Fundamentals of Wireless Communication - Teoria e Sistemas Sem Fio

Fundamentals of Wireless Communication The past decade has seen many advances in physicallayer wireless communi cation theory and their implementation in wireless systems This textbook takes a unified view of the fundamentals of wireless communication and explains the web of concepts underpinning these advances at a level accessible to an audience with a basic background in probability and digital communication TopicscoveredincludeMIMOmultipleinputmultipleoutputcommunication spacetime coding opportunistic communication OFDM and CDMA The concepts are illustrated using many examples from wireless systems such as GSM IS95 CDMA IS856 1 EVDO Flash OFDM and ArrayComm SDMA systems Particular emphasis is placed on the interplay between concepts and their implementation in systems An abundant supply of exercises and figures reinforce the material in the text This book is intended for use on graduate courses in electrical and computer engineering and will also be of great interest to practicing engineers David Tse is a professor at the Department of Electrical Engineering and Computer Sciences University of California at Berkeley Pramod Viswanath is an assistant professor at the Department of Electrical and Computer Engineering University of Illinois at UrbanaChampaign Fundamentals of Wireless Communication David Tse University of California Berkeley and Pramod Viswanath University of Illinois UrbanaChampaign c a m b r i d g e u n i v e r s i t y p r e s s Cambridge New York Melbourne Madrid Cape Town Singapore São Paulo c a m b r i d g e u n i v e r s i t y p r e s s The Edinburgh Building Cambridge CB2 2RU UK Published in the United States of America by Cambridge University Press New York wwwcambridgeorg Information on this title wwwcambridgeorg9780521845274 Cambridge University Press 2005 This book is in copyright Subject to statutory exception and to the provisions of relevant collective licensing agreements no reproduction of any part may take place without the written permission of Cambridge University Press First published 2005 Printed in the United Kingdom at the University Press Cambridge A catalog record for this book is available from the British Library ISBN13 9780521845274 hardback ISBN10 0521845270 hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or thirdparty internet websites referred to in this book and does not guarantee that any content on such websites is or will remain accurate or appropriate To my family DT To my parents and to Suma PV Contents Preface page xv Acknowledgements xviii List of notation xx 1 Introduction 1 11 Book objective 1 12 Wireless systems 2 13 Book outline 5 2 The wireless channel 10 21 Physical modeling for wireless channels 10 211 Free space fixed transmit and receive antennas 12 212 Free space moving antenna 13 213 Reflecting wall fixed antenna 14 214 Reflecting wall moving antenna 16 215 Reflection from a ground plane 17 216 Power decay with distance and shadowing 18 217 Moving antenna multiple reflectors 19 22 Input output model of the wireless channel 20 221 The wireless channel as a linear timevarying system 20 222 Baseband equivalent model 22 223 A discretetime baseband model 25 Discussion 21 Degrees of freedom 28 224 Additive white noise 29 23 Time and frequency coherence 30 231 Doppler spread and coherence time 30 232 Delay spread and coherence bandwidth 31 24 Statistical channel models 34 241 Modeling philosophy 34 242 Rayleigh and Rician fading 36 vii viii Contents 243 Tap gain autocorrelation function 37 Example 22 Clarkes model 38 Chapter 2 The main plot 40 25 Bibliographical notes 42 26 Exercises 42 3 Pointtopoint communication detection diversity and channel uncertainity 49 31 Detection in a Rayleigh fading channel 50 311 Noncoherent detection 50 312 Coherent detection 52 313 From BPSK to QPSK exploiting the degrees of freedom 56 314 Diversity 59 32 Time diversity 60 321 Repetition coding 60 322 Beyond repetition coding 64 Summary 31 Time diversity code design criterion 68 Example 31 Time diversity in GSM 69 33 Antenna diversity 71 331 Receive diversity 71 332 Transmit diversity spacetime codes 73 333 MIMO a 22 example 77 Summary 32 22 MIMO schemes 82 34 Frequency diversity 83 341 Basic concept 83 342 Singlecarrier with ISI equalization 84 343 Directsequence spreadspectrum 91 344 Orthogonal frequency division multiplexing 95 Summary 33 Communication over frequencyselective channels 101 35 Impact of channel uncertainty 102 351 Noncoherent detection for DS spreadspectrum 103 352 Channel estimation 105 353 Other diversity scenarios 107 Chapter 3 The main plot 109 36 Bibliographical notes 110 37 Exercises 111 4 Cellular systems multiple access and interference management 120 41 Introduction 120 42 Narrowband cellular systems 123 421 Narrowband allocations GSM system 124 422 Impact on network and system design 126 ix Contents 423 Impact on frequency reuse 127 Summary 41 Narrowband systems 128 43 Wideband systems CDMA 128 431 CDMA uplink 131 432 CDMA downlink 145 433 System issues 147 Summary 42 CDMA 147 44 Wideband systems OFDM 148 441 Allocation design principles 148 442 Hopping pattern 150 443 Signal characteristics and receiver design 152 444 Sectorization 153 Example 41 FlashOFDM 153 Chapter 4 The main plot 154 45 Bibliographical notes 155 46 Exercises 155 5 Capacity of wireless channels 166 51 AWGN channel capacity 167 511 Repetition coding 167 512 Packing spheres 168 Discussion 51 Capacityachieving AWGN channel codes 170 Summary 51 Reliable rate of communication and capacity 171 52 Resources of the AWGN channel 172 521 Continuoustime AWGN channel 172 522 Power and bandwidth 173 Example 52 Bandwidth reuse in cellular systems 175 53 Linear timeinvariant Gaussian channels 179 531 Single input multiple output SIMO channel 179 532 Multiple input single output MISO channel 179 533 Frequencyselective channel 181 54 Capacity of fading channels 186 541 Slow fading channel 187 542 Receive diversity 189 543 Transmit diversity 191 Summary 52 Transmit and recieve diversity 195 544 Time and frequency diversity 195 Summary 53 Outage for parallel channels 199 545 Fast fading channel 199 546 Transmitter side information 203 Example 53 Rate adaptation in IS856 209 547 Frequencyselective fading channels 213 x Contents 548 Summary a shift in point of view 213 Chapter 5 The main plot 214 55 Bibliographical notes 217 56 Exercises 217 6 Multiuser capacity and opportunistic communication 228 61 Uplink AWGN channel 229 611 Capacity via successive interference cancellation 229 612 Comparison with conventional CDMA 232 613 Comparison with orthogonal multiple access 232 614 General Kuser uplink capacity 234 62 Downlink AWGN channel 235 621 Symmetric case two capacityachieving schemes 236 622 General case superposition coding achieves capacity 238 Summary 61 Uplink and downlink AWGN capacity 240 Discussion 61 SIC implementation issues 241 63 Uplink fading channel 243 631 Slow fading channel 243 632 Fast fading channel 245 633 Full channel side information 247 Summary 62 Uplink fading channel 250 64 Downlink fading channel 250 641 Channel side information at receiver only 250 642 Full channel side information 251 65 Frequencyselective fading channels 252 66 Multiuser diversity 253 661 Multiuser diversity gain 253 662 Multiuser versus classical diversity 256 67 Multiuser diversity system aspects 256 671 Fair scheduling and multiuser diversity 258 672 Channel prediction and feedback 262 673 Opportunistic beamforming using dumb antennas 263 674 Multiuser diversity in multicell systems 270 675 A system view 272 Chapter 6 The main plot 275 68 Bibliographical notes 277 69 Exercises 278 7 MIMO I spatial multiplexing and channel modeling 290 71 Multiplexing capability of deterministic MIMO channels 291 711 Capacity via singular value decomposition 291 712 Rank and condition number 294 xi Contents 72 Physical modeling of MIMO channels 295 721 Lineofsight SIMO channel 296 722 Lineofsight MISO channel 298 723 Antenna arrays with only a lineofsight path 299 724 Geographically separated antennas 300 725 Lineofsight plus one reflected path 306 Summary 71 Multiplexing capability of MIMO channels 309 73 Modeling of MIMO fading channels 309 731 Basic approach 309 732 MIMO multipath channel 311 733 Angular domain representation of signals 311 734 Angular domain representation of MIMO channels 315 735 Statistical modeling in the angular domain 317 736 Degrees of freedom and diversity 318 Example 71 Degrees of freedom in clustered response models 319 737 Dependency on antenna spacing 323 738 Iid Rayleigh fading model 327 Chapter 7 The main plot 328 74 Bibliographical notes 329 75 Exercises 330 8 MIMO II capacity and multiplexing architectures 332 81 The VBLAST architecture 333 82 Fast fading MIMO channel 335 821 Capacity with CSI at receiver 336 822 Performance gains 338 823 Full CSI 346 Summary 81 Performance gains in a MIMO channel 348 83 Receiver architectures 348 831 Linear decorrelator 349 832 Successive cancellation 355 833 Linear MMSE receiver 356 834 Information theoretic optimality 362 Discussion 81 Connections with CDMA multiuser detection and ISI equalization 364 84 Slow fading MIMO channel 366 85 DBLAST an outageoptimal architecture 368 851 Suboptimality of VBLAST 368 852 Coding across transmit antennas DBLAST 371 853 Discussion 372 Chapter 8 The main plot 373 86 Bibliographical notes 374 87 Exercises 374 xii Contents 9 MIMO III diversitymultiplexing tradeoff and universal spacetime codes 383 91 Diversitymultiplexing tradeoff 384 911 Formulation 384 912 Scalar Rayleigh channel 386 913 Parallel Rayleigh channel 390 914 MISO Rayleigh channel 391 915 22 MIMO Rayleigh channel 392 916 nt nr MIMO iid Rayleigh channel 395 92 Universal code design for optimal diversitymultiplexing tradeoff 398 921 QAM is approximately universal for scalar channels 398 Summary 91 Approximate universality 400 922 Universal code design for parallel channels 400 Summary 92 Universal codes for the parallel channel 406 923 Universal code design for MISO channels 407 Summary 93 Universal codes for the MISO channel 410 924 Universal code design for MIMO channels 411 Discussion 91 Universal codes in the downlink 415 Chapter 9 The main plot 415 93 Bibliographical notes 416 94 Exercises 417 10 MIMO IV multiuser communication 425 101 Uplink with multiple receive antennas 426 1011 Spacedivision multiple access 426 1012 SDMA capacity region 428 1013 System implications 431 Summary 101 SDMA and orthogonal multiple access 432 1014 Slow fading 433 1015 Fast fading 436 1016 Multiuser diversity revisited 439 Summary 102 Opportunistic communication and multiple receive antennas 442 102 MIMO uplink 442 1021 SDMA with multiple transmit antennas 442 1022 System implications 444 1023 Fast fading 446 103 Downlink with multiple transmit antennas 448 1031 Degrees of freedom in the downlink 448 1032 Uplinkdownlink duality and transmit beamforming 449 1033 Precoding for interference known at transmitter 454 1034 Precoding for the downlink 465 1035 Fast fading 468 xiii Contents 104 MIMO downlink 471 105 Multiple antennas in cellular networks a system view 473 Summary 103 System implications of multiple antennas on multiple access 473 1051 Intercell interference management 474 1052 Uplink with multiple receive antennas 476 1053 MIMO uplink 478 1054 Downlink with multiple receive antennas 479 1055 Downlink with multiple transmit antennas 479 Example 101 SDMA in ArrayComm systems 479 Chapter 10 The main plot 481 106 Bibliographical notes 482 107 Exercises 483 Appendix A Detection and estimation in additive Gaussian noise 496 A1 Gaussian random variables 496 A11 Scalar real Gaussian random variables 496 A12 Real Gaussian random vectors 497 A13 Complex Gaussian random vectors 500 Summary A1 Complex Gaussian random vectors 502 A2 Detection in Gaussian noise 503 A21 Scalar detection 503 A22 Detection in a vector space 504 A23 Detection in a complex vector space 507 Summary A2 Vector detection in complex Gaussian noise 508 A3 Estimation in Gaussian noise 509 A31 Scalar estimation 509 A32 Estimation in a vector space 510 A33 Estimation in a complex vector space 511 Summary A3 Mean square estimation in a complex vector space 513 A4 Exercises 513 Appendix B Information theory from first principles 516 B1 Discrete memoryless channels 516 Example B1 Binary symmetric channel 517 Example B2 Binary erasure channel 517 B2 Entropy conditional entropy and mutual information 518 Example B3 Binary entropy 518 B3 Noisy channel coding theorem 521 B31 Reliable communication and conditional entropy 521 B32 A simple upper bound 522 B33 Achieving the upper bound 523 Example B4 Binary symmetric channel 524 Example B5 Binary erasure channel 525 B34 Operational interpretation 525 xiv Contents B4 Formal derivation of AWGN capacity 526 B41 Analog memoryless channels 526 B42 Derivation of AWGN capacity 527 B5 Spherepacking interpretation 529 B51 Upper bound 529 B52 Achievability 530 B6 Timeinvariant parallel channel 532 B7 Capacity of the fast fading channel 533 B71 Scalar fast fading channnel 533 B72 Fast fading MIMO channel 535 B8 Outage formulation 536 B9 Multiple access channel 538 B91 Capacity region 538 B92 Corner points of the capacity region 539 B93 Fast fading uplink 540 B10 Exercises 541 References 546 Index 554 Preface Why we wrote this book The writing of this book was prompted by two main developments in wireless communication in the past decade First is the huge surge of research activities in physicallayer wireless communication theory While this has been a subject of study since the sixties recent developments such as opportunistic and mul tiple input multiple output MIMO communication techniques have brought completely new perspectives on how to communicate over wireless channels Second is the rapid evolution of wireless systems particularly cellular net works which embody communication concepts of increasing sophistication This evolution started with secondgeneration digital standards particularly the IS95 Code Division Multiple Access standard continuing to more recent thirdgeneration systems focusing on data applications