another way of interpreting this fact is that given an index i0 N such that m l i0 is a multiple of N the decimation operation which appears in expression 232 retains the i0th term Hi0z Nzmli0 and decimates it as illustrated in Example 22 We also know that N 1 m l N 1 since m and l are within the set N Hence if m l 0 then expression 232 yields Hzml Hmlz N N Hmlz 233 On the other hand if m l 0 then Hzml zN HNlmzNN z1HNlmz 234 confirming the relations described in Equations 229 and 230 Figure 29 describes the transceiver through the polyphase decomposition of appropriate matrices including the pseudocirculant representation of the channel matrix It is worth noting that the descriptions of Figures 26 and 29 are equivalent As Figure 29 illustrates the transmitted and received vectors are denoted as sn s0n s1n sM1nT 235 sn s0n s1n sM1nT 236 42 2 TRANSMULTIPLEXERS vn yn Fz Hz Gz sn ˆsn un Figure 29 Blockbased transceivers in Zdomain employing polyphase decompositions From Figure 29 it is also possible to infer that the transfer matrix Tz of the transceiver can be expressed as Tz GzHzFz 237 where we considered the particular case in which vk 0 for all integer number k inspired by the zeroforcing ZF design A transceiver is zero forcing whenever Tz αzdIM for some α C and d N Notice that if there is no noise a zeroforcing solution is able to retrieve a scaled and delayed version of all transmitted signals An important observation about Figure 29 is that in order to be able to recover a block with M transmitted symbols one must send through the channel at least M elements in a data block ie we must necessarily have N M this fact explains the shapes of the boxes in Figure 29 Nonetheless if N M no redundancy is included then the matrices Fz Hz and Gz are square matrices and therefore a zeroforcing solution would not be achieved using only FIR filters considering that the channel model is not a simple delay as explained in 44 45 Hence some redundancy must be introduced in order to work with FIR transceivers3 Now let us assume that we choose N L ie the interpolationdecimation factor is greater than or equal to the channel order L a common situation in practice4 Based on Equation 230 we have that the only integer number j which satisfies the inequality constraint 0 jN i L is j 0 which lead us to conclude that Hiz hi for N i L On the other hand if there exists i L within the set N then Hiz 0 since there is no term to be added in order to form Hiz In other words we can say that for N L each element of the matrix Hz will consist of filters with a single possibly null coefficient In this case the pseudocirculant channel matrix in 3Employing IIR filters may bring about many drawbacks such as instability issuesThis is the reason why FIR transceivers are the prevalent choice 4Usually practical block transceivers use N M K where K is an integer number larger than or equal to L 23 FILTERBANK TRANSCEIVERS Equation 229 is represented by a firstorder FIR matrix described as Hz h0 0 0 0 h1 h0 0 0 hL hL 1 0 0 hL 0 hL h0 HISI z1 0 0 hL h1 0 0 0 hL 0 0 0 0 0 0 HIBI 238 Notice that Equation 238 implies the following relation in the time domain xk HISIuk HIBIuk 1 HIBI HISI uk 1 uk 239 where uk ukN N 1 ukN N 2 ukNT 240 xk xkN N 1 xkN N 2 xkNT 241 The relationship described in Equation 239 makes clear the roles of the matrices HISI and HIBI Indeed matrix HISI mixes the symbols transmitted in the current data block ie such a matrix introduces interferences among the current datablock symbols while matrix HIBI mixes some of the symbols transmitted in the past block The channel output vector xk is the result of adding both effects ISI and IBI Another way to derive Equation 239 is by analyzing what happens in the time domain when a signal uk passes through an FIR channel hk of order L In this case we know that the channel output xk is the linear convolution between the signals uk and hk that is xk h uk Hence if we look at a block of size N L containing the channel output signals in other words if we examine the elements of the vector xk then we can verify that the first L elements of this block are affected by the last L elements of the previous block due to the channel memory and the way the linear convolution is computed It is worth pointing out that generalizations of standard multicarrier communications systems may call for sophisticated transmultiplexer designs in which the transmitted signal is filtered by a precoder with memory consisting of a MIMO FIR filter The inherent memory at the transmitter can be viewed as a kind of redundancy since a given signal block is transmitted more than once along with neighboring blocks Sophisticated transmitters may require more complex receivers but they might allow a reduction in the amount of redundant signals necessary to attain zeroforcing solution for example All of these facts indicate that communication engineers should master the TMUXrelated tools in order to pursue new advances in communications systems especially regarding multicarrier transceivers The case of transceivers with memory will be addressed in Chapter 5 In this chapter we shall consider the widespread memoryless systems 24 MEMORYLESS BLOCKBASED SYSTEMS The particular and very important case where the transceivers are memoryless that is F z F and Gz G is addressed in this section This case encompasses the memoryless blockbased transceivers since these systems do not use data from previous or future blocks in the transmission and reception processing of the current data block That is only the current block takes part in the transceiver computations The traditional OFDM and SCFD transceivers are wellknown examples of memoryless blockbased systems The nonoverlapping behavior associated with memoryless transceivers is only possible if the lengths of the FIR causal subfilters fmkm M and gmkm M are less than or equal to N Indeed from Equations 216 and 217 we know that Fzim ΣjZ fmjN izj 242 Gzmi ΣjZ gmjN izj 243 for all pairs of numbers i m within the set N M Hence the matrix F z will have memory ie will depend on z if and only if there exists both a nonzero natural5 number j0 and a pair of numbers i0 m0 N M such that fm0j0 N i0 0 which occurs if and only if fm0k is a causal impulse response with length larger than N since j0 N i0 N The same conclusion can be drawn for the matrix Gz We shall briefly describe now the main memoryless LTI block transceivers which will be considered throughout this book Further details will be given in Chapters 3 and 4 5The index j0 cannot be negative because we are only interested in causal subfilters 241 CPOFDM The OFDM transceiver employing cyclic prefix as redundancy also known as cyclicprefix OFDM or just CPOFDM is described by the following transmitter and receiver matrices respectively F 0KM K IK IM WMH 244 G EWM 0MK IM RCPC MN 245 where the integer number K denotes the amount of redundant elements WM is the unitary M M discrete Fourier transform DFT matrix that is WMml wMl M 246 with m l M2 M M and WM ej2π M 247 In addition IM is the M M identity matrix 0MN is an M N matrix whose entries are zero and E is an M M diagonal equalizer matrix placed after the removal of the cyclic prefix and the application of the DFT matrix As can be noted the data block to be transmitted has length M however due to the prefix the transceiver actually transmits a block of length N M K in which K must be larger than or equal to the channel order L ie one must necessarily have M K L so that the CPOFDM system works properly The first K elements are repetitions of the last K elements of the inverse discrete Fourier transform IDFT output in order to implement the cyclic prefix Matrix ACP adds and matrix RCP removes the related cyclic prefix Note that based on Equation 238 the product RCPHzACP Hc CM M is given by Hc h0 0 0 hL h1 h1 h0 0 0 hL hL hL 1 0 0 hL 0 0 0 hL h0 248 2 TRANSMULTIPLEXERS where we can observe that RCP removes the IBI there is no dependency on z anymore whereas matrix ACP rightmultiplies the resulting memoryless matrix RCPHz CM N so that the overall matrix product is a circulant matrix of dimension M M Indeed one can observe that each row of matrix Hc can be obtained by circularshifting the related previous row After inclusion and removal of the cyclic prefix the resulting circulant matrix can be diagonalized by its rightmultiplication by the IDFT and leftmultiplication by the DFT matrices where these matrices are placed at the transmitter and receiver sides respectively Indeed we have WM Hc WMH WM h0 h1 hM 1 WMH WM h0 WM h1 WM hM1 WMH ĥ0 ĥ1 ĥM1 WMH 249 where hm is the mth column of matrix Hc and ĥm WM hm is its DFT Note that one can interpret the elements of vector hm as a periodic discretetime signal hmk whose period is M which respects the relation hmk h0 k m where h0 0 h0 h0 1 h1 h0L hL h0L 1 0 h0 M 1 0 h0M h0 and so forth Thus by remembering the circularshifting property of the DFT stating that given H0l DFTh0 k 1 M Σk M h0kWl k M 250 for all l M then one has DFT h0 k m Wml M H0l 251 Hence by applying this result we have Hml DFThmk DFTh0k m Wml M H0l 252 yielding ĥm diagWml M lM ĥ0 diagWl MlMm ĥ0 Dm ĥ0 253 in which D diagWlM lM denotes an M M diagonal matrix whose the l lth element is WlM for each l M We can therefore rewrite Equation 249 as WM HcWH M lm h0 Dh0 D2h0 DM1h0 WH Mlm H0lWl0M H0lWl1M H0lWlM1M WMm0 M WMm1 M WMmM1 M M H0lM iM WliM WimM H0lM iM WllmM 254 If l m then WllmM 1 for all i M implying WM Hc WH M ll H0lM M MH0l 255 while if l m then WlmM 1 implying WM Hc WH M lm H0lM WMlmM 1 WlmM 1 H0lM 1 1 WlmM 1 0 256 Therefore we can conclude that Λ WM Hc WH M WM RCP Hz Acp WH M diagMh0 diagM WM h0 diagM WM h 0ML11 diagλmmM 257 in which h h0 h1 hLT 2 TRANSMULTIPLEXERS Matrix Λ includes at its diagonal the distortion imposed by the channel on each symbol of the data block Hence the model of a CPOFDM transceiver is described by ŝ EΛs Ev 258 with v WM RCP v and for the sake of simplicity the time dependency of the expressions was omitted As can be observed the estimates of the transmitted symbols are uncoupled that is each symbol can be estimated independently of any other symbol within the related block avoiding intersymbol interference One can interpret this fact as if each symbol were transmitted through a flatfading subchannel From a signal processing perspective the model described in Equation 258 has a simple interpretation Indeed the addition and removal of the cyclic prefix turns the linear convolution described in Equation 239 into a circular convolution In this case the CPOFDM system loads each subcarrier in the frequency domain with a constellation symbol and after that performs the inverse discrete Fourier transformation generating a vector in the time domain The elements of this vector can be thought as a periodic signal which is processed by the channel through a linear convolution After that the signal is brought back to the original frequency domain A basic fact of digital signal processing is that the circular convolution of two signals can be implemented in the frequency domain by performing the product of the DFTs of the related signals Therefore the CPOFDM system can be further simplified if we take this fact into account All we have to do is to perform the entire processing in the frequency domain The symbols which are loaded at each subcarrier can be directly mapped to the received signals at each subcarrier by performing the product with the frequency response of the channel DFT of the zeropadded impulse response as in Equation 257 The equalizer E for this transceiver can be defined in several ways where the most popular are the zeroforcing ZF and the minimum mean square error MMSE equalizers In the ZF solution it is aimed to undo the distortions introduced by the channel Indeed when there is no noise the ZF solution is able to perfectly recover the transmitted vector It is assumed that matrix Λ can be inverted thus yielding EZF Λ1 259 As for the MMSE solution there is no requirement that matrix Λ be invertible since this latter operation is not needed In fact the linear MMSE equalizer matrix is the solution to the following optimization problem EMMSE argminVℂMM J E 260 where J is a realvalued function of a complexvalued matrix argument defined as J E Es EΛs v2 2 Es EΛs EvH s EΛs Ev trEs EΛs Ev s EΛs EvH trσs2IM σs2EΛΛ H EH σs2EΛ σs2Λ HEH σv2EEH 261 where E and tr are the expected value and trace operators respectively The derivation above assumes that the transmitted symbols and environment noise within a block are independent and identically distributed iid originating from a widesense stationary WSS white random sequences with zero means and uncorrelated These assumptions imply that EsvH EsEvH 0MM EvEsH EvsH and that EssH σs2IM and EvvH σv2IM where the positive real numbers σs2 and σv2 are the variances of the related WSS random sequences6 Now by using the following derivatives of scalar functions of complex matrices 83 trZAZHZ ZA 262 trAZHZ A 263 then we have J EE σs2EΛΛ H σs2Λ H σv2E 264 We know that the optimal solution EMMSE is such that J EMMSEE 0MM which implies that7 EMMSE Λ H ΛΛ H σv2σs2 IM1 diagλm λm2 σv2σs2mM 265 It is worth highlighting that the CPOFDM transceiver is the most popular type of OFDMbased techniques which are employed in practical applications 242 ZPOFDM An alternative OFDM system inserts zeros as redundancy and is called zeropadding OFDM ZPOFDM There are many variants of ZPOFDM One possible choice is the ZPOFDMOLA 6In this book we shall not employ distinct notations for deterministic and random variables 7We encourage the reader to justify why this is actually the minimum solution of the objective function 2 TRANSMULTIPLEXERS overlapandadd whose transmitter and receiver matrices are implemented as F IM 0KM AZP CNM WMH 266 G EWM IM IK 0MKK RZP CMN 267 where as in the CPOFDM case K L elements are inserted as redundancy and N M K The name OLA stems from the way the received signals are processed by the matrix RZP Matrices AZP and RZP perform the addition and removal of the guard period of zero redundancy respectively The matrix product RZPHzAZP CMM is given by RZPHzAZP h0 0 0 hL h1 h1 h0 0 0 hL hL hL1 0 hL 0 hL 0 h0 RCPHzACP 268 As can be observed matrix AZP removes the interblock interference whereas matrix RZP leftmultiplies the resulting memoryless Toeplitz matrix HzAZP CNM so