This book aims to present modern wireless communication concepts in a coherent and unified manner and to illustrate the concepts in the broader context of the wireless systems on which they have been applied Structure of the book This book is a web of interlocking concepts The concepts can be structured roughly into three levels 1 channel characteristics and modeling 2 communication concepts and techniques 3 application of these concepts in a system context A wireless communication engineer should have an understanding of the concepts at all three levels as well as the tight interplay between the levels We emphasize this interplay in the book by interlacing the chapters across these levels rather than presenting the topics sequentially from one level to the next xv xvi Preface Chapter 2 basic properties of multipath wireless channels and their mod eling level 1 Chapter 3 pointtopoint communication techniques that increase reliability by exploiting time frequency and spatial diversity 2 Chapter 4 cellular system design via a case study of three systems focusing on multiple access and interference management issues 3 Chapter 5 pointtopoint communication revisited from a more fundamental capacity point of view culminating in the modern concept of opportunistic communication 2 Chapter 6 multiuser capacity and opportunistic communication and its application in a thirdgeneration wireless data system 3 Chapter 7 MIMO channel modeling 1 Chapter 8 MIMO capacity and architectures 2 Chapter 9 diversitymultiplexing tradeoff and spacetime code design 2 Chapter 10 MIMO in multiuser channels and cellular systems 3 How to use this book This book is written as a textbook for a firstyear graduate course in wireless communication The expected background is solid undergraduatebeginning graduate courses in signals and systems probability and digital communica tion This background is supplemented by the two appendices in the book Appendix A summarizes some basic facts in vector detection and estimation in Gaussian noise which are used repeatedly throughout the book Appendix B covers the underlying information theory behind the channel capacity results used in this book Even though information theory has played a significant role in many of the recent developments in wireless communication in the main text we only introduce capacity results in a heuristic manner and use them mainly to motivate communication concepts and techniques No back ground in information theory is assumed The appendix is intended for the reader who wants to have a more indepth and unified understanding of the capacity results At Berkeley and UrbanaChampaign we have used earlier versions of this book to teach onesemester 15 weeks wireless communication courses We have been able to cover most of the materials in Chapters 1 through 8 and parts of 9 and 10 Depending on the background of the students and the time available one can envision several other ways to structure a course around this book Examples A senior level advanced undergraduate course in wireless communication Chapters 2 3 4 An advanced graduate course for students with background in wireless channels and systems Chapters 3 5 6 7 8 9 10 xvii Preface A short quarter course focusing on MIMO and spacetime coding Chap ters 3 5 7 8 9 The more than 230 exercises form an integral part of the book Working on at least some of them is essential in understanding the material Most of them elaborate on concepts discussed in the main text The exercises range from relatively straightforward derivations of results in the main text to back ofenvelope calculations for actual wireless systems to getyourhands dirty MATLAB types and to reading exercises that point to current research literature The small bibliographical notes at the end of each chapter provide pointers to literature that is very closely related to the material discussed in the book we do not aim to exhaust the immense research literature related to the material covered here Acknowledgements We would like first to thank the students in our research groups for the selfless help they provided In particular many thanks to Sanket Dusad Raúl Etkin and Lenny Grokop who between them painstakingly produced most of the figures in the book Aleksandar Joviˇcic who drew quite a few figures and proofread some chapters Ada Poon whose research shaped significantly the material in Chapter 7 and who drew several figures in that chapter as well as in Chapter 2 Saurabha Tavildar and Lizhong Zheng whose research led to Chapter 9 Tie Liu and Vinod Prabhakaran for their help in clarifying and improving the presentation of Costa precoding in Chapter 10 Several researchers read drafts of the book carefully and provided us with very useful comments on various chapters of the book thanks to Stark Draper Atilla Eryilmaz Irem Koprulu Dana Porrat and Pascal Vontobel This book has also benefited immensely from critical comments from stu dents who have taken our wireless communication courses at Berkeley and UrbanaChampaign In particular sincere thanks to Amir Salman Avestimehr Alex Dimakis Krishnan Eswaran Jana van Greunen Nils Hoven Shridhar Mubaraq Mishra Jonathan Tsao Aaron Wagner Hua Wang Xinzhou Wu and Xue Yang Earlier drafts of this book have been used in teaching courses at several universities Cornell ETHZ MIT Northwestern and University of Colorado at Boulder We would like to thank the instructors for their feedback Helmut Bölcskei Anna Scaglione Mahesh Varanasi Gregory Wornell and Lizhong Zheng We would like to thank Ateet Kapur Christian Peel and Ulrich Schus ter from Helmuts group for their very useful feedback Thanks are also due to Mitchell Trott for explaining to us how the ArrayComm systems work This book contains the results of many researchers but it owes an intellec tual debt to two individuals in particular Bob Gallagers research and teaching style have greatly inspired our writing of this book He has taught us that good theory by providing a unified and conceptually simple understanding of a morass of results should shrink rather than grow the knowledge tree This book is an attempt to implement this dictum Our many discussions with xviii xix Acknowledgements Rajiv Laroia have significantly influenced our view of the system aspects of wireless communication Several of his ideas have found their way into the system view discussions in the book Finally we would like to thank the National Science Foundation whose continual support of our research led to this book Notation Some specific sets R Real numbers C Complex numbers S A subset of the users in the uplink of a cell Scalars m Nonnegative integer representing discretetime L Number of diversity branches ℓ Scalar indexing the diversity branches K Number of users N Block length Nc Number of tones in an OFDM system Tc Coherence time Td Delay spread W Bandwidth nt Number of transmit antennas nr Number of receive antennas nmin Minimum of number of transmit and receive antennas hm Scalar channel complex valued at time m h Complex conjugate of the complex valued scalar h xm Channel input complex valued at time m ym Channel output complex valued at time m Nμσ² Real Gaussian random variable with mean μ and variance σ² CN0σ² Circularly symmetric complex Gaussian random variable the real and imaginary parts are iid N0σ²2 N₀ Power spectral density of white Gaussian noise wm Additive Gaussian noise process iid CN0N₀ with time m zm Additive colored Gaussian noise at time m P Average power constraint measured in joulessymbol P Average power constraint measured in watts SNR Signaltonoise ratio SINR Signaltointerferenceplusnoise ratio 𝜀b Energy per received bit Pe Error probability Capacities Cawgn Capacity of the additive white Gaussian noise channel Cϵ ϵOutage capacity of the slow fading channel Csum Sum capacity of the uplink or the downlink Csym Symmetric capacity of the uplink or the downlink Cϵsym ϵOutage symmetric capacity of the slow fading uplink channel pout Outage probability of a scalar fading channel poutAla Outage probability when employing the Alamouti scheme poutrep Outage probability with the repetition scheme poutul Outage probability of the uplink poutmimo Outage probability of the MIMO fading channel poutulmimo Outage probability of the uplink with multiple antennas at the basestation Vectors and matrices h Vector complex valued channel x Vector channel input y Vector channel output CN0K Circularly symmetric Gaussian random vector with mean zero and covariance matrix K w Additive Gaussian noise vector CN0N0I h Complex conjugatetranspose of h d Data vector 𝑑 Discrete Fourier transform of d H Matrix complex valued channel Kx Covariance matrix of the random complex vector x H Complex conjugatetranspose of H H Transpose of matrix H Q U V Unitary matrices In Identity n n matrix Λ Ψ Diagonal matrices diagp1pn Diagonal matrix with the diagonal entries equal to p1pn C Circulant matrix D Normalized codeword difference matrix Operations Ex Mean of the random variable x PA Probability of an event A TrK Trace of the square matrix K sinct Defined to be the ratio of sinπt to πt Qa a 12π expx22 dx L Lagrangian function This page is blank C H A P T E R 1 Introduction 11 Book objective Wireless communication is one of the most vibrant areas in the commu nication field today While it has been a topic of study since the 1960s the past decade has seen a surge of research activities in the area This is due to a confluence of several factors First there has been an explosive increase in demand for tetherless connectivity driven so far mainly by cellu lar telephony but expected to be soon eclipsed by wireless data applications Second the dramatic progress in VLSI technology has enabled smallarea and lowpower implementation of sophisticated signal processing algorithms and coding techniques Third the success of secondgeneration 2G digital wireless standards in particular the IS95 Code Division Multiple Access CDMA standard provides a concrete demonstration that good ideas from communication theory can have a significant impact in practice The research thrust in the past decade has led to a much richer set of perspectives and tools on how to communicate over wireless channels and the picture is still very much evolving There are two fundamental aspects of wireless communication that make the problem challenging and interesting These aspects are by and large not as significant in wireline communication First is the phenomenon of fading the time variation of the channel strengths due to the smallscale effect of multipath fading as well as largerscale effects such as path loss via dis tance attenuation and shadowing by obstacles Second unlike in the wired world where each transmitterreceiver pair can often be thought of as an isolated pointtopoint link wireless users communicate over the air and there is significant interference between them The interference can be between transmitters communicating with a common receiver eg uplink of a cellu lar system between signals from a single transmitter to multiple receivers eg downlink of a cellular system or between different transmitterreceiver pairs eg interference between users in different cells How to deal with fad ing and with interference is central to the design of wireless communication 1 2 Introduction systems and will be the central theme of this book Although this book takes a physicallayer perspective it will be seen that in fact the management of fading and interference has ramifications across multiple layers Traditionally the design of wireless systems has focused on increasing the reliability of the air interface in this context fading and interference are viewed as nuisances that are to be countered Recent focus has shifted more towards increasing the spectral efficiency associated with this shift is a new point of view that fading can be viewed as an opportunity to be exploited The main objective of the book is to provide a unified treatment of wireless communication from both these points of view In addition to traditional topics such as diversity and interference averaging a substantial portion of the book will be devoted to more modern topics such as opportunistic and multiple input multiple output MIMO communication An important component of this book is the system view emphasis the successful implementation of a theoretical concept or a technique requires an understanding of how it interacts with the wireless system as a whole Unlike the derivation of a concept or a technique this system view is less malleable to mathematical formulations and is primarily acquired through experience with designing actual wireless systems We try to help the reader develop some of this intuition by giving numerous examples of how the concepts are applied in actual wireless systems Five examples of wireless systems are used The next section gives some sense of the scope of the wireless systems considered in this book 12 Wireless systems Wireless communication despite the hype of the popular press is a field that has been around for over a hundred years starting around 1897 with Marconis successful demonstrations of wireless telegraphy By 1901 radio reception across the Atlantic Ocean had been established thus rapid progress in technology has also been around for quite a while In the intervening hundred years many types of wireless systems have flourished and often later disappeared For example television transmission in its early days was broadcast by wireless radio transmitters which are increasingly being replaced by cable transmission Similarly the pointtopoint microwave circuits that formed the backbone of the telephone network are being replaced by optical fiber In the first example wireless technology became outdated when a wired distribution network was installed in the second a new wired technology optical fiber replaced the older technology The opposite type of example is occurring today in telephony where wireless cellular technology is partially replacing the use of the wired telephone network particularly in parts of the world where the wired network is not well developed The point of these examples