that the overall product becomes a circulant matrix of dimension M M The reader should note that RZPHzAZP RCPHzACP Hc The ZPOFDMOLA transceiver discussed here is a simplified version of a more general transceiver proposed in 55 In fact the general transceiver allows the recovery of the transmitted symbols using zeroforcing equalizers independently of the locations of the channel zeros unlike the ZPOFDMOLA or CPOFDM that might have zero eigenvalues under certain channel conditions Unfortunately from the computational point of view this transceiver implementation is not as simple as for instance the CPOFDM since the equivalent channel matrix is not circulant turning impossible its diagonalization through fast transforms such as fast Fourier transform FFT Furthermore even for the design of a simple ZF equalizer the general ZPOFDM transceiver would require the inverse of a Toeplitz matrix being therefore more complex than the inversion of a circulant matrix required by a ZPOFDMOLA system 243 CPSCFD The cyclicprefix singlecarrier with frequencydomain CPSCFD equalization transceiver employs cyclic prefix as redundancy and it is closely related to the CPOFDM transceiver The CPSCFD system is described by the following transmitter and receiver matrices F 0KMK IK IM 269 G WMH EWM 0MK IM 270 244 ZPSCFD The zeropadding singlecarrier with frequencydomain ZPSCFD equalization transceiver inserts redundant zeros to the block to be transmitted as in the ZPOFDM transceiver The ZPSCFDOLA version may be modeled through the following transmitter and receiver matrices F IM 0KM 271 G WMH EWM IM IK 0MKK 272 245 ZPZJ TRANSCEIVERS Lin and Phoong 39 40 44 showed that the amount of redundancy guard samples K N M N required to eliminate IBI and ISI in memoryless blockbased transceivers must satisfy the inequality 2K L They proposed a family of memoryless discrete multitone transceivers with reduced redundancy A particular transceiver of interest to our studies here is the zeropadding zerojamming ZPZJ system which is characterized by the following transmitter and receiver matrices F Fbar 0KM NM G 0MLK Gbar MN in which Fbar CMM and Gbar CMM2KL The transfer matrix related to this transceiver is given by Tz GHzF G Hbar F Tbar 275 52 2 TRANSMULTIPLEXERS where after removing the redundancy the effective channel matrix is defined as H hL K h0 0 0 0 hK 0 h0 hL 0 hL K 0 0 0 hL hK CM2KLM 276 Considering vk 0 for all k Z we have ˆsn G H Fsn Tsn 277 Observe that the requirement of having 2K L 0 makes sense when we analyze the above expression Indeed in order to recover the M transmitted symbols the memoryless transfer matrix T of dimension M M must be fullrank This means that minM M 2K L M ie 2K L 0 K L 2 For this transceiver there are some constraints to be imposed upon the channel impulse response model so that a zeroforcing solution exists These constraints are related to the con cept of congruous zeros11 The congruous zeros of a transfer function Hz are the distinct zeros z0 z1 zμ1 C with μ N which meet the following condition zN i zN j with Hzi Hzj 0 for all i j 0 1 μ 1 Note that μ is a function of N As shown in 44 the channel model must satisfy the constraint μN K where μN denotes the cardinality number of elements of the largest set of congruous zeros with respect to N Therefore assuming the existence of minimumredundancy solutions for a given channel ie considering that μN L2 N then the ZF solution is such that its associated receiver matrix is given by G H F1 F 1H 1 278 where H CMM is given and F is predefined This solution is computationally intensive since it requires the inversions of M M matrices entailing OM3 arithmetic operations The conventional OFDM and SCFD transceivers need OM log M operations for the design of ZF and MMSE equalizers The equalization process associated with the minimumredundancy solution consists of multiplying the received vector by the receiver matrix entailing OM2 operations This complexity is high as compared to that of 11We shall address this topic in Chapter 5 in a more detailed manner 25 CONCLUDING REMARKS 53 OM log M required by traditional OFDM and SCFD transceivers This efficient equalization originates from the use of DFT matrices as well as the multiplication by memoryless diagonal matrices as explained in this chapter 25 CONCLUDING REMARKS This chapter has briefly reviewed the modeling of communications systems using the transmulti plexer frameworkThe LTI memoryless transceivers were the main focus of our presentationAmong these transceivers we particularly addressed the CPOFDM ZPOFDMOLA CPSCFD and ZPSCFDOLA transceivers highlighting their corresponding ZF and MMSE designs Some re sults taken from the open literature related to transceivers with reduced redundancy ZPZJ systems were also discussed Alessonlearnedfromthischapteristhatthe conventionalOFDMandSCFDtransceiversare very efficient since the receiver and the equalizer have very simple implementations These systems capitalize on the circulant structure of the effective channel matrix whenever a cyclic prefix of length at least L is inserted where L is the channel order The circulant matrices can be diagonalized using a pair of DFT and IDFT transformations Chapter 3 contains an indepth description of OFDM and SCFD techniques including details about the effects of employing different types of prefixessuffixes A further query is if it is possible to derive transceivers similar to the OFDM and SCFD while employing reduced redundancy and whose implementations rely on fast transforms as well The answer to such a query is yes as will be clarified later on in Chapter 4 Another relevant question is if it is possible to reduce even more the transmitted redundancy by working with timevarying transmultiplexers with memory Once again the answer is yes as described in Chapter 5 55 C H A P T E R 3 OFDM 31 INTRODUCTION As discussed in the previous chapters the orthogonal frequencydivision multiplexing OFDM is a transmission technique that is currently used in a number of wired and wireless systems This chapter describes OFDM in more detailstarting from its original conception in the continuoustime domain herein calledanalog OFDM and arriving at its current implementation in the discretetime domain In fact the discretetime description of OFDM has already been addressed in Section 24 of Chapter 2 However that description is solely based on the useful mathematical properties related to circulant matrices without necessarily calling for physical intuition of actual transmissions The focus of the present chapter on the other hand is to motivate the construction of the OFDM system by analyzing its very insightful analog version and to derive the discretetime implementation from this physically meaningful continuoustime system Indeed it was only with the widespread use of digital integrated circuit technology that the discretetime OFDM transmission technique became popular especially due to the existence of fast Fourier transform FFT algorithms which enable efficient computations of the discrete Fourier transform DFT employed for modulation From a historical perspective the origins of frequencydivision multiplexing FDM date back to the late nineteen century according to the review article by S B Weinstein 88 The analog version of OFDM was first proposed by R W Chang in 1966 10 who filed a patent that was granted in 1970 11 A major breakthrough was the perception that the use of analog subcarrier oscillators and their corresponding coherent demodulators could be avoided by replacing them by DFTbased transceivers In this context S B Weinstein and P M Ebert 89 were the originators of the DFTbased modulation and demodulation schemes Another key result related to the digital OFDM implementation was conceived by A Peled and A Ruiz 64 who advanced the use of cyclic prefix as solution for maintaining orthogonality among subcarriers at the receiver side Although the analog and digital versions of OFDM systems are closely related they are not always fully equivalent as discussed in 43 OFDM has become widely adopted in commercial applicationsthus explaining why there are so many works addressing its history 13 88 This chapter is organized as follows Section 32 describes the origins of OFDM in its analog version Such topic is particularly interesting for understanding the choices of some important parameters such as the OFDM symbol duration sampling period and guard period In addition Section 32 also introduces the importance of orthogonality in OFDM Section 33 describes the discretetime implementation of OFDM systems The idea of Section 33 is to connect what we have seen in Section 24 of Chapter 2 with the theory of analog OFDM Section 34 describes some variants of OFDMbased systems including singlecarrier with frequencydomain equalization SCFD zeropadding ZP schemes coded OFDM COFDM and discrete multitone DMT systems Finally some conclusions are drawn in Section 35 32 ANALOG OFDM Digital communications require the conversion of a discretetime signal to a continuoustime signal that is actually transmitted Such an operation is performed by a digitaltoanalog converter DAC If we assume that sm denotes the mth element of a discretetime signal with m M 0 1 M 1 and M N then the conversion to its related continuoustime baseband signal sDACt can be theoretically implemented by first multiplying each element sm by a continuoustime Dirac impulse δt mT and then passing the resulting signal through a linear timeinvariant LTI analog filter with impulse response pt In this context the positive realvalued parameter T denotes the sampling period of the DAC Mathematically we have sDACt mM sm δt mT pt mM sm δt mT pt mM sm pt mT 31 where represents linear convolution In other words the usual digitaltoanalog conversion which is always present in digital communications can be regarded as a timedivision multiplexing TDM operation of the elements which compose a discretetime signal A natural question arises at this point is there anything we can do to perform this conversion in a frequencydivision multiplexing FDM manner The answer is yes and we will show that such an FDMbased representation is a natural starting point to conceive the socalled analog OFDM 321 FROM TDM TO FDM In general the continuoustime signal st associated with a discretetime signal sm can be described as st mM sm pmt 32 where pmt is a continuoustime pulse signal The choice of the pulse signal determines how the elements of the discretetime signal are distributed over the timefrequency plane For example by choosing pmt in a TDM fashion so that pmt pt mT whose time support is the real interval mT m 1T we generate the following continuoustime signal see Equation 31 as well sTDMt mM sm pt mT 33 The former equation implies that sTDMt is a concatenation of pulses pt each of them starting at time t mT with duration of T seconds modulated by their corresponding symbol sm originating from a digital modulator From Fourier analysis we know that the Fourier transform FT of sTDMt is STDMω mM sm PωejωmT 34 where the FT of pt is represented by Pω whose bandwidth is Ω Since in TDM schemes each symbol is transmitted in a time slot with T seconds of duration then the transmission of M symbols lasts MT seconds In frequency domain each of these symbols occupies the entire available bandwidth Ω On the other hand in FDM schemes we utilize a dual strategy for signal transmission Indeed in FDM each symbol occupies a portion of the whole channel bandwidth Ω The frequency response of the transmitted signal is SFDMω mM sm Pω mΩ 35 in which the support of Pω mΩ is the real interval mΩ m 1Ω where Ω represents a fraction of the channel bandwidth Ω that is Ω MΩ A signal with such representation in the frequency domain can be written in time domain as sFDMt mM sm ptejmΩt mM sm ejmΩt pt 36 Thus FDM transmission is obtained when we choose pmt ptejmΩt which implies that the symbols sm are all superposed in time domain Equation 36 will be key to the forthcoming description of analog OFDM since it reveals clearly the central role that complex exponentials play in FDMbased transmissions Note that the existence of such complex exponentials is a natural consequence of the FDM characteristic In addition Equation 35 is related to the ideal concept of multicarrier systems which focus on dividing the available channel into many narrowband subchannels so that the channel frequency response can be considered constant in each individual subchannel In the remaining of this section it will be shown how analog OFDM exploits FDM transmissions in an efficient manner 322 ORTHOGONALITY AMONG SUBCARRIERS The starting point of analog OFDM is the term between parenthesis in expression 36 where each entry symbol of a data block modulates a subcarrier which can be interpreted as a tone In order for these symbols to be easily recovered at the receiver the subcarriers should be orthogonal The concept of orthogonality among subcarriers will be explored in the following discussion Let us consider that the OFDM subcarriers consist of equally spaced tones in frequency domain Indeed if we define fm fm1 R as the central frequencies corresponding to the mth and m 1th subcarriers respectively where m M 0 1 M 1 37 is the subcarrier index and M is a positive integer number representing the number of subcarriers then the frequency separation between two consecutive subcarriers is 1Δ fm1 fm 38 for all m M M 1 Note that by assuming that a subcarrier is comprised of a single tone let us say fm the timedomain representation of such subcarrier consists of a complex exponential at that frequency1 that is ej2πfmt The transmission of a block with M symbols belonging to a given constellation C C in which each symbol is denoted as smn C is performed by transmitting these symbols using subcarriers with distinct central frequencies In this context n is an integer number that identifies the block with M constellation symbols Such association between symbols and subcarriers is exemplified below as s0n f0 0 s1n f1 1Δ sM1n fM1 M 1Δ 39 Hence the mth symbol is associated with the subcarrier whose central frequency is fm mΔ for each m within M The nth data block to be transmitted usually called OFDM symbol2 is a complex signal denoted by ûnt in which t is a real variable representing time The OFDM symbol ûnt is generated as the superposition of the subcarriers each of them modulated by its 1Indeed a singletone signal whose tone is fm is a signal whose frequencydomain representation consists of an impulse centered at frequency fm From Fourier analysis we know that the inverse Fourier transform of an impulse at frequency fm corresponds to ej2πfmt 2The reader should not confuse the terms symbol and OFDM symbol While the former is a complex number generated at the output of a digital modulator see Section 12 the latter is associated with a collection of constellation symbols corresponding symbol smn yielding unt 1T mM smnej2πfmtnT 1T mM smnej2πΔ mtnT for nT t n1T where T is a positive real number representing the duration of an OFDM symbol In the time domain if we assume that each symbol smn represents an entry of a serial data then the time support of each symbol is TM since each OFDM symbol is comprised of M constellation symbols as described in Equation 310 The OFDM symbol duration T must be long enough to keep the subcarriers orthogonal to each other so that the individual data symbols can be extracted from the OFDM symbol Indeed for any two modulated subcarriers for example smnej2πΔmt and smlnej2πΔmlt where l is an integer number such that m l M their temporal crosscorrelation computed over the OFDM symbol duration is given by 1T nTn1T smnej2πΔmt smln ej2πΔmlt dt smnsmln 1T nTn1T ej2πΔmt ej2πΔmlt dt smnsmln 1T nTn1T ej2πΔ lt dt smn2 if l 0 0 if l 0 provided T is a multiple of Δ that is T κ Δ with κ being a positive integer Indeed the last equality follows easily by considering that ej2πΔ lt can be rewritten as ej2πΔ lt cos2πΔ lt j sin2πΔ lt and remembering that both sine and cosine functions integrate to zero in intervals corresponding to multiples of their fundamental period which in this case is given by ΔT Therefore we must choose T in Equation 311 in such a way that the crosscorrelation is equal to zero for all l M 0 This implies that the choice of T must be based on the slowest complex exponential which occurs when l 1 which in turn shows that T must be a multiple of ΔT Δ Therefore as pointed out in Equation 311 the orthogonality among subcarriers plays a key role in the choice of T and its relation with Δ Indeed this orthogonality can be obtained by choosing an OFDM symbol duration T κ Δ Note that as κ increases the OFDM symbol duration also increases but the amount of transmitted data is exactly the same that would be transmitted if κ 1 that is κ 1 reduces the system throughput That is why κ 1 is the natural choice In addition note that the orthogonality does not depend on the symbols that modulate the subcarriers This implies that the OFDM symbol unt can be redefined as unt 1T mM smnej2πT mtnT for nT t n1T 1T mM smnej2πT mt ej2πmnnT 1 for nT t n1T mM smn ptnT ej2πT mt φmtnT mM smn φmtnT where function φmt represents the mth subcarrier and the pulse signal is pt 1T for 0 t T 0 otherwise Let us interpret Equation 313 pictorially Each subcarrier φmt is a complex exponential multiplied by pt a rectangular window of duration T Figure 31a depicts a given pulse pt whose Fourier transform is the wellknown sinc function The square of the sinc represents the subcarrier spectrum as depicted in Figure 31b Figure 31c illustrates many OFDM subcarriers placed at their correct positions in order to show the distance between the central frequencies of neighboring subcarriers and to emphasize that at each subcarrier central frequency all other subcarriers have amplitude equal to zero as illustrated by the dotted lines Figure 31c illustrates several subcarriers belonging to a single OFDM symbol as if they were isolated ie we have not added the curves associated with each subcarrier However as given in Equation 313 an OFDM symbol is formed by the summation of all M subcarriers modulated by their corresponding symbol The result of such summation is represented in Figure 32 This figure depicts a frequencydomain representation of an OFDM symbol comprising three subcarriers Note that the support of such a representation is the entire real axis At this point it is worth mentioning that simple TDM and FDMbased transmissions can also yield orthogonal signals to the input of the communication channel as illustrated in the discussions of Section 15 of Chapter 1 within the framework of multipleaccess schemes Subcarrier Spectrum Pulse Shape pt 1T t T a Time domain 4T 2T 0 2T 4T f b Frequency domain c Subcarriers Figure 31 Representation of OFDM subcarriers a timedomain representation of pt b frequencydomain representation of pt and c a set of noninterfering subcarriers Equations 33 and 35 are examples of theoretically orthogonal TDM and FDM However in the case of FDM transmissions one must necessarily let empty spectral regions for separating the frequency content associated with each subcarrier otherwise the filters employed in such separation would be for certain noncausal filters due to the required sharp transitions in the frequency domain The aforementioned analog OFDM avoids this waste of spectrum that generally occurs in FDMbased systems by allowing spectrum superposition of the subcarriers In standard TDM transmissions when the transmitted signal crosses a frequencyselective channel the original timedomain orthogonality is lost A possible solution is adding guard intervals between the transmission 62 3 OFDM A B C Subcarrier C Subcarrier B Subcarrier A f Figure 32 Example of frequency representation of an OFDM symbol comprised of three subcarriers of each constellation symbol which also represents a waste of resources ActuallyTDMbased solu tions employ timedomain equalizers to decrease the interference among symbols due to the loss of orthogonality But even in this case if the interference level is too high then the order of the time domain equalizer can turn its implementation impractical Actual analog OFDM transmissions on the other hand are able to circumvent the interferences introduced by frequencyselective channels by using subcarriers that are orthogonal to each other at the receiver end thus justifying the name orthogonal FDM OFDM The key feature present in analog OFDM is the introduction of guard intervals between each OFDM symbol as explained in Subsection 323 323 ORTHOGONALITY AT RECEIVER THE ROLE OF GUARD INTERVAL So far we have seen how to design the OFDM transmitter in such a way that its subcarriers are orthogonal to each other which is an important feature that allows easy extraction of the symbols within an OFDM symbol However in practical communications systems this extraction is actually performed at the receiver endThuswe are interested in transmitting OFDM symbols that maintain their orthogonality among subcarriers when they reach the receiver end The task of keeping the subcarriers orthogonal at the receiver is very challenging especially when the OFDM symbol faces multipath fading channels As described in Section 17 this kind of channel has memory which means that delayed and attenuated versions of the transmitted signal arrive at the receiver This generates interference among transmitted constellation symbols and OFDM symbols assuming the transmission of more than one OFDM symbol An instinctive solution to maintain the orthogonality and combat interference is to extend the time support of the subcarriers Let us consider the simplest case in which a subcarrier is a simple sinusoid as depicted in Figure 33 where the solid line represents the original subcarrier with a finite time support and the Figure 33 Extending an OFDM carrier with a cyclic prefix initial dashed version represents its time extension This OFDM subcarrier extension is known as cyclic prefix CP Assuming this carrier energizes a frequencyselective channel whose timedelay spread spans up to τmem R seconds at the receiver end the first τmem seconds will be corrupted by the previous OFDM symbol thus generating the socalled interblock interference IBI After the period τmem the IBI is over and if we keep the subcarriers illuminating the channel for T seconds then the subcarriers will be orthogonal to each other as long as the time period T τmem is long enough and as long as we are able to eliminate the interference among constellation symbols within a given data block ie the socalled intersymbol interference ISI Figure 34 illustrates how OFDM symbols are concatenated taking into consideration a guard interval⁵ of duration τ R seconds The useful symbol time 𝑇 see Figure 34 corresponds to the original duration of the OFDM symbol whose subcarriers are orthogonal to each other only at the transmitter side Thus the extended OFDM symbol has duration T 𝑇 τ where the guard interval τ is longer than the longest multipath delay τ τmem and is used to avoid the harmful interferences introduced by frequencyselective channels Figure 34 Using a guard period to avoid interblock interference ⁵ A guard interval or guard period is a more general concept which includes the cyclic prefix as a special case Mathematically if we define M subcarriers separated in frequency by 1Tτ Hz as φmt ptej 2πTτ mt 315 for m M where the subcarrier index m is within the set M and the pulse signal is⁶ pt 1 T τ for τ t T τ 0 otherwise 316 then the transmitted signal which is the concatenation of extended OFDM symbols can be written as ut n Z unt n Z m M smn φmt nT 317 unt where smn is the mth symbol within the nth block representing an extended OFDM symbol An extended OFDM symbol obtained by the extension of each subcarriertime support is equivalent to a timedomain signal with a cyclic prefix of length τ ie in which the first τ seconds of the data block coincide with the last τ seconds Indeed consider the nth extended OFDM symbol whose time support is nT τ nT T τ Let t be an arbitrary real number within the first τ seconds of that interval ie t nT τ nT In addition let t be a real number defined as t t T τ which denotes a time instant within the last τ seconds of the referred block Thus we have unt m M smnpt nT ej 2πTτ mtnT m M smn 1T τ ej 2πTτ mtTτnT m M smn 1T τ ej 2πTτ mtnT ej2πm 1 unt 318 A detailed representation of an extended OFDM symbol⁷ using cyclic prefix is depicted in Figure 35 The reader should note the similarities and differences between Equations 313 and 317 The signal ûnt in Equation 313 is simply an OFDM symbol ie no guard period is ⁶ We considered the interval τ T τ instead of 0 T since this choice simplifies the forthcoming notations used in Section 33 ⁷ In some texts especially in standards the extended OFDM symbol is simply referred to as OFDM symbol Figure 35 Cyclic prefix in an OFDM symbol being used and as a consequence the received signal corresponding to this ûnt after crossing a multipath fading channel would not yield orthogonal subcarriers On the other hand the signal ut in Equation 317 is the concatenation of infinitely many extended OFDM symbols using cyclic prefix in order to ensure orthogonality among subcarriers at the receiver side Indeed the orthogonality among subcarriers is maintained since the interference between OFDM symbols ie the IBI can be eliminated by discarding the first τ seconds out of T seconds of each received data block As for the remaining interference ISI due to constellationsymbol superpositions within the resulting block of duration T τ we can eliminate it using the subcarrier orthogonality In order to verify mathematically these facts let us analyze the received signal yt assuming a noiseless baseband channel model whose timedelay spread is τmem as follows yt H ut H n Z m M smnφmt nT n Z m M smnH φmt nT m M n Z smnH φmt nT 319 where H represents the linear system that models the referred baseband channel Now by considering that the channel model remains constant during the interval of an OFDM symbol we can compute the quantity H φmt nT through a convolution integral of φmt nT with the channel impulse response hnt associated with the nth symbol Therefore we have yt summ in M sumn in mathbbZ smnej frac2 piT au mtnT intinfty aumem hn au ptnT au ej frac2 piT au m au d au summ in M sumn in mathbbZ smn ej frac2 piT au mtnT int0 aumem hn au ptnT au ej frac2 piT au m au d au 320 where hn au 0 for all au otin 0 aumem and ptnT au left beginarraycc frac1sqrtT au extfor t aun1T leq au t aunT 0 extotherwise endarray right 321 As we are studying both interblock and intersymbol interferences associated with the transmission of blocks unt with n in mathbbZ of length T it is convenient to separate the received signal into blocks with the same length T Thus we can write yt sumn in mathbbZ ynTt underset riangle ynt sumn in mathbbZ ynt 322 where the time instant t is in the interval au T au This way it follows from Equation 320 that the nth received block can be expressed by ynt summ in M sumn in mathbbZ smn ej frac2 piT au mnnTt Innt 323 with Innt riangleq int0 aumem hn au p left nnT t au right ej frac2 piT au m au d au 324 Observe that for n geq n1 we have pnnT t au eq 0 only for some values of au outside the integration interval 0 aumem The same occurs for n leq n2 Indeed based on Equation 321 one has that the pulse signal pnnT t au assumes nonzero values only for values of au such that nn1T t au au leq nnT t au Therefore if n geq n1 then the upper bound for au will be such that nnT t au leq t au T tT au 0 since t in au T au which means that au cannot be within the interval 0 aumem for this 8Actually it would be more appropriate to state that hn au is approximately zero for all au otin 0 aumem condition Similarly if n leq n2 then the lower bound for au will be such that nn1T t au geq t au T aumem which means that au cannot be within the interval 0 aumem in this case as well Hence the integrand which appears in the above integral Innt is always zero except for n n and n n1 When n n then the pulse signal pnnT t au is nonzero whenever au is such that t au T 0 au leq t Similarly when n n1 the interval is 0 t au au leq t au T aumem Therefore we have Innt fracdelta nnsqrtT au int0min t au aumem hn au ej frac2 piT au m au d au fracdelta nn1sqrtT au intmin t au aumem aumem hn au ej frac2 piT au m au d au 325 where delta n denotes the Kronecker delta which is defined as delta n riangleq left beginarraycc 1 extif n0 0 extotherwise endarray right 326 Now if we consider only time instants t within the interval 0 T au with au aumem then min t au aumem aumem and therefore the second integral which appears at the