is that there are many situations in which there is a choice 3 12 Wireless systems between wireless and wire technologies and the choice often changes when new technologies become available In this book we will concentrate on cellular networks both because they are of great current interest and also because the features of many other wireless systems can be easily understood as special cases or simple generalizations of the features of cellular networks A cellular network consists of a large number of wireless subscribers who have cellular telephones users that can be used in cars in buildings on the street or almost anywhere There are also a number of fixed basestations arranged to provide coverage of the subscribers The area covered by a basestation ie the area from which incoming calls reach that basestation is called a cell One often pictures a cell as a hexagonal region with the basestation in the middle One then pictures a city or region as being broken up into a hexagonal lattice of cells see Figure 11a In reality the basestations are placed somewhat irregularly depending on the location of places such as building tops or hill tops that have good communication coverage and that can be leased or bought see Figure 11b Similarly mobile users connected to a basestation are chosen by good communication paths rather than geographic distance When a user makes a call it is connected to the basestation to which it appears to have the best path often but not always the closest basestation The basestations in a given area are then connected to a mobile telephone switching office MTSO also called a mobile switching center MSC by high speed wire connections or microwave links The MTSO is connected to the public wired telephone network Thus an incoming call from a mobile user is first connected to a basestation and from there to the MTSO and then to the wired network From there the call goes to its destination which might be an ordinary wire line telephone or might be another mobile subscriber Thus we see that a cellular network is not an independent network but rather an appendage to the wired network The MTSO also plays a major role in coordinating which basestation will handle a call to or from a user and when to handoff a user from one basestation to another When another user either wired or wireless places a call to a given user the reverse process takes place First the MTSO for the called subscriber is found Figure 11 Cells and basestations for a cellular network a An oversimplified view in which each cell is hexagonal b A more realistic case where basestations are irregularly placed and cell phones choose the best basestation a b 4 Introduction then the closest basestation is found and finally the call is set up through the MTSO and the basestation The wireless link from a basestation to the mobile users is interchangeably called the downlink or the forward channel and the link from the users to a basestation is called the uplink or a reverse channel There are usually many users connected to a single basestation and thus for the downlink channel the basestation must multiplex together the signals to the various connected users and then broadcast one waveform from which each user can extract its own signal For the uplink channel each user connected to a given basestation transmits its own waveform and the basestation receives the sum of the waveforms from the various users plus noise The basestation must then separate out the signals from each user and forward these signals to the MTSO Older cellular systems such as the AMPS advanced mobile phone service system developed in the USA in the eighties are analog That is a voice waveform is modulated on a carrier and transmitted without being trans formed into a digital stream Different users in the same cell are assigned different modulation frequencies and adjacent cells use different sets of fre quencies Cells sufficiently far away from each other can reuse the same set of frequencies with little danger of interference Secondgeneration cellular systems are digital One is the GSM global system for mobile communication system which was standardized in Europe but now used worldwide another is the TDMA timedivision multiple access standard developed in the USA IS136 and a third is CDMA code division multiple access IS95 Since these cellular systems and their standards were originally developed for telephony the current data rates and delays in cellular systems are essentially determined by voice requirements Third generation cellular systems are designed to handle data andor voice While some of the thirdgeneration systems are essentially evolution of second generation voice systems others are designed from scratch to cater for the specific characteristics of data In addition to a requirement for higher rates data applications have two features that distinguish them from voice Many data applications are extremely bursty users may remain inactive for long periods of time but have very high demands for short periods of time Voice applications in contrast have a fixedrate demand over long periods of time Voice has a relatively tight latency requirement of the order of 100 ms Data applications have a wide range of latency requirements realtime applications such as gaming may have even tighter delay requirements than voice while many others such as http file transfers have a much laxer requirement In the book we will see the impact of these features on the appropriate choice of communication techniques 5 13 Book outline As mentioned above there are many kinds of wireless systems other than cellular First there are the broadcast systems such as AM radio FM radio TV and paging systems All of these are similar to the downlink part of cellular networks although the data rates the sizes of the areas covered by each broadcasting node and the frequency ranges are very different Next there are wireless LANs local area networks These are designed for much higher data rates than cellular systems but otherwise are similar to a single cell of a cellular system These are designed to connect laptops and other portable devices in the local area network within an office building or similar environment There is little mobility expected in such systems and their major function is to allow portability The major standards for wireless LANs are the IEEE 80211 family There are smallerscale standards like Bluetooth or a more recent one based on ultrawideband UWB communication whose purpose is to reduce cabling in an office and simplify transfers between office and handheld devices Finally there is another type of LAN called an ad hoc network Here instead of a central node basestation through which all traffic flows the nodes are all alike The network organizes itself into links between various pairs of nodes and develops routing tables using these links Here the network layer issues of routing dissemination of control information etc are important concerns although problems of relaying and distributed cooperation between nodes can be tackled from the physicallayer as well and are active areas of current research 13 Book outline The central object of interest is the wireless fading channel Chapter 2 intro duces the multipath fading channel model that we use for the rest of the book Starting from a continuoustime passband channel we derive a discretetime complex baseband model more suitable for analysis and design Key physical parameters such as coherence time coherence bandwidth Doppler spread and delay spread are explained and several statistical models for multipath fading are surveyed There have been many statistical models proposed in the literature we will be far from exhaustive here The goal is to have a small set of example models in our repertoire to evaluate the performance of basic communication techniques we will study Chapter 3 introduces many of the issues of communicating over fading channels in the simplest pointtopoint context As a baseline we start by look ing at the problem of detection of uncoded transmission over a narrowband fading channel We find that the performance is very poor much worse than over the additive white Gaussian noise AWGN channel with the same average signaltonoise ratio SNR This is due to a significant probability that the channel is in deep fade Various diversity techniques to mitigate this adverse effect of fading are then studied Diversity techniques increase 6 Introduction reliability by sending the same information through multiple independently faded paths so that the probability of successful transmission is higher Some of the techniques studied include interleaving of coded symbols over time to obtain time diversity intersymbol equalization multipath combining in spreadspectrum systems and coding over subcarriers in orthogonal frequency division multiplexing OFDM systems to obtain frequency diversity use of multiple transmit andor receive antennas via spacetime coding to obtain spatial diversity In some scenarios there is an interesting interplay between channel uncer tainty and the diversity gain as the number of diversity branches increases the performance of the system first improves due to the diversity gain but then subsequently deteriorates as channel uncertainty makes it more difficult to combine signals from the different branches In Chapter 4 the focus is shifted from pointtopoint communication to studying cellular systems as a whole Multiple access and intercell interfer ence management are the key issues that come to the forefront We explain how existing digital wireless systems deal with these issues The concepts of frequency reuse and cell sectorization are discussed and we contrast nar rowband systems such as GSM and IS136 where users within the same cell are kept orthogonal and frequency is reused only in cells far away and CDMA systems such as IS95 where the signals of users both within the same cell and across different cells are spread across the same spectrum ie frequency reuse factor of 1 Due to the full reuse CDMA systems have to manage intracell and intercell interference more efficiently in addition to the diversity techniques of timeinterleaving multipath combining and soft handoff power control and interference averaging are the key interference management mechanisms All the five techniques strive toward the same sys tem goal to maintain the channel quality of each user as measured by the signaltointerferenceandnoise ratio SINR as constant as possible This chapter is concluded with the discussion of a wideband OFDM system which combines the advantages of both the CDMA and the narrowband systems Chapter 5 studies the capacity of wireless channels This provides a higher level view of the tradeoffs involved in the earlier chapters and also lays the foundation for understanding the more modern developments in the subse quent chapters The performance over the nonfaded AWGN channel as a baseline for comparison We introduce the concept of channel capacity as the basic performance measure The capacity of a channel provides the fun damental limit of communication achievable by any scheme For the fading channel there are several capacity measures relevant for different scenarios Two distinct scenarios provide particular insight 1 the slow fading channel where the channel stays the same random value over the entire timescale 7 13 Book outline of communication and 2 the fast fading channel where the channel varies significantly over the timescale of communication In the slow fading channel the key event of interest is outage this is the situation when the channel is so poor that no scheme can communicate reliably at a certain target data rate The largest rate of reliable communication at a certain outage probability is called the outage capacity In the fast fading channel in contrast outage can be avoided due to the ability to average over the time variation of the channel and one can define a positive capacity at which arbitrarily reliable communication is possible Using these capacity measures several resources associated with a fading channel are defined 1 diversity 2 number of degrees of freedom 3 received power These three resources form a basis for assessing the nature of performance gain by the various communication schemes studied in the rest of the book Chapters 6 to 10 cover the more recent developments in the field In Chapter 6 we revisit the problem of multiple access over fading channels from a more fundamental point of view Information theory suggests that if both the transmitters and the receiver can track the fading channel the optimal strategy to maximize the total system throughput is to allow only the user with the best channel to transmit at any time A similar strategy is also optimal for the downlink Opportunistic strategies of this type yield a systemwide multiuser diversity gain the more users in the system the larger the gain as there is more likely to be a user with a very strong channel To implement this concept in a real system three important considerations are fairness of the resource allocation across users delay experienced by the individual user waiting for its channel to become good and measurement inaccuracy and delay in feeding back the channel state to the transmitters We discuss how these issues are addressed in the context of IS865 also called HDR or CDMA 2000 1 EVDO a thirdgeneration wireless data system A wireless system consists of multiple dimensions time frequency space and users Opportunistic communication maximizes the spectral efficiency by measuring when and where the channel is good and only transmits in those degrees of freedom In this context channel fading is beneficial in the sense that the fluctuation of the channel across the degrees of freedom ensures that there will be some degrees of freedom in which the channel is very good This is in sharp contrast to the diversitybased approach in Chapter 3 where channel fluctuation is always detrimental and the design goal is to average out the fading to make the overall channel as constant as possible Taking this philosophy one step further we discuss a technique