righthand side of Equation 325 will be zero In other words when we discard the first au aumem seconds of each received block ynt then there is no interference between the nth and n1th transmitted OFDM symbols ie for any t in 0 T au we have Innt fracdelta nnsqrtT au int0 aumem hn au ej frac2 piT au m au d au underset riangle Hn left frac2 pi mT au right fracdelta nnsqrtT au Hn left frac2 pi mT au right 327 so that see Equation 323 as well ynt frac1sqrtT au summ in M Hn left frac2 pi mT au right smn ej frac2 piT au m t 328 in which once again we highlight that t must be in the interval 0 T au From Equation 328 it is clear that for each fixed time interval nT au nTT au if one discards the first au aumem seconds of the received signal in that interval then one ends up with a continuoustime signal that is composed of a sum of M complex exponentials modulated by complex numbers smn Hn left frac2 pi mT au right Hence if we compute the temporal crosscorrelation over a useful symbol duration hatT T au for any of those two modulated subcarriers we would reach a similar result to Equation 311 This means that the orthogonality between subcarriers is achieved at the receiver side This is the main feature of analog OFDM since the IBI is eliminated and at the same time the ISI can be eliminated by using the resulting orthogonality among subcarriers at the receiver Figure 36 Timefrequency map of analog OFDM signals Therefore analog OFDM transmissions illuminate the channel at each T hatT au period of time with au aumem using M subcarriers whose central frequencies of neighboring subcarriers are separated by Deltaf riangleq frac1hatT frac1T au as illustrated in Figure 36 Each subcarrier will be responsible for transporting a single symbol during a time slot taking into consideration the longest path propagation time aumem Note that the extended OFDM symbols do not have orthogonal subcarriers at the transmitter due to the insertion of the guard interval This can be verified through different ways For example if one computes the temporal crosscorrelation between two distinct subcarriers described in Equation 315 one would end up with a nonzero temporal crosscorrelation in general This occurs because we do not integrate over an integer multiple of the period T au Another way of verifying this fact is to observe Figure 37 in which the timedomain and frequencydomain representations of the pulse pt in extended OFDM symbols are depicted As the subcarrier central frequencies are proportional to frac1T au and the zeros of the subcarrier spectra are proportional to frac1T then there 32 ANALOG OFDM 69 1 T pt Pulse Shape t T a Time domain 2 T Subcarrier Spectrum 0 4 T 4 T f 2 T T sin πfT πfT 2 b Frequency domain Figure 37 Representation of extended OFDM subcarriers a timedomain representation of pt and b frequencydomain representation of pt will exist interferences from adjacent subcarriers at those central frequencies This means that the subcarriers are not orthogonal at the transmitter side However this is not an issue since the infor mation extraction occurs at the receiver Thus the orthogonality among subcarriers at the receiver end allows a proper extraction of the symbols associated with different subcarriers within a received extended OFDM symbol even when the received signal has been severely distorted by the channel That is why we exchange the original orthogonality present in OFDM symbols by an orthogonality of the received extended OFDM symbols with cyclic prefix after discarding the first τ seconds 324 SPECTRAL EFFICIENCY PAPR CFO AND IQ IMBALANCE Before we move on to describe an implementation sketch of analog OFDM let us briefly comment on its spectral efficiency and some of the OFDM issues namely peaktoaverage power ratio PAPR carrierfrequency offset CFO and inphasequadraturephase IQ imbalance Let us start with the spectral efficiency We know that during a period of T seconds OFDM transmits M symbols from a given constellation with 2b pointswhere b is a natural number denoting the number of bits required to represent a single symbol Thus the OFDM bit rate BR is BR Mb T 329 in bits per second bps If we add the spectra of all subcarriers and consider that side lobes below 20 dB from the main lobe are negligible corresponding to the second side lobes on each side of the main lobe then the total bandwidth is⁹ BW M 1 T τ 2 3 T The ratio BR BW Mb M1T Tτ 6 b M1M T Tτ 6 M is the socalled spectral efficiency which tends to b1 τ T for large M That means the OFDM is an optimal modulation in terms of spectral efficiency as long as τ T 1 However when τ T very dispersive environment then the spectral efficiency of OFDM transmissions is quite small Other multicarrier and singlecarrier transmissions which address this drawback of spectral efficiency will be described in Chapters 4 and 5 Figure 38 depicts the instantaneous power of a given OFDM symbol unt A dotted line is used to represent the average power of the OFDM symbol In this figure we can observe that there exist some peaks in the power of unt that are much higher than the average power ie they are well above the dotted line Indeed it is well known that the peaktoaverage power ratio PAPR of OFDM transmissions is higher than the PAPR of singlecarrier transmissions see Subsection 341 for further details in the discretetime domain High PAPR is undesirable because it implies a wide dynamic range of the signal to be transmitted which in turn requires power amplifiers with linear response over a wide range increasing the cost of such devices This is one of the main reasons why in LTE the use of OFDMA in the uplink was avoided Therefore PAPR is an important impairment related to OFDM transmissions In cases where the carrier frequency of the received signal does not match the carrier frequency of the transmitted signal we have the socalled carrierfrequency offset CFO Thus CFO is the offset difference between two numbers representing carrier frequencies one at the transmitter and the other at the receiver end Ideally CFO should be close to zero but there are many practical cases in which nonnegligible CFO occurs For example when the transmitter andor receiver are moving which usually happens in mobile communications the Doppler effect acts as a source of CFO Note that from our previous discussion about the importance of orthogonality in OFDM systems it is rather intuitive that CFO has the potential to severely degrade the quality of OFDM transmissions Most of the solutions to the CFO issue rely on blind estimation of the frequency offset and are a bit complex andor applicable to very particular cases Some of the lowcomplexity solutions to CFO are presented in 46 92 ⁹The bandwidth must be computed by considering that the central frequencies of the M subcarriers are separated by 1 T τ Hz and each subcarrier is a sinc whose second side lobes decay more than 20 dB at 3 T Hz as can be observed in Figure 37b Figure 38 Instantaneous power of a single OFDM symbol Moreover digital transmissions usually employ two branches an inphase I and a quadraturephase Q branch These branches are associated with the real and imaginary parts of the transmitted signal respectively IQ imbalance occurs when there is phase andor amplitude mismatches between I and Q branches Such mismatches are usually due to the imperfections in the process of the radiofrequency signal downconversion to baseband signal and are therefore unavoidable in the analog frontend 74 In most cases IQ imbalance can only deteriorate the biterror rate BER performance of OFDM systems when they are employing highorder modulation schemes such as 64QAM quadrature amplitude modulation When IQ imbalance is a major issue one can use digital signal processing techniques to compensate such mismatch Indeed there already exists several techniques to compensate for the IQ imbalance without increasing significantly the computational burden For more details see 6 78 85 and references therein 325 IMPLEMENTATION SKETCH Figure 39 depicts the implementation sketch of an analog OFDM transmission scheme First the symbols smn modulate their corresponding timeextended subcarriers φmt nT This operation is represented by the filtering of the continuoustime signal smnδt nT through the analog filter φmt The signals resulting from each of these modulations are then added forming an extended OFDM symbol which crosses an analog channel and is then corrupted by the environment noise vt Here we assume that the length of the cyclic prefix is greater than the channel timedelay spread At the receiver end there are filters ψmt which are responsible for discarding the cyclic prefix as well as extracting the symbol smn from the corrupted OFDM symbol Indeed by defining the receiver filters as ψmt qtej2πTτmt 332 for m M where qt 1sqrtTτ for 0 t T τ 0 otherwise the output of the filter ψmt will be equivalent to the temporal crosscorrelation between the OFDM symbol and the mth receiver filter Following the same steps performed in Equation 311 and remembering that the fundamental period of the slowest subcarrier is Δ T τ it is easy to verify that ψmt will remove all subcarriers except φmt due to orthogonality In fact the pair of functions φmt and ψmt are biorthogonal 17 In addition note that the basis function at the receiver ψmt has a time support shorter than the basis function at the transmitter φmt Figure 39 Analog OFDM implementation sketch Even though analog OFDM can be derived from a very insightful view of the digitaltoanalog conversion implemented in an orthogonal FDMbased fashion the resulting implementation sketch depicted in Figure 39 also summarizes its main drawback in general practical solutions entail the use of a large number of orthogonal subcarriers thus hindering the applicability of this structure in practice Indeed if M is large then we would have to implement a large amount of different oscillatorsmodulators which may not be practical This is one of the main reasons why this analog version of OFDM was not employed in commercial applications after its proposal by R W Chang 10 11 However many of its properties and interpretations are still useful as S B Weinstein and P M Ebert 89 as well as A Peled and A Ruiz 64 noticed when they realized that OFDM could be efficiently implemented in the discretetime domain This implementation will be addressed in the next section 33 DISCRETETIME OFDM 73 33 DISCRETETIME OFDM Digital signal processing DSP has emerged as a powerful and efficient tool in a growing number of applications Indeed there are many situations where the use of DSPbased techniques has either greatly simplified the implementation of practical systems or simply enhanced their performance The discretetime implementation of OFDM systems is an example of how DSP can even enable the practical usage of a given technique which could be quite hard to implement otherwise In this section we will apply sampling to the OFDM symbols and show a discretetime implementation of OFDM systems As will be shown the discrete Fourier transform DFT plays a central role in this process enabling efficient implementations of OFDM by means of fast Fourier transform FFT algorithms 331 DISCRETIZATION OF THE OFDM SYMBOL As in the continuoustime case let us start with OFDM symbols without guard intervals described in Subsection 322 The first challenge that one faces while trying to derive a discretetime imple mentation of OFDM is that the OFDM symbols are not bandlimited Indeed due to the use of a timedomain rectangular window ˆpt as illustrated in Figure 31a each subcarrier in an OFDM symbol and the entire OFDM symbol itself contain spectral components at infinitely large frequen ciesas depicted in Figures 31b and 31cThereforea uniform sampling of such continuoustime signal would invariably result in aliasing effects regardless of the particular sampling frequency em ployed In other words no discretetime implementation of OFDM obtained by sampling the analog OFDM can be used to recover the same original continuoustime signal associated with analog OFDM which employs rectangular windows In general the recovering process is implemented as a digitaltoanalog conversion such as the one described in expression 31 with the constellation symbols being replaced by the elements that compose an OFDM symbol Such impossibility is also pointed out in 43 In principle the presence of aliasing effects could hinder any interpretation based on the intuitive results of analog OFDM presented in Section 32 Fortunately the aliasing effects are not determinant here and the reason is quite simple to understand the original signal to be transmitted smn has a discrete nature which means that we are not interested in the particular form of the continuoustime OFDM symbol as long as we can detect the original discretetime signal10 It turns out that each element of this discrete signal is the scaling factor of the subcarrier spectra at their central frequencies Hence if the spectral components at the subcarrier central frequencies do not suffer from aliasing effects then we could recover the discretetime signal smn Once again the orthogonality of the subcarriers in an OFDM symbol plays an important role here see Figure 31c 10This is true only for detection purposes If one is interested in studying the spectral rolloff related to the output of OFDM systems or the effect of CFO just to mention a few examples then one should work directly with the discretetime OFDM model after the digitaltoanalog conversion 43 3 OFDM Indeed when we sample an OFDM symbol the spectral repetitions due to the sampling process are spaced apart by fs Hz where fs 1 T denotes the sampling frequency and Ts R denotes the sampling period First of all fs must be larger than M 1 T otherwise we would have at least one central frequency being shared by two distinct subcarriers constituting a harmful type of aliasing in this case But if we consider any integer multiple of 1 T larger than M 1 then we would not have any kind of interference at the subcarrier central frequencies For simplicity reasons it is better to use the smallest sampling frequency that does not cause interference at the subcarrier central frequencies ie fs M T Thus we have Ts T M 334 Without loss of generality since the OFDM symbols are nonoverlapping in time we can analyze each block of symbols separately The