called opportunistic beamforming in which channel fluctuation can be induced in situations when the natural fading has small dynamic range andor is slow From the cellular system point of view this technique also increases the fluctuations of the interference imparted on adjacent cells and presents an opposing philosophy to the notion of interference averaging in CDMA systems 8 Introduction Chapters 7 8 9 and 10 discuss multiple input multiple output MIMO communication It has been known for a while that the uplink with multiple receive antennas at the basestation allow several users to simultaneously communicate to the receiver The multiple antennas in effect increase the number of degrees of freedom in the system and allow spatial separation of the signals from the different users It has recently been shown that a similar effect occurs for pointtopoint channels with multiple transmit and receive antennas ie even when the antennas of the multiple users are colocated This holds provided that the scattering environment is rich enough to allow the receive antennas to separate out the signal from the different transmit antennas allowing the spatial multiplexing of information This is yet another example where channel fading is beneficial to communication Chapter 7 studies the properties of the multipath environment that determine the amount of spatial multiplexing possible and defines an angular domain in which such properties are seen most explicitly We conclude with a class of statistical MIMO channel models based in the angular domain which will be used in later chapters to analyze the performance of communication techniques Chapter 8 discusses the capacity and capacityachieving transceiver archi tectures for MIMO channels focusing on the fast fading scenario It is demon strated that the fast fading capacity increases linearly with the minimum of the number of transmit and receive antennas at all values of SNR At high SNR the linear increase is due to the increase in degrees of freedom from spatial multiplexing At low SNR the linear increase is due to a power gain from receive beamforming At intermediate SNR ranges the linear increase is due to a combination of both these gains Next we study the transceiver architectures that achieve the capacity of the fast fading channel The focus is on the VBLAST architecture which multiplexes independent data streams one onto each of the transmit antennas A variety of receiver structures are considered these include the decorrelator and the linear minimum mean squareerror MMSE receiver The performance of these receivers can be enhanced by successively canceling the streams as they are decoded this is known as successive interference cancellation SIC It is shown that the MMSESIC receiver achieves the capacity of the fast fading MIMO channel The VBLAST architecture is very suboptimal for the slow fading MIMO channel it does not code across the transmit antennas and thus the diversity gain is limited by that obtained with the receive antenna array A modifi cation called DBLAST where the data streams are interleaved across the transmit antenna array achieves the outage capacity of the slow fading MIMO channel The boost of the outage capacity of a MIMO channel as compared to a single antenna channel is due to a combination of both diversity and spatial multiplexing gains In Chapter 9 we study a fundamental tradeoff between the diversity and multiplexing gains that can be simultaneously har nessed over a slow fading MIMO channel This formulation is then used as a unified framework to assess both the diversity and multiplexing performance 9 13 Book outline of several schemes that have appeared earlier in the book This framework is also used to motivate the construction of new tradeoffoptimal spacetime codes In particular we discuss an approach to design universal spacetime codes that are tradeoffoptimal Finally Chapter 10 studies the use of multiple transmit and receive antennas in multiuser and cellular systems this is also called spacedivision multi ple access SDMA Here in addition to providing spatial multiplexing and diversity multiple antennas can also be used to mitigate interference between different users In the uplink interference mitigation is done at the base station via the SIC receiver In the downlink interference mitigation is also done at the basestation and this requires precoding we study a precoding scheme called Costa or dirtypaper precoding that is the natural analog of the SIC receiver in the uplink This study allows us to relate the performance of an SIC receiver in the uplink with a corresponding precoding scheme in a reciprocal downlink The ArrayComm system is used as an example of an SDMA cellular system C H A P T E R 2 The wireless channel A good understanding of the wireless channel its key physical parameters and the modeling issues lays the foundation for the rest of the book This is the goal of this chapter A defining characteristic of the mobile wireless channel is the variations of the channel strength over time and over frequency The variations can be roughly divided into two types Figure 21 Largescale fading due to path loss of signal as a function of distance and shadowing by large objects such as buildings and hills This occurs as the mobile moves through a distance of the order of the cell size and is typically frequency independent Smallscale fading due to the constructive and destructive interference of the multiple signal paths between the transmitter and receiver This occurs at the spatialscaleoftheorderofthecarrierwavelengthandisfrequencydependent We will talk about both types of fading in this chapter but with more emphasis on the latter Largescale fading is more relevant to issues such as cellsite planning Smallscale multipath fading is more relevant to the design of reliable and efficient communication systems the focus of this book We start with the physical modeling of the wireless channel in terms of elec tromagnetic waves We then derive an inputoutput linear timevarying model for the channel and define some important physical parameters Finally we introduce a few statistical models of the channel variation over time and over frequency 21 Physical modeling for wireless channels Wireless channels operate through electromagnetic radiation from the trans mitter to the receiver In principle one could solve the electromagnetic field equations in conjunction with the transmitted signal to find the 10 Figure 21 Channel quality varies over multiple timescales At a slow scale channel varies due to largescale fading effects At a fast scale channel varies due to multipath effects Channel quality Time electromagnetic field impinging on the receiver antenna This would have to be done taking into account the obstructions caused by ground buildings vehicles etc in the vicinity of this electromagnetic wave Cellular communication in the USA is limited by the Federal Communication Commission FCC and by similar authorities in other countries to one of three frequency bands one around 09 GHz one around 19 GHz and one around 58 GHz The wavelength λ of electromagnetic radiation at any given frequency f is given by λ cf where c 3 108 ms is the speed of light The wavelength in these cellular bands is thus a fraction of a meter so to calculate the electromagnetic field at a receiver the locations of the receiver and the obstructions would have to be known within submeter accuracies The electromagnetic field equations are therefore too complex to solve especially on the fly for mobile users Thus we have to ask what we really need to know about these channels and what approximations might be reasonable One of the important questions is where to choose to place the basestations and what range of power levels are then necessary on the downlink and uplink channels To some extent this question must be answered experimentally but it certainly helps to have a sense of what types of phenomena to expect Another major question is what types of modulation and detection techniques look promising Here again we need a sense of what types of phenomena to expect To address this we will construct stochastic models of the channel assuming that different channel behaviors appear with different probabilities and change over time with specific stochastic properties We will return to the question of why such stochastic models are appropriate but for now we simply want to explore the gross characteristics of these channels Let us start by looking at several overidealized examples 1 By obstructions we mean not only objects in the lineofsight between transmitter and receiver but also objects in locations that cause nonnegligible changes in the electromagnetic field at the receiver we shall see examples of such obstructions later The wireless channel 211 Free space fixed transmit and receive antennas First consider a fixed antenna radiating into free space In the far field the electric field and magnetic field at any given location are perpendicular both to each other and to the direction of propagation from the antenna They are also proportional to each other so it is sufficient to know only one of them just as in wired communication where we view a signal as simply a voltage waveform or a current waveform In response to a transmitted sinusoid cos 2πft we can express the electric far field at time t as Ef t r θ ψ αsθ ψ f cos 2πft rc r 21 Here r θ ψ represents the point u in space at which the electric field is being measured where r is the distance from the transmit antenna to u and where θ ψ represents the vertical and horizontal angles from the antenna to u respectively The constant c is the speed of light and αsθ ψ f is the radiation pattern of the sending antenna at frequency f in the direction θ ψ it also contains a scaling factor to account for antenna losses Note that the phase of the field varies with frc corresponding to the delay caused by the radiation traveling at the speed of light We are not concerned here with actually finding the radiation pattern for any given antenna but only with recognizing that antennas have radiation patterns and that the free space far field behaves as above It is important to observe that as the distance r increases the electric field decreases as r1 and thus the power per square meter in the free space wave decreases as r2 This is expected since if we look at concentric spheres of increasing radius r around the antenna the total power radiated through the sphere remains constant but the surface area increases as r2 Thus the power per unit area must decrease as r2 We will see shortly that this r2 reduction of power with distance is often not valid when there are obstructions to free space propagation Next suppose there is a fixed receive antenna at the location u r θ ψ The received waveform in the absence of noise in response to the above transmitted sinusoid is then Erf t u αθ ψ f cos 2πf t rc r 22 where αθ ψ f is the product of the antenna patterns of transmit and receive antennas in the given direction Our approach to 22 is a bit odd since we started with the free space field at u in the absence of an antenna Placing a The far field is the field sufficiently far away from the antenna so that 21 is valid For cellular systems it is a safe assumption that the receiver is in the far field 21 Physical modeling for wireless channels receive antenna there changes the electric field in the vicinity of u but this is taken into account by the antenna pattern of the receive antenna Now suppose for the given u that we define Hf αθ ψ fej2πfrc r 23 We then have Erf t u ℜ Hfej2πft We have not mentioned it yet but 21 and 22 are both linear in the input That is the received field waveform at u in response to a weighted sum of transmitted waveforms is simply the weighted sum of responses to those individual waveforms Thus Hf is the system function for an LTI linear timeinvariant channel and its inverse Fourier transform is the impulse response The need for understanding electromagnetism is to determine what this system function is We will find in what follows that linearity is a good assumption for all the wireless channels we consider but that the time invariance does not hold when either the antennas or obstructions are in relative motion 212 Free space moving antenna Next consider the fixed antenna and free space model above with a receive antenna that is moving with speed v in the direction of increasing distance from the transmit antenna That is we assume that the receive antenna is at a moving location described as ut rt θ ψ with rt r0 vt Using 21 to describe the free space electric field at the moving point ut for the moment with no receive antenna we have Ef t r0 vt θ ψ αsθ ψ f cos 2πf t r0c vtc r0 vt 24 Note that we can rewrite ft r0c vtc as f1 vct fr0c Thus the sinusoid at frequency f has been converted to a sinusoid of frequency f1 vc there has been a Doppler shift of fvc due to the motion of the observation point Intuitively each successive crest in the transmitted sinusoid has to travel a little further before it gets observed at the moving observation point If the antenna is now placed at ut and the change of field due to the antenna presence is again represented by the receive antenna pattern the received waveform in analogy to 22 is Erf t r0 vt θ ψ αθ ψ f cos 2πf1 vct r0c r0 vt 25 The reader should be familiar with the Doppler shift associated with moving cars When an ambulance is rapidly moving toward us we hear a higher frequency siren When it passes us we hear a rapid shift toward a lower frequency The wireless channel This channel cannot be represented as an LTI channel If we ignore the timevarying attenuation in the denominator of 25 however we can represent the channel in terms of a system function followed by translating the frequency f by the Doppler shift fvc It is important to observe that the amount of shift depends on the frequency f We will come back to discussing the importance of this Doppler shift and of the timevarying attenuation after considering the next example The above analysis does not depend on whether it is the transmitter or the receiver or both that are moving So long as rt is interpreted as the distance between the antennas and the relative orientations of the antennas are constant 24 and 25 are valid 213 Reflecting wall fixed antenna Consider Figure 22 in which there is a fixed antenna transmitting the sinusoid cos 2πft a fixed receive antenna and a single perfectly reflecting large fixed wall We assume that in the absence of the receive antenna the electromagnetic field at the point where the receive antenna will be placed is the sum of the free space field coming from the transmit antenna plus a reflected wave coming from the wall As before in the presence of the receive antenna the perturbation of the field due to the antenna is represented by the antenna pattern An additional assumption here is that the presence of the receive antenna does not appreciably affect the plane wave impinging on the wall In essence what we have done here is to approximate the solution of Maxwells equations by a method called ray tracing The assumption here is that the received waveform can be approximated by the sum of the free space wave from the transmitter plus the reflected free space waves from each of the reflecting obstacles In the present situation if we assume that the wall is very large the reflected wave at a given point is the same except for a sign change as the free space wave that would exist on the opposite side of the wall if the wall were not present see Figure 23 This means that the reflected wave from the wall has the intensity of a free space wave at a distance equal to the distance to the wall and then Transmit antenna Wall Receive antenna Figure 22 Illustration of a direct path and a reflected path By basic electromagnetics this sign is a consequence of the fact that the electric field is parallel to the plane of the wall for this example Figure 23 Relation of reflected wave to wave without wall Transmit antenna Wall back to the receive antenna ie 2d r Using 22 for both the direct and the reflected wave and assuming the same antenna gain α for both waves we get Erft α cos 2πftrcr α cos 2πft2drc2dr 26 The received signal is a superposition of two waves both of frequency f The phase difference between the two waves is Δθ 2πf2drc π 2πfrc 4πfcdr π 27 When the phase difference is an integer multiple of 2π the two waves add constructively and the received signal is strong When the phase difference is an odd integer multiple of π the two waves add destructively and the received signal is weak As a function of r this translates into a spatial pattern of constructive and destructive interference of the waves The distance from a peak to a valley is called the coherence distance Δxc λ4 28 where λ cf is the wavelength of the transmitted sinusoid At distances much smaller than Δxc the received signal at a particular time does not change appreciably The constructive and destructive interference pattern also depends on the frequency f for a fixed r if f changes by 122drc rc1 29 we move from a peak to a valley The quantity Td 2drc rc 210 is called the delay spread of the channel it is the difference between the propagation delays along the two signal paths The constructive and destructive interference pattern does not change appreciably if the frequency changes by an amount much smaller than 1Td This parameter is called the coherence bandwidth 214 Reflecting wall moving antenna Suppose the receive antenna is now moving at a velocity v Figure 24 As it moves through the pattern of constructive and destructive interference created by the two waves the strength of the received signal increases and decreases This is the phenomenon of multipath fading The time taken to travel from a peak to a valley is c4fv this is the timescale at which the fading occurs and it is called the coherence time of the channel An equivalent way of seeing this is in terms of the Doppler shifts of the direct and the reflected waves Suppose the receive antenna is at location r0 at time 0 Taking r r0 vt in 26 we get Erft α cos 2πf1vctr0cr0vt α cos 2πf1vctr02dc2dr0vt 211 The first term the direct wave is a sinusoid at frequency f1vc experiencing a Doppler shift D1 fvc The second is a sinusoid at frequency f1vc with a Doppler shift D2 fvc The parameter Ds D2D1 212 is called the Doppler spread For example if the mobile is moving at 60 kmh and f 900 MHz the Doppler spread is 100 Hz The role of the Doppler spread can be visualized most easily when the mobile is much closer to the wall than to the transmit antenna In this case the attenuations are roughly the same for both paths and we can approximate the denominator of the second term by r r0 vt Then combining the two sinusoids we get Erft 2α sin 2πfvtc r0dc sin 2πftdcr0 vt 213 This is the product of two sinusoids one at the input frequency f which is typically of the order of GHz and the other one at fvc Ds2 which might be of the order of 50 Hz Thus the response to a sinusoid at f is another sinusoid at f with a timevarying envelope with peaks going to zeros around every 5 ms Figure 25 The envelope is at its widest when the mobile is at a peak of the Transmit antenna d Wall rt v Figure 24 Illustration of a direct path and a reflected path Figure 25 The received waveform oscillating at frequency f with a slowly varying envelope at frequency Ds2 Ert t interference pattern and at its narrowest when the mobile is at a valley Thus the Doppler spread determines the rate of traversal across the interference pattern and is inversely proportional to the coherence time of the channel We now see why we have partially ignored the denominator terms in 211 and 213 When the difference in the length between two paths changes by a quarter wavelength the phase difference between the responses on the two paths changes by π2 which causes a very significant change in the overall received amplitude Since the carrier wavelength is very small relative to the path lengths the time over which this phase effect causes a significant change is far smaller than the time over which the denominator terms cause a significant change The effect of the phase changes is of the order of milliseconds whereas the effect of changes in the denominator is of the order of seconds or minutes In terms of modulation and detection the timescales of interest are in the range of milliseconds and less and the denominators are effectively constant over these periods The reader might notice that we are constantly making approximations in trying to understand wireless communication much more so than for wired communication This is partly because wired channels are typically timeinvariant over a very long timescale while wireless channels are typically timevarying and appropriate models depend very much on the timescales of interest For wireless systems the most important issue is what approximations to make Thus it is important to understand these modeling issues thoroughly 215 Reflection from a ground plane Consider a transmit and a receive antenna both above a plane surface such as a road Figure 26 When the horizontal distance r between the antennas becomes very large relative to their vertical displacements from the ground 18 The wireless channel Figure 26 Illustration of a direct path and a reflected path off a ground plane Transmit antenna Groud plane Receive antenna hr hs r2 r r1 plane ie height a very surprising thing happens In particular the differ ence between the direct path length and the reflected path length goes to zero as r1 with increasing r Exercise 25 When r is large enough this difference between the path lengths becomes small relative to the wavelength cf Since the sign of the electric field is reversed on the reflected path5 these two waves start to cancel each other out The electric wave at the receiver is then attenu ated as r2 and the received power decreases as r4 This situation is partic ularly important in rural areas where basestations tend to be placed on roads 216 Power decay with distance and shadowing The previous example with reflection from a ground plane suggests that the received power can decrease with distance faster than r2 in the presence of disturbances to free space In practice there are several obstacles between the transmitter and the receiver and further the obstacles might also absorb some power while scattering the rest Thus one expects the power decay to be considerably faster than r2 Indeed empirical evidence from experimental field studies suggests that while power decay near the transmitter is like r2 at large distances the power can even decay exponentially with distance The ray tracing approach used so far provides a high degree of numerical accuracy in determining the electric field at the receiver but requires a precise physical model including the location of the obstacles But here we are only looking for the order of decay of power with distance and can consider an alternative approach So we look for a model of the physical environment with the fewest parameters but one that still provides useful global information about the field properties A simple probabilistic model with two parameters of the physical environment the density of the obstacles and the fraction of energy each object absorbs is developed in Exercise 26 With each obstacle 5 This is clearly true if the electric field is parallel to the ground plane It turns out that this is also true for arbitrary orientations of the electric field as long as the ground is not a perfect conductor and the angle of incidence is small enough The underlying electromagnetics is analyzed in Chapter 2 of Jakes 62 absorbing the same fraction of the energy impinging on it the model allows us to show that the power decays exponentially in distance at a rate that is proportional to the density of the obstacles With a limit on the transmit power either at the basestation or at the mobile the largest distance between the basestation and a mobile at which communication can reliably take place is called the coverage of the cell For reliable communication a minimal received power level has to be met and thus the fast decay of power with distance constrains cell coverage On the other hand rapid signal attenuation with distance is also helpful it reduces the interference between adjacent cells As cellular systems become more popular however the major determinant of cell size is the number of mobiles in the cell In engineering jargon the cell is said to be capacity limited instead of coverage limited The size of cells has been steadily decreasing and one talks of micro cells and pico cells as a response to this effect With capacity limited cells the intercell interference may be intolerably high To alleviate the intercell interference neighboring cells use different parts of the frequency spectrum and frequency is reused at cells that are far enough Rapid signal attenuation with distance allows frequencies to be reused at closer distances The density of obstacles between the transmit and receive antennas depends very much on the physical environment For example outdoor plains have very little by way of obstacles while indoor environments pose many obstacles This randomness in the environment is captured by modeling the density of obstacles and their absorption behavior as random numbers the overall phenomenon is called shadowing The effect of shadow fading differs from multipath fading in an important way The duration of a shadow fade lasts for multiple seconds or minutes and hence occurs at a much slower timescale compared to multipath fading 217 Moving antenna multiple reflectors Dealing with multiple reflectors using the technique of ray tracing is in principle simply a matter of modeling the received waveform as the sum of the responses from the different paths rather than just two paths We have seen enough examples however to understand that finding the magnitudes and phases of these responses is no simple task Even for the very simple large wall example in Figure 22 the reflected field calculated in 26 is valid only at distances from the wall that are small relative to the dimensions of the wall At very large distances the total power reflected from the wall is proportional to both d2 and to the area of the cross section of the wall The power reaching the receiver is proportional to d rt2 Thus the power attenuation from transmitter to receiver for the large distance case is proportional to ddrt2 rather than to 2d rt2 This shows that ray tracing must be used with some caution Fortunately however linearity still holds in these more complex cases Another type of reflection is known as scattering and can occur in the atmosphere or in reflections from very rough objects Here there are a very large number of individual paths and the received waveform is better modeled as an integral over paths with infinitesimally small differences in their lengths rather than as a sum Knowing how to find the amplitude of the reflected field from each type of reflector is helpful in determining the coverage of a basestation although ultimately experimentation is necessary This is an important topic if our objective is trying to determine where to place basestations Studying this in more depth however would take us afield and too far into electromagnetic theory In addition we are primarily interested in questions of modulation detection multiple access and network protocols rather than location of basestations Thus we turn our attention to understanding the nature of the aggregate received waveform given a representation for each reflected wave This leads to modeling the inputoutput behavior of a channel rather than the detailed response on each path 22 Inputoutput model of the wireless channel We derive an inputoutput model in this section We first show that the multipath effects can be modeled as a linear timevarying system We then obtain a baseband representation of this model The continuoustime channel is then sampled to obtain a discretetime model Finally we incorporate additive noise 221 The wireless channel as a linear timevarying system In the previous section we focused on the response to the sinusoidal input phit cos 2 pi f t The