nth OFDM symbol is nonzero only for time instants within the interval n T n 1 T This means that for all t 0 T we have from Equation 313 that ûnt n T m M smn φmt n T n T m M smn φmt 335 Now let ûnk be the discretetime signal stemming from sampling the continuoustime signal ûnt n T described in the above equation at each time instant t kTs where due to the definition of Ts in Equation 334 and as t 0 T we must have k within the set M Hence the resulting discretetime representation of an OFDM symbol is ûnk ûnkTs n T m M smn φmkTs φmk m M smn φmk 336 where based on Equation 313 we have that the discretetime version of the mth subcarrier in an OFDM symbol is given as φmk 1 T ej 2π T mk T M 1 T ej 2π M mk 337 11 Here sampling means to multiply the continuoustime function by a train of Dirac impulses 33 DISCRETETIME OFDM Thus we can rewrite the discrete OFDM symbol in a more convenient manner as follows ûnk 1 T m M smn ej 2π M mk MIDFT smnk M T IDFT smnk 338 for each k M where the inverse DFT IDFT of the discretetime signal smn with m M is also a sequence with length M Hence the following relation also holds smn T M DFT ûnkm 339 in which the DFT of the sequence ûnk is also a sequence whose mth element is defined as DFT ûnkm 1 M k M ûnk ej 2π M mk 340 thus implying that the symbols smn can be actually written as smn T M k M ûnk φmk 341 In summary the discretetime version of OFDM symbols without guard intervals is easily computed through an IDFT of the transmitted discretetime signal which can be efficiently implemented by using an FFT algorithm Thus assuming the channel introduces no distortion on the transmitted signal the sequence of symbols smn could be recovered at the receiver end by taking the DFT of the sequence ûnk and scaling the result as shown in Equation 339 This is a rather obvious conclusion since the DFT and IDFT are inverse operations Note that the discretetime OFDM symbols ûnk in expression 336 can be thought as the superposition of M subcarriers φmk modulated by the symbols smn It is worth mentioning that the original orthogonality present in the analog OFDM symbol is preserved in the discretetime case Indeed we can see that the following relation between any two subcarriers with indexes i j M is valid k M φjk φik 1 T k M ej 2π M kj i 1 T ej 2π M j i 1 ej 2π M j i 1 if i j 0 if i j M T otherwise M T δi j 342 Equation 342 is nothing but the orthogonality between subcarriers which are synchronized and have the same duration T This expression stems from the projection of a transmitted signal onto a subcarrier for detection purposes at the receiver end as exemplified in Equation 341 For analog OFDM we have shown that the subcarrier orthogonality at the transmitter side is not sufficient to allow the detection of the transmitted symbols in practical situations ie when the data faces a frequencyselective channel This occurs since frequencyselective fading channels extend the time support of the transmitted signals generating both IBI and ISI as explained in Subsection 17 In Subsection 323 we ensure subcarrier orthogonality at the receiver by inserting a cyclic prefix before transmission In the following subsection we shall generate a discretetime version of the results related to extended analog OFDM symbols using cyclic prefix 332 DISCRETIZATION AT RECEIVER THE CPOFDM Our starting point in this subsection is the expression 328 of the received block after removing the first τ seconds As pointed out before τ is chosen in such a manner that it is larger than the delay spread of the channel τmem Let L be a positive integer number defined as L τmem Ts 343 where Ts T M is the sampling rate associated with the discretization process In addition let K be a positive integer number such that K L Thus by choosing the length of the cyclic prefix as τ K Ts 344 then we have that τ τmem and therefore expression 328 holds for any t 0 T τ Now by remembering from Subsection 323 that the useful symbol time T is given by T τ then the sampling rate can be written as Ts T τ M 345 which means that 0 T τ 0 MTS Let ynk be the discretetime signal originating from sampling the continuoustime signal yn t described in Equation 328 at each time instant t kTs with k M As explained before this continuoustime signal is the n th received block after removing the first τ KTs seconds Thus the resulting discretetime representation of the n th received extended OFDM symbol after removing the first K elements12 is ynk Δ yn kTs 1 T τ mM Hn 2π mT τ sm ne j 2π M τ τ T mk 1 T τ mM Hn 2π mT τ sm ne j 2π M mk mM Hn 2π mT τ sm n e j 2π M mkT τ mM λm nsm nφmk 346 Equation 346 means that the received signal after removing the first K elements associated with the guard interval introduced at the transmitter side is composed of M modulated subcarriers φmk Even though the kth element of such signal is affected by all transmitted symbols sm n with m M one can use the subcarrier orthogonality expressed in Equation 342 in order to recover a scaled version of the transmitted symbols without ISI Therefore by projecting the received signal onto the m th subcarrier one gets λm nsm n T τ M kM ynkφmk 347 The former projection process can be implemented in a much more efficient way Indeed based on Equation 346 and following a similar approach which was employed in Subsection 331 we can rewrite the received discrete OFDM symbol in a more convenient manner as follows ynk MT τ IDFT λm nsm nk 348 for each k M Therefore the following relation also holds λm nsm n T τM DFT ynkm 349 12If we are ignoring the first τ KTs seconds of the continuoustime received block then we are ignoring the first K elements of the related discretetime signal with corresponding sampling rate of 1Ts Hz Hence in order to recover sm n all we need is to multiply the m th element of the DFT of the received OFDM symbol by 1λm n assuming λm n 0 for all m M This is the socalled zeroforcing ZF equalizer In fact there are many other ways to perform equalization in order to estimate sm n The choice of the equalizer depends on the types of distortion faced by the transmitted signal For instance in our previous discussion we have neglected the existence of additive noise In the presence of such type of noise an equalizer that minimizes the mean square error MSE would be more appropriate than an equalizer that eliminates only the ISI such as the ZF equalizer In this case the equalization would consist of multiplying the m th element of the DFT of the received OFDM symbol by λ m nλm n2 σv2 nσs2 n 350 where σs2 n and σv2 n represent the variance of symbols and noise respectively13 This type of equalizer is known as minimum MSE MMSE equalizer as previously discussed in Subsection 241 As explained in Subsection 323 the subcarrier orthogonality at the receiver end is obtained by including a cyclic prefix of length τ at the transmitter end generating the extended OFDM symbols Thus by following the same steps employed in Subsection 331 but now considering the nth extended OFDM symbol un t of Equation 317 we have un t nT mM sm nφm t nT nT mM sm nφm t 351 for all t τ T τ KTs MTS Hence for each k K 1 0 M 1 we can define the discretetime representation of an extended OFDM symbol as see Equation 315 unk Δ un kTs nT 1T τ mM sm ne j 2π M mk 352 where for k M one has mM sm ne j 2π M mk M IDFT sm nk 353 13It is assumed here that all constellation symbols within the nth OFDM symbol have the same variance σs2 n In addition it was assumed that all noise components have the same variance σv2 n as well Those assumptions are not necessary but they simplify the notation See Subsection 344 for the case where we do not consider those equalpower assumptions whereas for k K 1 one has mM sm ne j 2π M mk mM sm ne j 2π M mk e m M m 1 mM sm ne j 2π M M km M IDFT sm nMk 354 with M k M K M 1 M assuming that M K ie the useful symbol duration is larger than or equal to the duration of the guard interval Therefore we can use these results in Equation 352 so that for each k K 1 0 M 1 we have unk MT τ IDFT sm nk 355 in which the above notation considers the inherent periodicity property of the IDFT ie IDFT sm nk IDFT sm nMk for k 1 K Therefore the first K elements of the signal unk are equal to its last K elements thus characterizing the type of guard interval as cyclic prefix CPOFDM In the discretetime domain this guard interval is also called redundancy since it corresponds to entries that do not carry additional information So far we described the discretetime versions of the transmitter and receiver We shall address the channel model in Subsection 333 333 DISCRETETIME MULTIPATH CHANNEL The term λm n which appears in expression 346 is associated with the frequency response of the analog channel evaluated at the central frequency mTτ Hz for each subcarrier index m M and for each block n Z The aim of this subsection is to show the relation between λm n and the discretetime model of the analog baseband channel In actual discretetime OFDM implementations one must always associate the discrete OFDM signal unk with a related continuoustime signal let us say ūt As already mentioned due to aliasing effects this continuoustime version is not the analog OFDM symbol presented in Section 32 The standard way to perform such conversion is through a DAC process as briefly described in Subsection 321 so that ūt Δ nZ M1 kK unk pT t k nNTs 356 where N Δ M K 357 34 OTHER OFDMBASED SYSTEMS 91 P S S P T D F T D F I Cyclic Prefix Digital Channel Equalizer Receiver Transmitter Prefix Remove s0n s1n s2n sM1n sM2n ˆs0n ˆs1n ˆs2n ˆsM2n ˆsM1n Figure 311 SCFD digital transceiver with CP properly designed equalizer Thus in general ZPbased transceivers do not yield so simple one tap equalizers as CPbased systems do These observations are valid for the general ZP versions of the OFDM and SCFD herein called ZPOFDM and ZPSCFD respectively In fact there are some tools related to structured matrix representations which can work around these difficulties yielding general ZPtransceivers which are still based on DFTs and onetap equalizers see Figure 43 of Chapter 4 However there are particular versions of ZPOFDM and ZPSCFD that have the same simple equalizers as the ones used in CPOFDM and CPSCFD systems These particular versions known as ZPOFDMOLA and ZPSCFDOLA perform overlapandadd OLA operations at the receiver side and they are depicted in Figures 312 and 313 The mathematical details regarding these particular versions of ZPOFDM and ZPSCFD were already explained in Subsections 242 and 243 respectively The following topics are research results concerning the differences between the general ZP OFDM and ZPSCFD and their cyclicprefix counterparts 55 87 ZPOFDM introduces more nonlinear distortion than CPOFDM transceivers 92 3 OFDM P S T D F I S P T D F Processing Post Digital Channel Equalizer Transmitter Receiver Zero s0n s1n s2n sM1n sM2n ˆs0n ˆs1n ˆs2n ˆsM2n ˆsM1n Padding Figure 312 OFDM digital transceiver with ZP ZPOFDM has better performance in terms of BER or MSE than CPOFDM for a given averagebitenergytonoise power ratio EbN0 The ZPSCFD has lower PAPR presents robustness to CFO and has also better uncoded performance However the equalization is a bit more complex to implement In the case some kind of channel coding is included COFDM the coded version is better when code rate is low and the error correcting coding capability is enhanced In the coded case ZPSCFD is better than COFDM for high code rate ZPSCFD only has clear performance advantages over uncoded OFDM Uncoded CPOFDM is inferior to zeroforcing equalized CPSCFD transceiver CPOFDM with equalgain power allocation has the same performance as zeroforcing equal ized CPSCFD transceiver 34 OTHER OFDMBASED SYSTEMS 93 P S S P T D F T D F I Processing Post Digital Channel Equalizer Receiver Transmitter Zero s1n s2n s0n sM2n sM1n ˆs0n ˆs1n ˆs2n ˆsM2n ˆsM1n Padding Figure 313 SCFD digital transceiver with ZP CPOFDM with AMBER an approximately minimum BER power allocation is better than zeroforcing equalized CPSCFD 343 CODED OFDM The usual complete coded OFDM COFDM system utilized in some broadcast systems is depicted in Figure 314 where two levels of coding and an interleaving are employed These building blocks are required to protect the transmitted information against distortions and deep fadings in some subcarriers 13 20 COFDM includes a few building blocks as described in 13 The outer encoder coder 1 is meant to insert redundancy in the data stream eg using ReedSolomon codes 38 90 This will increase the required bandwidth for transmission but on the other hand allows more reliable receptionThe aim is either to obtain block codes with increased Hamming distances or to encode the data in a continuous way such that a correlation between the data is induced to help detection at the receiver end The inner encoder coder 2 is meant to increase the Euclidean distance among the symbols of the constellation usually employing a trellis coded modulationThe idea is to increase the number of possible symbols in comparison with the number of points in the given constellation and subdivide 94 3 OFDM Deinterleaver Inner Decoder OFDM Demodulator Interleaver OFDM Modulator Outer Encoder Outer Decoder Inner Encoder OFDM Transmitter bitstream bitstream Channel OFDM Receiver Figure 314 Coded OFDM COFDM system them in subsets with greater Euclidean distances while maintaining the transmission energy The subsets and the points in the subsets are tied using a trellis diagram derived with the convolutional code If a Viterbi decoder 38 is used at the receiver some coding gain is expected 13 Error bursts could occur if the inner decoder decoder 2 chooses a wrong decoding path which could then be corrected by the combination of the interleaver and the outer decoder decoder 1 In the final analysis the reason for coding is to provide a link among the symbols transmitted on different subcarriers such that a symbol transmitted in a strongly faded subcarrier ie λmn 0 can be recovered at the receiver by estimating the symbols transmitted by other subcarriers The bottom line is that we are distributing and mixing the transmitted information to increase the chance of proper detection at the receiver end 344 DMT A discrete multitone DMT transceiver is essentially an OFDM system comprised of three particular features i there is no passband conversion to a higher carrier frequencywhich