received signal can be written as sum over i of aift phi t tauift where aift and tauift are respectively the overall attenuation and propagation delay at time t from the transmitter to the receiver on path i The overall attenuation is simply the product of the attenuation factors due to the antenna pattern of the transmitter and the receiver the nature of the reflector as well as a factor that is a function of the distance from the transmitting antenna to the reflector and from the reflector to the receive antenna We have described the channel effect at a particular frequency f If we further assume that the aift and the tauift do not depend on the frequency f then we can use the principle of superposition to generalize the above inputoutput relation to an arbitrary input xt with nonzero bandwidth yt sum over i of aitxttauit In practice the attenuations and the propagation delays are usually slowly varying functions of frequency These variations follow from the timevarying path lengths and also from frequencydependent antenna gains However we are primarily interested in transmitting over bands that are narrow relative to the carrier frequency and over such ranges we can omit this frequency dependence It should however be noted that although the individual attenuations and delays are assumed to be independent of the frequency the overall channel response can still vary with frequency due to the fact that different paths have different delays For the example of a perfectly reflecting wall in Figure 24 then a1t alpha r0 v t a2t alpha 2 d r0 v t tau1t r0 vt c angle phi1 2 pi f tau2t 2d r0 vt c angle phi2 2 pi f where the first expression is for the direct path and the second for the reflected path The term angle phij here is to account for possible phase changes at the transmitter reflector and receiver For the example here there is a phase reversal at the reflector so we take phi1 0 and phi2 pi Since the channel 214 is linear it can be described by the response htaut at time t to an impulse transmitted at time t tau In terms of htaut the inputoutput relationship is given by yt integral from infinity to infinity of htaut xt tau d tau Comparing 217 and 214 we see that the impulse response for the fading multipath channel is htaut sum over i of ait delta tau tauit This expression is really quite nice It says that the effect of mobile users arbitrarily moving reflectors and absorbers and all of the complexities of solving Maxwells equations finally reduce to an inputoutput relation between transmit and receive antennas which is simply represented as the impulse response of a linear timevarying channel filter The effect of the Doppler shift is not immediately evident in this representation From 216 for the single reflecting wall example tauit vi c where vi is the velocity with which the ith path length is increasing Thus the Doppler shift on the ith path is f tauit In the special case when the transmitter receiver and the environment are all stationary the attenuations ait and propagation delays tauit do not depend on time t and we have the usual linear timeinvariant channel with an impulse response hτ i ai δτ τi 219 For the timevarying impulse response hτ t we can define a timevarying frequency response Hf t hτ tej2πfτ dτ i aitej2πfτit 220 In the special case when the channel is timeinvariant this reduces to the usual frequency response One way of interpreting Hf t is to think of the system as a slowly varying function of t with a frequency response Hf t at each fixed time t Corresponding hτ t can be thought of as the impulse response of the system at a fixed time t This is a legitimate and useful way of thinking about many multipath fading channels as the timescale at which the channel varies is typically much longer than the delay spread ie the amount of memory of the impulse response at a fixed time In the reflecting wall example in Section 214 the time taken for the channel to change significantly is of the order of milliseconds while the delay spread is of the order of microseconds Fading channels which have this characteristic are sometimes called underspread channels 222 Baseband equivalent model In typical wireless applications communication occurs in a passband fc W2 fc W2 of bandwidth W around a center frequency fc the spectrum having been specified by regulatory authorities However most of the processing such as codingdecoding modulationdemodulation synchronization etc is actually done at the baseband At the transmitter the last stage of the operation is to upconvert the signal to the carrier frequency and transmit it via the antenna Similarly the first step at the receiver is to downconvert the RF radiofrequency signal to the baseband before further processing Therefore from a communication system design point of view it is most useful to have a baseband equivalent representation of the system We first start with defining the baseband equivalent representation of signals Consider a real signal st with Fourier transform Sf bandlimited in fc W2 fc W2 with W 2fc Define its complex baseband equivalent sbt as the signal having Fourier transform Sbf 2 Sf fc f fc 0 0 f fc 0 221 Figure 27 Illustration of the relationship between a passband spectrum Sf and its baseband equivalent Sbf Since st is real its Fourier transform satisfies Sf Sf which means that sbt contains exactly the same information as st The factor of 2 is quite arbitrary but chosen to normalize the energies of sbt and st to be the same Note that sbt is bandlimited in W2 W2 See Figure 27 To reconstruct st from sbt we observe that 2 Sf Sbf fc Sb f fc 222 Taking inverse Fourier transforms we get st 12 sbtej2πfc t sb tej2πfc t 2 ℜsbtej2 πfc t 223 In terms of real signals the relationship between st and sbt is shown in Figure 28 The passband signal st is obtained by modulating ℜsbt by 2 cos 2π fc t and 𝕴sbt by 2 sin 2π fc t and summing to get 2ℜsbtej2πfc t upconversion The baseband signal ℜsbt respectively 𝕴sbt is obtained by modulating st by 2 cos 2π fc t respectively 2 sin 2π fc t followed by ideal lowpass filtering at the baseband W2 W2 downconversion Let us now go back to the multipath fading channel 214 with impulse response given by 218 Let xbt and ybt be the complex baseband equivalents of the transmitted signal xt and the received signal yt respectively Figure 29 shows the system diagram from xbt to ybt This implementation of a passband communication system is known as quadrature amplitude modulation QAM The signal ℜxbt is sometimes called the Figure 28 Illustration of upconversion from sbt to st followed by downconversion from st back to sbt Figure 29 System diagram from the baseband transmitted signal xbt to the baseband received signal ybt inphase component I and 𝕴xbt the quadrature component Q rotated by π2 We now calculate the baseband equivalent channel Substituting xt 2 ℜxbtej2πfc t and yt 2ℜybtej2πfc t into 214 we get ℜybtej2πfc t i aitℜxbt τitej2πfct τit ℜi aitxbt τitej2πfc τit ej2πfc t 224 Similarly one can obtain Exercise 213 𝕴ybtej2πfc t 𝕴i aitxbt τitej2πfc τit ej2πfc t 225 Hence the baseband equivalent channel is ybt i aibtxbt τit 226 22 Inputoutput model of the wireless channel where aibt aitej2pi fc auit 227 The inputoutput relationship in 226 is also that of a linear timevarying system and the baseband equivalent impulse response is hb au t sumi aibt delta au auit 228 This representation is easy to interpret in the time domain where the effect of the carrier frequency can be seen explicitly The baseband output is the sum over each path of the delayed replicas of the baseband input The magnitude of the i th such term is the magnitude of the response on the given path this changes slowly with significant changes occurring on the order of seconds or more The phase is changed by pi2 ie is changed significantly when the delay on the path changes by 14fc or equivalently when the path length changes by a quarter wavelength ie by c4fc If the path length is changing at velocity v the time required for such a phase change is c4fc v Recalling that the Doppler shift D at frequency f is fvc and noting that f approx fc for narrowband communication the time required for a pi2 phase change is 14D For the single reflecting wall example this is about 5 ms assuming fc 900 MHz and v 60 kmh The phases of both paths are rotating at this rate but in opposite directions Note that the Fourier transform Hbf t of hb au t for a fixed t is simply Hf fc t ie the frequency response of the original system at a fixed t shifted by the carrier frequency This provides another way of thinking about the baseband equivalent channel 223 A discretetime baseband model The next step in creating a useful channel model is to convert the continuoustime channel to a discretetime channel We take the usual approach of the sampling theorem Assume that the input waveform is bandlimited to W The baseband equivalent is then limited to W 2 and can be represented as xbt sumn xn mathrmsincW t n 229 where xn is given by xbn W and mathrmsinct is defined as mathrmsinct fracsinpi tpi t 230 This representation follows from the sampling theorem which says that any waveform bandlimited to W 2 can be expanded in terms of the orthogonal The wireless channel basis mathrmsincW t nn with coefficients given by the samples taken uniformly at integer multiples of 1 W Using 226 the baseband output is given by ybt sumn xn sumi aibt mathrmsincW t W auit n 231 The sampled outputs at multiples of 1 W ym ybm W are then given by ym sumn xn sumi aibm W mathrmsincm n auim W W 232 The sampled output ym can equivalently be thought of as the projection of the waveform ybt onto the waveform W mathrmsincW t m Let ell m n Then ym sumell xm ell sumi aibm W mathrmsincell auim W W 233 By defining hellm sumi aibm W mathrmsincell auim W W 234 233 can be written in the simple form ym sumell hellm xm ell 235 We denote hellm as the ell th complex channel filter tap at time m Its value is a function of mainly the gains aibt of the paths whose delays auit are close to ell W Figure 210 In the special case where the gains aibt and the delays auit of the paths are timeinvariant 234 simplifies to hell sumi aib mathrmsincell aui W 236 and the channel is linear timeinvariant The ell th tap can be interpreted as the sample ell W th of the lowpass filtered baseband channel response hb au cf 219 convolved with mathrmsincW au We can interpret the sampling operation as modulation and demodulation in a communication system At time n we are modulating the complex symbol xm inphase plus quadrature components by the sinc pulse before the upconversion At the receiver the received signal is sampled at times m W Figure 210 Due to the decay of the sinc function the i th path contributes most significantly to the ell th tap if its delay falls in the window ell W 1 2 W ell W 1 2 W at the output of the lowpass filter Figure 211 shows the complete system In practice other transmit pulses such as the raised cosine pulse are often used in place of the sinc pulse which has rather poor timedecay property and tends to be more susceptible to timing errors This necessitates sampling at the Nyquist sampling rate but does not alter the essential nature of the model Hence we will confine to Nyquist sampling Due to the Doppler spread the bandwidth of the output ybt is generally slightly larger than the bandwidth W 2 of the input xbt and thus the output samples ym do not fully represent the output waveform This problem is usually ignored in practice since the Doppler spread is small of the order of tens to hundreds of Hz compared to the bandwidth W Also it is very convenient for the sampling rate of the input and output to be the same Alternatively it would be possible to sample the output at twice the rate of the input This would recapture all the information in the received waveform Figure 211 System diagram from the baseband transmitted symbol xm to the baseband sampled received signal ym The number of taps would be almost doubled because of the reduced sample interval but it would typically be somewhat less than doubled since the representation would not spread the path delays so much 224 Additive white noise As a last step we include additive noise in our inputoutput model We make the standard assumption that wt is zeromean additive white Gaussian noise AWGN with power spectral density N02 ie Ew0wt N02δt The model 214 is now modified to be yt Σi aitxt τit wt 237 See Figure 212 The discretetime basebandequivalent model 235 now becomes ym Σℓ hemxm ℓ wm 238 where wm is the lowpass filtered noise at the sampling instant mW Just like the signal the white noise wt is downconverted filtered at the baseband and ideally sampled Thus it can be verified Exercise 211 that ℜwm wtψm1tdt 239 ℑwm wtψm2tdt 240 where ψm1t 2W cos2πfctsincWt m ψm2t 2W sin2πfctsincWt m 241 It can further be shown that ψm1t ψm2tm forms an orthonormal set of waveforms ie the waveforms are orthogonal to each other Exercise 212 In Appendix A we review the definition and basic properties of white Gaussian random vectors ie vectors whose components are independent and identically distributed iid Gaussian random variables A key property is that the projections of a white Gaussian random vector onto any orthonormal vectors are independent and identically distributed Gaussian random variables Heuristically one can think of continuoustime Gaussian white noise as an infinitedimensional white random vector and the above property carries through the projections onto orthogonal waveforms are uncorrelated and hence independent Hence the discretetime noise process wm is white ie independent over time moreover the real and imaginary components are iid Gaussians with variances N02 A complex Gaussian random variable X whose real and imaginary components are iid satisfies a circular symmetry property ejϕX has the same distribution as X for any ϕ We shall call such a random variable circular symmetric complex Figure 212 A complete system diagram Gaussian denoted by CN0 σ2 where σ2 EX2 The concept of circular symmetry is discussed further in Section A13 of Appendix A The assumption of AWGN essentially means that we are assuming that the primary source of the noise is at the receiver or is radiation impinging on the receiver that is independent of the paths over which the signal is being received This is normally a very good assumption for most communication situations 23 Time and frequency coherence 231 Doppler spread and coherence time An important channel parameter is the timescale of the variation of the channel How fast do the taps hem vary as a function of time m Recall that hem Σi abi mWsincℓ τimWW Σi aimWej2πfcτimWsincℓ τimWW 242 Let us look at this expression term by term From Section 222 we gather that significant changes in ai occur over periods of seconds or more Significant changes in the phase of the ith path occur at intervals of 14Di where Di fcτit is the Doppler shift for that path When the different paths contributing to the ℓth tap have different Doppler shifts the magnitude of hem changes significantly This is happening at the timescale inversely proportional to the largest difference between the Doppler shifts the Doppler spread Ds Ds maxij fc τit τjt 243 23 Time and frequency coherence where the maximum is taken over all the paths that contribute significantly to a tap7 Typical intervals for such changes are on the order of 10 ms Finally changes in the sinc term of 242 due to the time variation of each τit are proportional to the bandwidth whereas those in the phase are proportional to the carrier frequency which is typically much larger Essentially it takes much longer for a path to move from one tap to the next than for its phase to change significantly Thus the fastest changes in the filter taps occur because of the phase changes and these are significant over delay changes of 14Ds The coherence time Tc of a wireless channel is defined in an order of magnitude sense as the interval over which hem changes significantly as a function of m What we have found then is the important relation Tc 1 4Ds 244 This is a somewhat imprecise relation since the largest Doppler shifts may belong to paths that are too weak to make a difference We could also view a phase change of π4 to be significant and thus replace the factor of 4 above by 8 Many people instead replace the factor of 4 by 1 The important thing is to recognize that the major effect in determining time coherence is the Doppler spread and that the relationship is reciprocal the larger the Doppler spread the smaller the time coherence In the wireless communication literature channels are often categorized as fast fading and slow fading but there is little consensus on what these terms mean In this book we will call a channel fast fading if the coherence time Tc is much shorter than the delay requirement of the application and slow fading if Tc is longer The operational significance of this definition is that in a fast fading channel one can transmit the coded symbols over multiple fades of the channel while in a slow fading channel one cannot Thus whether a channel is fast or slow fading depends not only on the environment but also on the application voice for example typically has a short delay requirement of less than 100 ms while some types of data applications can have a laxer delay requirement 232 Delay spread and coherence bandwidth Another important general parameter of a wireless system is the multipath delay spread Td defined as the difference in propagation time between the 7 The Doppler spread can in principle be different for different taps Exercise 210 explores this possibility The wireless channel longest and shortest path counting only the paths with significant energy Thus Td maxij τit τjt 245 This is defined as a function of t but we regard it as an order of magnitude quantity like the time coherence and Doppler spread If a cell or LAN has a linear extent of a few kilometers or less it is very unlikely to have path lengths that differ by more than 300 to 600 meters This corresponds to path delays of one or two microseconds As cells become smaller due to increased cellular usage Td also shrinks As was already mentioned typical wireless channels are underspread which means that the delay spread Td is much smaller than the coherence time Tc The bandwidths of cellular systems range between several hundred kilohertz and several megahertz and thus for the above multipath delay spread values all the path delays in 234 lie within the peaks of two or three sinc functions more often they lie within a single peak Adding a few extra taps to each channel filter because of the slow decay of the sinc function we see that cellular channels can be represented with at most four or five channel filter taps On the other hand there is a recent interest in ultrawideband UWB communication operating from 31 to 106 GHz These channels can have up to a few hundred taps When we study modulation and detection for cellular systems we shall see that the receiver must estimate the values of these channel filter taps The taps are estimated via transmitted and received waveforms and thus the receiver makes no explicit use of and usually does not have any information about individual path delays and path strengths This is why we have not studied the details of propagation over multiple paths with complicated types of reflection mechanisms All we really need is the aggregate values of gross physical mechanisms such as Doppler spread coherence time and multipath spread The delay spread of the channel dictates its frequency coherence Wireless channels change both in time and frequency The time coherence shows us how quickly the channel changes in time and similarly the frequency coherence shows how quickly it changes in frequency We first understood about channels changing in time and correspondingly about the duration of fades by studying the simple example of a direct path and a single reflected path That same example also showed us how channels change with frequency We can see this in terms of the frequency response as well Recall that the frequency response at time t is Hf t i aitej2πfτit 246 The contribution due to a particular path has a phase linear in f For multiple paths there is a differential phase 2πfτit τkt This differential 23 Time and frequency coherence phase causes selective fading in frequency This says that Erf t changes significantly not only when t changes by 14Ds but also when f changes by 12Td This argument extends to an arbitrary number of paths so the coherence bandwidth Wc is given by Wc 1 2Td 247 This relationship like 244 is intended as an order of magnitude relation essentially pointing out that the coherence bandwidth is reciprocal to the multipath spread When the bandwidth of the input is considerably less than Wc the channel is usually referred to as flat fading In this case the delay spread Td is much less than the symbol time 1W and a single channel filter tap is sufficient to represent the channel When the bandwidth is much larger than Wc the channel is said to be frequencyselective and it has to be represented by multiple taps Note that flat or frequencyselective fading is not a property of the channel alone but of the relationship between the bandwidth W and the coherence bandwidth Td Figure 213 The physical parameters and the timescale of change of key parameters of the discretetime baseband channel model are summarized in Table 21 The different types of channels are summarized in Table 22 Figure 213 a A channel over 200 MHz is frequencyselective and the impulse response has many taps b The spectral content of the same channel c The same channel over 40 MHz is flatter and has for fewer taps d The spectral contents of the same channel limited to 40 MHz bandwidth At larger bandwidths the same physical paths are resolved into a finer resolution Table 21 A summary of the physical parameters of the channel and the timescale of change of the key parameters in its discretetime baseband model Table 22 A summary of the types of wireless channels and their defining characteristics 24 Statistical channel models 241 Modeling philosophy We defined Doppler spread and multipath spread in the previous section as quantities associated with a given receiver at a given location velocity and time However we are interested in a characterization that is valid over some range of conditions That is we recognize that the channel filter taps hℓm must be measured but we want a statistical characterization of how many taps are necessary how quickly they change and how much they vary Such a characterization requires a probabilistic model of the channel tap values perhaps gathered by statistical measurements of the channel We are familiar with describing additive noise by such a probabilistic model as a Gaussian random variable We are also familiar with evaluating error probability while communicating over a channel using such models These error probability evaluations however depend critically on the independence and Gaussian distribution of the noise variables It should be clear from the description of the physical mechanisms generating Doppler spread and multipath spread that probabilistic models for the channel filter taps are going to be far less believable than the models for additive noise On the other hand we need such models even if they are quite inaccurate Without models systems are designed using experience and experimentation and creativity becomes somewhat stifled Even with highly oversimplified models we can compare different system approaches and get a sense of what types of approaches are worth pursuing To a certain extent all analytical work is done with simplified models For example white Gaussian noise WGN is often assumed in communication models although we know the model is valid only over sufficiently small frequency bands With WGN however we expect the model to be quite good when used properly For wireless channel models however probabilistic models are quite poor and only provide orderofmagnitude guides to system design and performance We will see that we can define Doppler spread multipath spread etc much more cleanly with probabilistic models but the underlying problem remains that these channels are very different from each other and cannot really be characterized by probabilistic models At the same time there is a large literature based on probabilistic models for wireless channels and it has been highly useful for providing insight into wireless systems However it is important to understand the robustness of results based on these models There is another question in deciding what to model Recall the continuoustime multipath fading channel yt i aitxt τit wt 248 This contains an exact specification of the delay and magnitude of each path From this we derived a discretetime baseband model in terms of channel filter taps as ym ℓ hℓmxm ℓ wm 249 where hℓm i aimWej2πfc τimWsincℓ τimWW 250 We used the sampling theorem expansion in which xm xbmW and ym ybmW Each channel tap hℓm contains an aggregate of paths with the delays smoothed out by the baseband signal bandwidth Fortunately it is the filter taps that must be modeled for inputoutput descriptions and also fortunately the filter taps often contain a sufficient path aggregation so that a statistical model might have a chance of success 242 Rayleigh and Rician fading The simplest probabilistic model for the channel filter taps is based on the assumption that there are a large number of statistically independent reflected and scattered paths with random amplitudes in the delay window corresponding to a single tap The phase of the ith path is 2πfc τi modulo 2π Now fc τi diλ where di is the distance travelled by the ith path and λ is the carrier wavelength Since the reflectors and scatterers are far away relative to the carrier wavelength ie di λ it is reasonable to assume that the phase for each path is uniformly distributed between 0 and 2π and that the phases of different paths are independent The contribution of each path in the tap gain hℓm is aimWej2πfc τimWsincℓ τimWW 251 and this can be modeled as a circular symmetric complex random variable⁸ Each tap hℓm is the sum of a large number of such small independent circular symmetric random variables It follows that ℜhℓm is the sum of many small independent real random variables and so by the Central Limit Theorem it can reasonably be modeled as a zeromean Gaussian random variable Similarly because of the uniform phase ℜhℓmejφ is Gaussian with the same variance for any fixed φ This assures us that hℓm is in fact circular symmetric CN0 σℓ see Section A13 in Appendix A for an elaboration It is assumed here that the variance of hℓm is a function of the tap ℓ but independent of time m there is little point in creating a probabilistic model that depends on time With this assumed Gaussian probability density we know that the magnitude hℓm of the ℓth tap is a Rayleigh random variable with density cf A20 in Appendix A and Exercise 214 x σℓ² expx²2σℓ² x 0 252 and the squared magnitude hℓm² is exponentially distributed with density 1σℓ² expxσℓ² x 0 253 This model which is called Rayleigh fading is quite reasonable for scattering mechanisms where there are many small reflectors but is adopted primarily for its simplicity in typical cellular situations with a relatively small number of reflectors The word Rayleigh is almost universally used for this ⁸ See Section A13 in Appendix A for a more indepth discussion of circular symmetric random variables and vectors model but the assumption is that the tap gains are circularly symmetric complex Gaussian random variables There is a frequently used alternative model in which the lineofsight path often called a specular path is large and has a known magnitude and that there are also a large number of independent paths In this case hℓm at least for one value of ℓ can be modeled as hℓm sqrtκκ1σℓ ejθ sqrt1κ1 𝒞𝒩0σℓ2 254 with the first term corresponding to the specular path arriving with uniform phase θ and the second term corresponding to the aggregation of the large number of reflected and scattered paths independent of θ The parameter κ socalled Kfactor is the ratio of the energy in the specular path to the energy in the scattered paths the larger κ is the more