means that the actual transmittedsignalsarebasebandsignalsiisinceanyactualtransmissionemploysrealvaluedsignals then the baseband transmitted signals must be realvalued This means that the input constellation symbols smn must have the conjugate symmetric propertyiesmn s Mmnfor all m M and iii there is some kind of channelstate information CSI at the transmitter side so that the transceiver can use some smart techniques in order to cope with possible channel impairments in advance The third DMT property above is indeed its key feature since it enables transmissions with higher data rates Nevertheless it is usually applicable in wired connections in which the channel state does not change too often As described in Subsection 131 of Chapter 1 the DMT system is currently employed in many digital subscriber line xDSL applications The aim of this subsection is to describe how DMTbased systems take into account the availability of information about the channel at the transmitter side in order to enhance the overall transmission performance As described in Subsection 334 OFDM systems can be thought as M parallel uncou pled subchannels whose mth received signal of the nth transmitted OFDM symbol is given by 102 3 OFDM Subchannel index 6 2 0 3 7 4 1 5 λmn2 00 20 40 20 00 20 40 20 σ 2vmnλmn2 004 002 004 004 002 004 p0n 004 Initial σ 2smn 0005 0005 0005 0005 0005 0005 0005 0005 Initial SNRmn 000 012 025 012 000 012 025 012 Initial Cmn 000 017 032 017 000 017 032 017 Initial Cn 01654 Optimized σ 2smn 000 000 002 000 000 000 002 000 Optimized SNRmn 000 000 100 000 000 000 100 000 Optimized Cmn 000 000 100 000 000 000 100 000 Optimized Cn 02500 p0n 040 Initial σ 2smn 005 005 005 005 005 005 005 005 Initial SNRmn 000 125 250 125 000 125 250 125 Initial Cmn 000 117 181 117 000 117 181 117 Initial Cn 10368 Optimized σ 2smn 000 006 008 006 000 006 008 006 Optimized SNRmn 000 150 400 150 000 150 400 150 Optimized Cmn 000 132 232 132 000 132 232 132 Optimized Cn 12414 2 A NOTE ABOUT OFDM SYSTEMS AND BEYOND Before closing this chapter let us comment on the OFDMbased schemes that we have just pre sented There is a growing demand for transmission resources which shows no sign of settling In the case of wireless data services for instance it is possible to predict that spectrum shortage is a sure event in the near future As we mentioned before the first step to address this problem is to choose a modulation scheme that is efficient in terms of channel capacity in bitstransmission particularly in broadband transmissions where the channel presents a frequencyselective model In such cases multicarrier communication systems is a very smart solution given that this modulation scheme is efficient for data transmission through channels with moderate and severe ISI The most widely used and simplest multicarrier system is the OFDM comprising its variants presented in this chapter Whenever the subchannels are narrow enough it is possible to consider that each subchannelfrequency range is flat avoiding the use of sophisticated equalizers In appli cations where there is a return channel it is also possible to exploit the SNR in different subchannels in order to select the subchannelmodulation order so that subchannels with high SNR utilize highorder modulation whereas loworder modulation should be employed in subchannels with moderate SNR This loading strategy enables more effective usage of frequencyselective channels 35 CONCLUDING REMARKS 103 In OFDM a serial data stream is divided into blocks where each block is appended with redundancy guard interval in order to avoid IBIThe special features of the OFDM system are the elimination of IBI ISI and its inherent low computational complexity OFDM system utilizes IDFT at the transmitter and DFT at the receiver enabling the use of computationally efficient FFT algorithms The key strategy to avoid IBI in transmissions through frequencyselective channels is to include some redundancy at the transmitter thus reducing the data rate The SCFD scheme has also emerged as an alternative solution to overcome some drawbacks inherent to OFDMbased systems such as PAPR and CFO Also for some frequencyselective channels the BER of an SCFD system can be lower than for the OFDM particularly if some subchannels have high attenuation In this solution the data stream is inserted with redundancy to avoid IBI and at the receiver the equalization is performed in the frequency domain keeping the efficient equalization scheme inherited from OFDM A way to postpone the spectrum shortage is to increase the data throughput for a given bandwidth by utilizing some smart technology Assuming we are employing block transmission it is worth discussing how to reduce the amount of redundancy in multicarrier systemwhile constraining the transceiver to employ fast algorithms A possible solution is to employ reducedredundancy transceivers utilizing DFTs and diagonal matrices The reducedredundancy transceiver allows for higher data throughput than OFDM in a number of practical situations A computationally efficient solution for reducedredundancy transceiver is presented in Chapter 4 35 CONCLUDING REMARKS In this chapter we addressed distinct aspects of OFDM systems for example continuoustime analog OFDM discretetime OFDM coded OFDM COFDM and DMT We derived the discretetime OFDM as the sampled version of the analog OFDM and discussed the issue of ensur ing orthogonality among subcarriers in frequencyselective channelsThe COFDM was introduced as a discretetime OFDM in which we include some channel encoding elements inner and outer coders interleavers andor scramblers in order to protect the information to be transmitted In addition the DMT can be seen as a discretetime OFDM system in which the transmitter has channel state information and therefore it is capable of performing optimal power allocation thus increasing the channel capacity given a power constraint All these systems require the use of a guard period whose time extension is greater or equal to the channel memory or time delay spread for the analog OFDM Therefore the guard period τ increases with the channel memory τmem As a consequence for an OFDM symbol of a given duration T the useful symbol time T τ decreases thus reducing the system capacity In other words when the durations of the OFDM symbol and the channel impulse response are of the same order the throughput of OFDM systems decreases In order to address this issue the next chapter describes systems whose computational complexity are comparable to the OFDM complexity with the advantage of employing a reduced number of redundant elements that is their guard time is shorter 105 C H A P T E R 4 Memoryless LTI Transceivers with Reduced Redundancy 41 INTRODUCTION Multicarriermodulation methods are of paramount importance to many datatransmission systems whose channels induce severe or moderate intersymbol interference ISI As previously discussed the key idea behind the success of multicarrier techniques is the partition of the physical channel into ideally noninterfering narrowband subchannels If the bandwidths associated with these subchannels are narrow enough then the related channelfrequency response of each subchannel appears to be flat thus avoiding the use of sophisticated equalizers In addition the subchannel division allows whenever possible the exploitation of signaltonoise ratios SNRs in the different subchannels by managing the data load in each individual subchannel The orthogonal frequencydivision multiplexing OFDM described in Chapter 3 is the most popular multicarriermodulation technique OFDMbased systems feature lots of good properties regarding their performance and implementation simplicity However the OFDM has some draw backs such as high peaktoaverage power ratio PAPR high sensitivity to carrierfrequency offset CFO and possibly significant loss in spectral efficiency due to the redundancy insertion re quired to eliminate the interblock interference IBI The singlecarrier with frequencydomain equalization SCFD is an efficient transmission technique which reduces both PAPR and CFO as compared to the OFDM system These advantages are attained without changing drastically the overall complexity of the transceiver as shown in Chapter 3 Alternatively one could consider the general transmultiplexer TMUX framework described in Chapter 2 in order to conceive new multicarriermodulation techniques which are able to cir cumvent some of the OFDM limitations For example intuition might suggest that we can reduce intercarrier interference ICI caused by the loss of orthogonality between the subcarriers as long as we are able to design TMUXes containing a large number of subchannels with sharp transitions Therefore it would be possible to achieve better solutions in terms of biterror rate for example for filterbank transceivers to be employed in both future generations of wireless systems and current developments of localbroadband wireless networksThis strategy however remains to be proved as a viable solution in practice In fact one should always take into account the fundamental tradeoff between performance gains and cost effectiveness from a practical perspective The computational complexity1 is among the factors that directly affects the cost effectiveness of new advances in com 1Number of arithmetical operations employed in the related processing 106 4 MEMORYLESS LTI TRANSCEIVERS WITH REDUCED REDUNDANCY munications TMUXes with sharp transitions must necessarily have long memory which might hinder their use in practiceThis explains why memoryless linear timeinvariant LTI TMUXes are still preferred in many practical applications The price paid for using memoryless LTI TMUXes is that some redundancy must be nec essarily included at the transmitter end in order to allow the elimination of the ISI induced by the channel Redundancy plays a central role in communications systems Channelcoding schemes are good examples of how to apply redundancy in order to achieve reliable transmissions In addition redundancy is also employed in many blockbased transceivers in practice such as OFDM and SCFD systems in order to eliminate the inherent IBI and to yield simple equalizer structures Regarding the spectralresource usage the amount of redundancy employed in both OFDM andSCFDsystemsdependsonthedelayspreadofthechannelimplyingthatbothtransceiverswaste the same bandwidth on redundant data Nevertheless there are many ways to increase the spectral efficiency of communications systems such as by decreasing the overall symbolerror probability in the physical layer so that less redundancy needs to be inserted in upperlayers by means of channel coding In general this approach increases the costs in the physical layer since it leads to more computationally complex transceivers hindering its implementation in some practical applications Other means to improve spectral efficiency are therefore highly desirable Reducing the amount of transmitted redundancy inserted in the physical layer is a possible solution2 Although reliability and simplicity are rather important in practical applications the amount of redundancy should be reduced to potentially increase the spectral efficiency In the context of fixed and memo ryless TMUXes it is possible to show that the minimum redundancy required to eliminate IBI and still allow the design of zeroforcing ZF solutions is only half the amount of redundancy used by standard OFDM and SCFD systems In this context an important question arises can we design memoryless LTI transceivers with reduced redundancy whose computational complexity is comparable to OFDM and SCFD systems That is these transceivers should be amenable to superfast implementations in order to keep their computational complexities competitive with practical OFDMbased systems If this is possible the resulting transceivers would probably allow higher data throughputs in broadband channels This chapter describes how transceivers with reduced redundancy Section 42 can be imple mented employing superfast algorithms based on the concepts of structured matrix representations Section 43To achieve this objective we describe some mathematical decompositions of a special class of structured matrix the socalled Bezoutian matrices Section 44The resulting structures of multicarrier and singlecarrier reducedredundancy systems are then presented and analyzed through some examples Section 45 2The bottom line here is that many distinct and interesting ways of designing multicarrier systems are available We will focus on a particular type of solution reducedredundancy system that allows us to present to the reader a set of tools related to structured matrix representations which can be eventually employed in many other contexts 114 4 MEMORYLESS LTI TRANSCEIVERS WITH REDUCED REDUNDANCY In such systems the equivalent channel matrix H is no longer circulant rather it is an M 2K L M Toeplitz matrix as described in Equation 46 Nevertheless we could take into account the Toeplitz structure in order to decompose the generalized inverse of H maybe using only DFT and diagonal matricesSuch an approach employs the same basic ideas present in CPOFDMbased systems except for two main features present only in OFDMbased systems i the inverse of the equivalent channel matrix has exactly the same structure as the equivalent channel matrix itself circulant structure and ii the efficient decomposition of the inverse of the equivalent channel matrix corresponds to its diagonalization 43 STRUCTURED MATRIX REPRESENTATIONS A matrix is considered structured if its entries follow some predefined pattern The origin of this pattern can be elementary mathematical relations among the matrix coefficients or simply the way certain coefficients appear repeatedly as entries of the given matrix Some of these patterns happen naturally in the matrices describing the behavior of many practical systems eg linear and circular discretetime convolutions can be modeled by structured matrices or can be easily induced by applying simple matrix transformations In any case a structured matrix can be described by using a reduced set of distinct parameters that is the number of parameters required to describe the matrix is much smaller than its number of entries There are several examples of structured matrices which are usually found in signal processing and communication applications such as diagonal circulant pseudocirculantToeplitz among others Such structural patterns may bring about efficient means for exploiting features of the related problems Besides computations involving structured matrices can be further simplified by taking into account these structural patterns Consider for instance the sum of two M2 M1 Toeplitz matrices If one ignores the structural patterns present in such matrices then this operation will require M2M1 additions since there are M2M1 entries in each matrix However these matrices are completely defined by up to M2 M1 1 elements since the first row M1 elements along with the first column M2 elements in which the first element pertains to the first row as well are enough to define a given Toeplitz matrix This way it would be quite reasonable to expect that matrix operations may be performed faster by using a reduced amount of parameters Indeed if one considers the structure of the matrices then this operation will require only M2 M1 1 additions corresponding to the sum of the first row and the first column of each matrixThe resulting Toeplitz matrix can be built by rearranging the elements of the resulting vectors accordingly As previously mentioned the widespread use of OFDM and SCFD transceivers relies on their key feature of transforming the original description of theToeplitz channelconvolution matrix into a circulant matrix for the case where the channel is linear and timeinvariant7 Since a circulant matrix has eigenvectors comprising the columns of the unitary DFT matrix the diagonalization 7The timeinvariance assumption only needs to hold during the transmission of one data block 43 STRUCTURED MATRIX REPRESENTATIONS 117 Example 42 λCirculant Operator Matrices This example illustrates the effects of right and leftmultiplying a given matrix by λcirculant matrices Consider a 3 3 matrix C given by C 1 2 3 4 5 6 7 8 9 425 In addition assume that λ 1 and Z1 0 0 1 1 0 0 0 1 0 426 Hence we have CZ1 2 3 1 5 6 4 8 9 7 427 Z1C 7 8 9 1 2 3 4 5 6 428 Therefore rightmultiplication by a λcirculant matrix shifts all columns to the left where the first original column multiplied by λ is moved to the last column of the resulting matrix On the other hand leftmultiplying by a λcirculant matrix shifts down all rows where the last original row multiplied by λ is moved to the first row of the resulting matrix Usually structured matrices can be associated with some linear displacement operator These operators might reveal if a given structured matrix can be represented by a reduced number of parameters This representation is the key feature that allows the derivation of superfast algorithms9 for inverting as well as performing matrixtovector multiplication involving the related structured matrix The fast implementation of the reducedredundancy transceivers rely on the displacement rank of the matrices involved The procedure entails the following steps 1 Compression If the rank of the displacement matrix of a given M1 M2 structured matrix C is lower than the dimensions of C then it is possible to represent this matrix with a reduced number of coefficients Indeed the displacement operator applied to the original matrix can be compressed and represented by the socalled displacementgenerator pair of matrices P Q with the following features assuming we are dealing with a Sylvester displacement operator 9That is algorithms that require OM logd M numerical operations where d 3 61 43 STRUCTURED MATRIX REPRESENTATIONS 121 the equality νm2m1 νm1 m2 In this case VVV ννν presents the following form VVV ννν ν0 0 ν1 0 νM2 0 νM1 0 ν0 1 ν1 1 νM2 1 νM1 1 ν0 M2 ν1 M2 νM2 M2 νM1 M2 ν0 M1 ν1 M1 νM2 M1 νM1 M1 νm1 m2 m2m1M2 438 As in the Toeplitz case see Equation 437 a similar kind of compression can be applied to Vandermonde matrices as well Let us consider the application of the Sylvester displacement operator DνννZ0 in which Dννν diagννν CMM on a given M M Vandermonde matrix VVV ννν In this case we have DνννZ0VVV ννν DνννVVV ννν VVV νννZ0 ν0 νM1 0 νM 0 ν1 νM1 1 νM 1 νM1 νM1 M1 νM M1 ν0 νM1 0 0 ν1 νM1 1 0 νM1 νM1 M1 0 0 0 νM 0 0 0 νM 1 0 0 νM M1 439 which consists of a rank1 matrix with M degrees of freedom Note that even though the original Vandermonde matrixVVV ννν is comprised of M2 entriesthe M elements which compose the vectorννν are enough to completely define VVV νννThis compression example therefore shows that the displacement approach is able to reveal analytically this reduced number of degrees of freedom In addition there is a close relation between Vandermonde and DFT matrices In order to derive such a useful relation let us first remember that the Mth roots of a given complex number ξ consist of M distinct complex numbers ξm with m M such that ξM m ξ ξej ξ where j2 1 and ξ π π R represents the principal10 phase of the complex number ξ when 10Remember that if ξ is a phase of a given complex number ξ then ξ 2iπ is also a phase of ξ for any integer number i The principal phase is the unique phase of ξ within the interval π π 136 4 MEMORYLESS LTI TRANSCEIVERS WITH REDUCED REDUNDANCY can be straightforwardly acquired using Equation 437 as follows ˆP 1 4 0 6 0 1 0 2 482 ˆQ 1 0 0 0 3 1 483 and ˇP 1 2 0 1 0 0 484 ˇQ 3 0 2 0 4 0 6 1 485 Now Equation 457 gives us P 01224 01633 01633 20816 00816 00028 00028 01122 00204 00334 00334 06531 486 Q 20288 01002 02635 00334 13525 00668 01577 00223 02254 00111 01929 00037 03006 00148 09239 03284 487 Thus using Equation 472 we have R 4 and P 00520 01067 01067 11864 01577 01171j 01915 00429j 01915 00429j 25293 10203j 01577 01171j 01915 00429j 01915 00429j 25293 10203j 488 Q 12022 00594 06957 03135 16531 18033j 00816 00891j 10816 04564j 03061 00297j 33061 01633 08367 03878 16531 18033j 00816 00891j 10816 04564j 03061 00297j 489 144 4 MEMORYLESS LTI TRANSCEIVERS WITH REDUCED REDUNDANCY 0 5 10 15 20 25 30 0 50 100 150 200 250 300 350 SNR dB Throughput Mbps ZFOFDM MMSEOFDM ZFMCMRBT MMSEMCMRBT Figure 44 Throughput Mbps as a function of SNR dB for random Rayleigh channels considering multicarrier transmissions 12 dB in the ZF solutions In this example we use a convolutional code with constraint length 7 rc 12 and generators g0 133 octal and g1 165 octal This configuration is adapted from the LTE longterm evolution specifications 94 In addition for the BLER computation we consider that a block 16 bits is lost if at least one of its received bits is incorrect We have employed a MATLAB implementation of a harddecision Viterbi decoder Note that such favor able result stems from the choices for M and L representing delay constrained applications in very dispersive environment These types of applications are suitable for the ZPZJ transceivers In the cases where M L the traditional OFDM and SCFD solutions are more adequate Example 45 ReducedRedundancy Transceivers In Example 44 we have shown that minimumredundancy systems may significantly improve the throughput performance of multi carrier and singlecarrier transmissions Nevertheless minimumredundancy transceivers may also incur in high noise enhancements induced by the inversion of the Toeplitz effective channel matrix in the equalization process In this example we first chose a fourthorder channel model see 83 45 REDUCEDREDUNDANCY SYSTEMS 145 pp 306307 HAz 01659 03045z1 01159z2 00733z3 00015z4 4114 for which the throughput performance of the minimumredundancy systems is poorFor this channel Channel A we transmit 50000 data blocks carrying M 16 symbols of a 64QAM constellation b 6 bits per symbol In fact each data block stems from 48 data bits that after channel coding yield 96 bits to be baseband modulated The channel coding is the same as in Example 44 and we assume that the sampling frequency is fs 100 MHz Figure 46 depicts the obtained throughput results We compare four different transceivers the ZPOFDMOLA and the three possible multicarrier reducedredundancy block transceivers MCRRBTThere are three possible MCRRBT systems since the amount of redundant elements respects the inequality L 2 K L ie K 2 3 4 From Figure 46 one can observe that the minimumredundancy multicarrier system MCRRBT for K 2 that employs an MMSE equalizer is not able to produce a reliable estimate for the transmitted bits However if just one 0 5 10 15 20 25 30 0 50 100 150 200 250 300 350 SNR dB Throughput Mbps ZFSCFD MMSESCFD ZFSCMRBT MMSESCMRBT Figure 45 Throughput Mbps as a function of SNR dB for random Rayleigh channels considering singlecarrier transmissions 146 4 MEMORYLESS LTI TRANSCEIVERS WITH REDUCED REDUNDANCY 15 20 25 30 35 0 50 100 150 200 250 300 SNR dB Throughput Mbps MMSEOFDM MMSEMCRRBT K 2 MMSEMCRRBT K 3 MMSEMCRRBT K 4 Figure 46 Throughput Mbps as a function of SNR dB for Channel A considering multicarrier transmissions additional redundant element is included in the transmissionthe resulting MCRRBT system K 3 is enough to outperform the MMSEOFDM One should bear in mind that such throughput gains are attained without increasing substantially the computational complexity related to OFDM based systems Moreover the MCRRBT system with K 3 also outperforms the MCRRBT system with K 4 in terms of throughput especially for large SNR values ie adding another redundant element in the transmission MCRRBT for K 4 does not contribute to improving the throughput performance in this case Now we will consider an FIRchannel model Channel B whose zeros are 0999 0999 07j 07j and 04jThis channel has zeros very close to the unit circleWe there fore expect that the performance of the traditional SCFD system should be rather poor Apart from the channel model all simulation parameters are the same of the previous experiment Figure 47 depicts the throughput results One can observe that the SCRRBT systems always outperform the traditional SCFD system Another important fact is that the throughput performance does not necessarily improves as the number of transmitted redundant elements is increased For example for low SNR valuesit is better to use a reducedredundancy system that transmits with a large number of 46 CONCLUDING REMARKS 147 15 20 25 30 35 20 40 60 80 100 120 140 160 180 200 SNR dB Throughput Mbps MMSESCFD MMSESCRRBT K 3 MMSESCRRBT K 4 MMSESCRRBT K 5 Figure 47 Throughput Mbps as a function of SNR dB for Channel B considering singlecarrier transmissions redundant elements K 5 whereas for large SNR values it is better to use a reducedredundancy system that transmits with a small number of redundant elements K 3 Once again it is im portant to highlight that the superfast ZPZJ systems described in this chapter are just examples of how to transmit with a small number of redundant elements while using superfast transforms and singletap equalizers 46 CONCLUDING REMARKS This chapter described how to design memoryless LTI singlecarrier and multicarrier transceivers with reduced redundancy for both ZF and MMSE optimal receivers The block transceivers pre sented here are computationally efficient We also introduced the mathematical tools which allow structured matrices to be represented through displacement operators In particular we emphasized the representation of Bezoutian matrices employing DFT IDFT and diagonal matrices It is worth mentioning that similar decomposition is possible by using transforms with real entries such as those 148 4 MEMORYLESS LTI TRANSCEIVERS WITH REDUCED REDUNDANCY described in 48 52 based on discrete Hartley transform DHT diagonal and antidiagonal ma trices In addition the complexity of the efficient transceivers described here can be further reduced by employing suboptimal solutions as shown in 50 As described in previous chapters OFDMbased transceivers are rather efficient systems incorporating two desired features in practical systems namely goodfair performance and compu tational simplicity Nevertheless there is always room for improvements We showed in this chapter how to increase the transmission datarates in block transceivers while keeping the computational complexity close to the complexity of OFDMbased systems We believe that the strategy followed in this chapter is even more important than the described transceivers themselves since it illus trates how one can deduce new systems using a set of efficient tools related to structured matrix representations An important question still remains what happens when we move from memoryless LTI to timevarying FIR systems Is it possible to reduce even further the amount of transmitted redun dancyThese and other questions related to what is beyond OFDMbased systems will be addressed in Chapter 5 149 C H A P T E R 5 FIR LTV Transceivers with Reduced Redundancy 51 INTRODUCTION Nowadays many practical communications systems employ the orthogonal frequencydivision mul tiplexing OFDM as their core physicallayer modulation as previously highlighted in Section 13 Such widespread adoption is due to many good properties that OFDMbased systems enjoy as thoroughly explained in Chapter 3 However one of the main drawbacks of OFDM is its related loss of spectral efficiency caused by the insertion of redundant elements in the transmission This is an important issue considering the current trend of increasing the demand for data transmissions which shows no sign of settling For example the amount of wireless data services is more than doubling each year 37 leading to spectrum shortage as a sure event soon It turns out that all efforts to maximize the spectrum usage are therefore highly justifiable at this point The memoryless linear timeinvariant LTI transceivers with reduced redundancy described in Chapter 4 are a possible way of tackling the problem of increasing the spectral efficiency Indeed we show in Chapter 4 thatas compared to OFDMbased systemsreducedredundancy LTI systems may decrease the amount of transmitted redundant elements in up to fifty percent thus allowing a better use of the available spectrum for transmissions However this amount of reduction may still not be enough especially in delayconstrained applications in very dispersive environments in which the size of the transmitted data block cannot be too large and in addition the channel model is lengthy Hence we should ask ourselves if the memoryless LTIbased solutions could be further improved The transceivers described in Chapters 2 3 and 4 are concrete examples of memoryless linear and timeinvariant systems Linearity plays a central role in the overall system design since it yields simpler transceivers a very desirable feature in practice A step ahead would be to investigate how to design simple timevarying transceivers with memory implemented through timevarying finiteimpulse response FIR filters1 The memory and timevarying properties would introduce additional degrees of freedom in the design of systems that could help us circumvent some limitations inherent to memoryless LTI transceivers In fact the timevarying FIR transceiver increases the transmission diversity since more than one version of the transmitted message is received due to the memory of the system and in addition 1In this case we consider that the orders of the related filters are larger than or equal to 1 A memoryless system can be regarded as an FIR system whose order is equal to 0 150 5 FIR LTV TRANSCEIVERS WITH REDUCED REDUNDANCY these versions are more likely to differ from each other due to the timevarying characteristic If the receiver can take these facts into accountthen it could trade off the amount of transmitted redundant elements with the amount of memory and degree of timevariance in the systems Moreover FIR linear timevarying LTV transceivers allow the design of highselective transmitter filters and the interpretation of codedivision multiple access CDMA schemes with long codes as a particular type of LTV transmultiplexer TMUX as will be explained later on It is worth recalling that intersymbol interference ISI is one of the most harmful effects inherent to broadband communications That is why communication engineers often constrain their designs to eliminate ISI especially when noise and other types of interference are negligible leading to the socalled zeroforcing ZF transceiversThis chapter will describe some conditions that FIR LTV systems must satisfy in order to achieve ZF solutions It turns out that some redundant elements must always be introduced in the ZF designs On the other hand ZF systems can be regarded as suboptimal solutions in the mean square error MSE sense In this case we can give up imposing the ZF constraint upon the transceiver thus allowing the transmission without adding any redundant element as long as pure minimum MSEbased solutions are employed By pure minimum MSE MMSE we mean that there is no ZF constraint imposed upon the transceiver design Hence the way to reduce even further the amount of redundancy inserted in the physical layer is either to design LTV transceivers andor to allow the transmitter and receiver multipleinput multipleoutput MIMO filters to have memory In such cases it is possible to derive ZF solutions using only one redundant element Although timevarying transceivers with memory bring about a reduction in redundancy to allow for ZF solutions their fast implementations are not known In addition the numerical accuracy of transceivers with reduced redundancy is not fully exploited in the open literature and for certain is a crucial issue to be addressed before any attempt to include these solutions in practical implementations or standards This chapter adopts a rather different approach as compared to Chapters 2 3 and 4 The previous chapters focus on the derivationdescription of either structures or performances of OFDM and beyondOFDM systems whereas the main focus of this chapter is on the limits of some parameters related to FIR LTV transceiversWe start by describing carefully how to use our previous knowledge about FIR LTI transceivers developed in Chapter 2 in order to model FIR LTV systems Section 52 Then we study the fundamental limits concerning some parameters inherent to FIR LTV transceivers that satisfy the ZF constraint namely memory of receiver MIMO matrix and number of transmitted redundant elements Section 53 After showing that when no redundancy is employed in the transmission ZF solutions cannot be achieved then we present pure MMSE based solutions Section 54 Some examples are also given in this chapter Section 55 including the interpretation of the ZF conditions within the framework of CDMA systems with long codes 55 EXAMPLES 173 Q 1 curves employ K 6 and Q 1 which means that the orders of the related receiver matrices are not satisfying the ZF constraint The obtained results show that by not following the ZF conditions either by choosing K 5 or Q 1 a floor in the BER curves appears starting from 20 dB of SNR On the other hand for the designs following the ZF conditions the BER tends to zero as the SNR increases This example also illustrates that when alternative design criteria are used such as the min imization of the MSE the ZF conditions described in Section 53 are still useful in order to avoid performance loss due to errors in the reconstruction of the signal This eventually means that even for very high SNRs it is still possible to have a BER floor due to the nonexistence of the ZF so lution On the other hand this BER floor does not appear by observing the conditions presented in this chapter Example 56 LongCode CDMA Systems As previously illustrated in Figure 121 of Chap ter 1 MIMO models encompass many blocktransmission configurations ranging from singleuser pointtopoint communications employing multiple antennas to multipleaccess schemes in mul tiuser systems This example illustrates an application of the theoretical results related to FIR LTV transceivers with reduced redundancy within the framework of CDMA systems Indeed CDMA systems can be described using the concept of MIMO transceivers By using such a description it is possible to derive some theoretical conditions for designing equalizers that guarantee the perfect reconstruction of the transmitted signal at the receiver end For example we can apply the theoretical analysis in order to obtain multiuser detection in CDMA systems with long codes ie codes which last for more than one symbol duration as described in the following reasoning Consider the TMUX structure of Figure 26 of Chapter 2 Assume that smn is the symbol associated with the mth user at the time instant n It is possible to imagine that the impulse re sponses of the synthesis filters fmk with m M that appear in that figure can play the role of a spreading sequence of a CDMA system whereas the impulse responses of analysis filters gmk with m M can be thought as the despreading codes all of them associated with the mth user If in addition we assume that the synthesis and analysis filters are actually timevarying filters then we can consider that each subfilter implements a piece of a given CDMA spreadingdespreading long code associated with a user In other words CDMA with long codes can be interpreted as CDMA with timevarying short codes As a result it is possible to verify that CDMA with long codes can be represented by a timevarying structure ie it is an example of FIR LTV system Each signal block faces a time varying codewhich is implemented using timevarying transmit filtersThusby adapting the results concerning the amount of redundancy required to allow a ZF solution it is possible to establish the conditions for the existence of zeroforcing multiuser detectors In this context N represents the spreading factor M denotes the number of codes that are going to be used whereas K represents the number of unused codes The main conclusion of this 174 5 FIR LTV TRANSCEIVERS WITH REDUCED REDUNDANCY analysis is that ZF equalization is always possible in CDMA systems using long codes as long as the system is not at full capacity ie one must have at least one unused code K 1 In addition it can be shown that both the complexity of the receiver and its performance depend directly on the number of unused codesThe conditions derived serve as useful guidelines for the design of communications systems allowing the tradeoff between performance and complexity of the receiver In conclusion the existence of ZF equalizers is guaranteed if the amount of redundancy is greater than or equal to the number of congruous zeros or if there are enough different transmission filters at the transmitter For practical channels approximately congruous zeros may cause numerical instability in equalizer design A CDMA system with long codes may be interpreted as a transmul tiplexer with memoryless timevarying filters In the uplink direction the system can be modeled as a transmultiplexer with timevarying filters with memory In any case for the CDMA system the block length is determined by the spreading factor implying that the redundancy is equal to the number of unused codes 56 CONCLUDING REMARKS This chapter addressed the problem of further reducing transmission redundancy in block transceivers by relieving their designs from the memoryless and timeinvariance constraints The use of timevarying transceivers with memory allows ZF solutions whose amount of required re dundancy can be reduced to a single element We discussed several results related to the amount of memory at the transmitter and receiver as well as the time variance of the transceiver in order to understand what are available to achieve ZF solutions with as 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93 91 X G Xia New precoding for intersymbol interference cancellation using nonmaximally dec imated multirate filterbanks with ideal FIR equalizers IEEE Trans on Signal Processing vol 45 no 10 pp 24312441 October 1997 DOI 10110978640709 Cited on pages 92 Y Yao and G B Giannakis Blind carrier frequency offset estimation in SISO MIMO and multiuser OFDM systems IEEE Trans on Communications vol 53 no 1 pp 173183 January 2005 DOI 101109TCOMM2004840623 Cited on pages 70 93 W Zhang X Ma B Gestner and D V Anderson Designing lowcomplexity equalizers for wireless systems IEEE Communications Magazine vol 47 no 1 pp 5662 January 2009 DOI 101109MCOM20094752677 Cited on pages 94 Evolved Universal Terrestrial Radio Access EUTRAN Multiplexing and Channel Coding 3GPP TS 36212 ver 870 3rd Generation Partnership Project May 2009 Cited on pages 144 183 Authors Biographies PAULO S R DINIZ Paulo S R Diniz was born in Niterói Brazil He received the Electronics Eng degree Cum Laude from the Federal Uni versity of Rio de Janeiro UFRJ in 1978 a MSc degree from COPPEUFRJ in 1981 and a PhD from Concordia University Montreal PQ Canada in 1984 all in Electrical Engineering Since 1979 he has been with the Department of Electronic Engineering undergraduate at UFRJ He has also been with the Program of Electrical Engineering the graduate studies dept COPPEUFRJ since 1984 where he is presently a Professor He served as Undergraduate Course Coordinator and Chairman of the Graduate Department He has received the Rio de Janeiro State Scientist award from the Governor of Rio de Janeiro From January 1991 to July 1992 he was a visiting Research Associate in the Department of Electrical and Computer Engineering of University of Victoria Victoria BC Canada He also held a Docent position at Helsinki University of Technology From January 2002 to June 2002 he was a Melchor Chair Professor in the Department of Electrical Engineering of University of Notre Dame Notre Dame IN USA His teaching and research interests are in analog and digital signal processing adaptive signal processing digital communications wireless communications multirate systems stochastic processes and electronic circuits He has published several refereed papers in some of these areas and wrote the books Adaptive Filtering Algorithms and Practical Implementation 4th ed Springer NY 2012 and Digital Signal Processing System Analysis and Design 2nd ed Cambridge University Press Cambridge UK 2010 with EAB da Silva and SL Netto He has served as the Technical Program Chair of the 1995 MWSCAS held in Rio de Janeiro Brazil He was the General coChair of the IEEE ISCAS2011 and Technical Program coChair of the IEEE SPAWC2008He has been on the technical committee of several international conferences including ISCAS ICECS EUSIPCO and MWSCAS He has served as Vice President for region 9 of the IEEE Circuits and Systems Society and as Chairman of the DSP technical committee of the same Society He is also a Fellow of IEEE He has served as associate editor for the following Journals IEEE Transactions on Circuits and Systems II Analog and Digital Signal Processing from 19961999 IEEE Transactions on Signal Processing from 19992002 and the Circuits Systems and Signal Processing Journal from 19982002 He was a distinguished lecturer of the IEEE Circuits and Systems Society from 20002001 In 2004 he served as distinguished lecturer of the IEEE Signal 184 AUTHORS BIOGRAPHIES Processing Society and received the 2004 Education Award of the IEEE Circuits and Systems Society He also holds some bestpaper awards from conferences and from an IEEE journal WALLACE A MARTINS Wallace A Martins was born in Brazil in 1983 He received an Electronics Engineering degree Cum Laude from the Fed eral University of Rio de Janeiro UFRJ in 2007 and MSc and DSc degrees in Electrical Engineering from COPPEUFRJ in 2009 and 2011 respectively He worked as a technical consul tant for Nokia Institute of Technology INDT Brazil and for TechKnowledge Training Brazil In 2008 he was a research vis itor at the Department of Electrical Engineering University of Notre Dame Notre Dame IN Since 2010 he has been with the Department of Control and Automation Industrial Engineering Federal Center for Technological Education Celso Suckow da Fonseca CEFETRJ UnEDNI where he is presently a Lecturer of Engineering His research interests are in the fields of digital communication microphone ar ray signal processing and adaptive signal processing Dr Martins received the Best Student Paper Award from EURASIP at EUSIPCO2009 Glasgow Scotland MARKUS V S LIMA Markus V S Lima was born in Rio de Janeiro Brazil in 1984 He received an Electronics Engineering degree from the Federal University of Rio de Janeiro UFRJ in 2008 an MSc degree in Electrical Engineering from COPPEUFRJ in 2009 and is cur rently pursuing his DScdegree at COPPEUFRJHe has served as a teaching assistant for the following undergraduate courses taught at UFRJ Digital Transmission Digital Signal Process ing and Linear Systems His main interests are in adaptive signal processing microphone array signal processing digital commu nications wireless communications statistical signal processing and linear algebra