deterministic is the channel The magnitude of such a random variable is said to have a Rician distribution Its density has quite a complicated form it is often a better model of fading than the Rayleigh model 243 Tap gain autocorrelation function Modeling each hℓm as a complex random variable provides part of the statistical description that we need but this is not the most important part The more important issue is how these quantities vary with time As we will see in the rest of the book the rate of channel variation has significant impact on several aspects of the communication problem A statistical quantity that models this relationship is known as the tap gain autocorrelation function Rℓn It is defined as Rℓn Ehℓm hℓmn 255 For each tap ℓ this gives the autocorrelation function of the sequence of random variables modeling that tap as it evolves in time We are tacitly assuming that this is not a function of time m Since the sequence of random variables hℓm for any given ℓ has both a mean and covariance function that does not depend on m this sequence is widesense stationary We also assume that as a random variable hℓm is independent of hℓm for all ℓ ℓ and all m m This final assumption is intuitively plausible since paths in different ranges of delay contribute to hℓm for different values of ℓ9 The coefficient Rℓ0 is proportional to the energy received in the ℓth tap The multipath spread Td can be defined as the product of 1W times the range of ℓ which contains most of the total energy ℓ0 Rℓ0 This is 9 One could argue that a moving reflector would gradually travel from the range of one tap to another but as we have seen this typically happens over a very large timescale somewhat preferable to our previous definition in that the statistical nature of Td becomes explicit and the reliance on some sort of stationarity becomes explicit Now we can also define the coherence time Tc more explicitly as the smallest value of n 0 for which Rℓn is significantly different from Rℓ0 With both of these definitions we still have the ambiguity of what significant means but we are now facing the reality that these quantities must be viewed as statistics rather than as instantaneous values The tap gain autocorrelation function is useful as a way of expressing the statistics for how tap gains change given a particular bandwidth W but gives little insight into questions related to choice of a bandwidth for communication If we visualize increasing the bandwidth we can see several things happening First the ranges of delay that are separated into different taps ℓ become narrower 1W seconds so there are fewer paths corresponding to each tap and thus the Rayleigh approximation becomes poorer Second the sinc functions of 250 become narrower and Rℓ0 gives a finer grained picture of the amount of power being received in the ℓth delay window of width 1W In summary as we try to apply this model to larger W we get more detailed information about delay and correlation at that delay but the information becomes more questionable Example 22 Clarkes model This is a popular statistical model for flat fading The transmitter is fixed the mobile receiver is moving at speed v and the transmitted signal is scattered by stationary objects around the mobile There are K paths the ith path arriving at an angle θi 2πiK i 0K1 with respect to the direction of motion K is assumed to be large The scattered path arriving at the mobile at the angle θ has a delay of τθt and a timeinvariant gain aθ and the inputoutput relationship is given by yt i0K1 aθi xt τθit 256 The most general version of the model allows the received power distribution pθ and the antenna gain pattern αθ to be arbitrary functions of the angle θ but the most common scenario assumes uniform power distribution and isotropic antenna gain pattern ie the amplitudes aθ aK for all angles θ This models the situation when the scatterers are located in a ring around the mobile Figure 214 We scale the amplitude of each path by K so that the total received energy along all paths is a2 for large K the received energy along each path is a small fraction of the total energy Suppose the communication bandwidth W is much smaller than the reciprocal of the delay spread The complex baseband channel can be represented by a single tap at each time ym h0mxm wm 257 Figure 214 The onering model The phase of the signal arriving at time 0 from an angle θ is 2π fc τθ0 mod 2π where fc is the carrier frequency Making the assumption that this phase is uniformly distributed in 0 2π and independently distributed across all angles θ the tap gain process h0m is a sum of many small independent contributions one from each angle By the Central Limit Theorem it is reasonable to model the process as Gaussian Exercise 217 shows further that the process is in fact stationary with an autocorrelation function R0n given by R0n 2a2π J0 n π TDsW 258 where J0 is the zerothorder Bessel function of the first kind J0x 1π 0π ej x cos θ dθ 259 and Ds 2 fc vc is the Doppler spread The power spectral density Sf defined on 12 12 is given by Sf 4 a2 WDs 12 f WDs2 Ds2W f Ds2W 0 else 260 This can be verified by computing the inverse Fourier transform of 260 to be 258 Plots of the autocorrelation function and the spectrum for are shown in Figure 215 If we define the coherence time Tc to be the value of nW such that R0n 005 R00 then Tc J01005 π Ds 261 ie the coherence time is inversely proportional to Ds Figure 215 Plots of the autocorrelation function and Doppler spectrum in Clarkes model In Exercise 217 you will also verify that Sfdf has the physical interpretation of the received power along paths that have Doppler shifts in the range ff df Thus Sf is also called the Doppler spectrum Note that Sf is zero beyond the maximum Doppler shift Chapter 2 The main plot Largescale fading Variation of signal strength over distances of the order of cell sizes Received power decreases with distance r like 1r2 free space 1r4 reflection from ground plane Decay can be even faster due to shadowing and scattering effects Smallscale fading Variation of signal strength over distances of the order of the carrier wavelength due to constructive and destructive interference of multipaths Key parameters Doppler spread Ds coherence time Tc 1Ds Doppler spread is proportional to the velocity of the mobile and to the angular spread of the arriving paths delay spread Td coherence bandwidth Wc 1Td Delay spread is proportional to the difference between the lengths of the shortest and the longest paths Inputoutput channel models Continuoustime passband 214 yt Σi ait xt τit Continuoustime complex baseband 226 ybt Σi aitej2πfctτit xbt τit Discretetime complex baseband with AWGN 238 ym Σℓ hem xm ℓ wm The ℓth tap is the aggregation of the physical paths with delays in ℓW 12W ℓW 12W Statistical channel models hemm is modeled as circular symmetric processes independent across the taps If for all taps hem CN 0 σe2 the model is called Rayleigh If for one tap hem κκ 1 σe ejθ 1κ 1 CN 0 σe2 the model is called Rician with Kfactor κ The tap gain autocorrelation function Rℓn Ehℓ0hℓn models the dependency over time The delay spread is 1W times the range of taps ℓ which contains most of the total gain Σℓ0 Rℓ0 The coherence time is 1W times the range of n for which Rℓn is significantly different from Rℓ0 25 Bibliographical notes This chapter was modified from R G Gallagers MIT 6450 course notes on digital communication The focus is on smallscale multipath fading Largescale fading models are discussed in many texts see for example Rappaport 98 Clarkes model was introduced in 22 and elaborated further in 62 Our derivation here of the Clarke power spectrum follows the approach of 111 26 Exercises Exercise 21 Gallager Consider the electric field in 24 1 It has been derived under the assumption that the motion is in the direction of the lineofsight from sending antenna to receive antenna Find the electric field assuming that ϕ is the angle between the lineofsight and the direction of motion of the receiver Assume that the range of time of interest is small enough so that changes in θ ψ can be ignored 2 Explain why and under what conditions it is a reasonable approximation to ignore the change in θ ψ over small intervals of time Exercise 22 Gallager Equation 213 was derived under the assumption that rt d Derive an expression for the received waveform for general rt Break the first term in 211 into two terms one with the same numerator but the denominator 2d r0 vt and the other with the remainder Interpret your result Exercise 23 In the twopath example in Sections 213 and 214 the wall is on the right side of the receiver so that the reflected wave and the direct wave travel in opposite directions Suppose now that the reflecting wall is on the left side of transmitter Redo the analysis What is the nature of the multipath fading both over time and over frequency Explain any similarity or difference with the case considered in Sections 213 and 214 Exercise 24 A mobile receiver is moving at a speed v and is receiving signals arriving along two reflected paths which make angles θ1 and θ2 with the direction of motion The transmitted signal is a sinusoid at frequency f 1 Is the above information enough for estimating i the coherence time Tc ii the coherence bandwidth Wc If so express them in terms of the given parameters If not specify what additional information would be needed 2 Consider an environment in which there are reflectors and scatterers in all directions from the receiver and an environment in which they are clustered within a small angular range Using part 1 explain how the channel would differ in these two environments Exercise 25 Consider the propagation model in Section 215 where there is a reflected path from the ground plane 1 Let r1 be the length of the direct path in Figure 26 Let r2 be the length of the reflected path summing the path length from the transmitter to the ground plane and the path length from the ground plane to the receiver Show that r2 r1 is asymptotically equal to br and find the value of the constant b Hint Recall that for x small sqrt1 x 1 x2 in the sense that sqrt1 x 1x 12 as x 0 2 Assume that the received waveform at the receive antenna is given by Erf t α cos 2πft fr1c r1 α cos 2πft fr2c r2 262 Approximate the denominator r2 by r1 in 262 and show that Er β r2 for r1 much smaller than cf Find the value of β 3 Explain why this asymptotic expression remains valid without first approximating the denominator r2 in 262 by r1 Exercise 26 Consider the following simple physical model in just a single dimension The source is at the origin and transmits an isotropic wave of angular frequency ω The physical environment is filled with uniformly randomly located obstacles We will model the interobstacle distance as an exponential random variable ie it has the density10 η eη r r 0 263 Here 1η is the mean distance between obstacles and captures the density of the obstacles Viewing the source as a stream of photons suppose each obstacle independently from one photon to the other and independent of the behavior of the other obstacles either absorbs the photon with probability γ or scatters it either to the left or to the right both with equal probability 1γ2 Now consider the path of a photon transmitted either to the left or to the right with equal probability from some fixed point on the line The probability density function of the distance denoted by r to the first obstacle the distance can be on either side of the starting point so r takes values on the entire line is equal to qr η eηr 2 r ℜ 264 So the probability density function of the distance at which the photon is absorbed upon hitting the first obstacle is equal to f1r γ qr r ℜ 265 10 This random arrangement of points on a line is called a Poisson point process 1 Show that the probability density function of the distance from the origin at which the second obstacle is met is f2r 1γ qx f1r x dx r ℜ 266 2 Denote by fkr the probability density function of the distance from the origin at which the photon is absorbed by exactly the kth obstacle it hits and show the recursive relation fk1r 1γ qx fkr x dx r ℜ 267 3 Conclude from the previous step that the probability density function of the distance from the source at which the photon is absorbed by some obstacle denoted by fr satisfies the recursive relation fr γ qr 1γ qx fr x dx r ℜ 268 Hint Observe that fr Σk1 fkr 4 Show that fr sqrtγ η2 eη sqrtγ r 269 is a solution to the recursive relation in 268 Hint Observe that the convolution between the probability densities q and f in 268 is more easily represented using Fourier transforms 5 Now consider the photons that are absorbed at a distance of more than r from the source This is the radiated power density at a distance r and is found by integrating fx over the range r if r 0 and r if r 0 Calculate the radiated power density to be eγ sqrtη r 2 270 and conclude that the power decreases exponentially with distance r Also observe that with very low absorption γ 0 or very few obstacles η 0 the power density converges to 05 this is expected since the power splits equally on either side of the line Exercise 27 In Exercise 26 we considered a singledimensional physical model of a scattering and absorption environment and concluded that power decays exponentially with distance A reading exercise is to study 42 which considers a natural extension of this simple model to two and threedimensional spaces Further it extends the analysis to two and threedimensional physical models While the analysis is more complicated we arrive at the same conclusion the radiated power decays exponentially with distance Exercise 28 Gallager Assume that a communication channel first filters the transmitted passband signal before adding WGN Suppose the channel is known and the channel filter has an impulse response ht Suppose that a QAM scheme with symbol duration T is developed without knowledge of the channel filtering A baseband filter θt is developed satisfying the Nyquist property that θt kTk is an orthonormal set The matched filter θt is used at the receiver before sampling and detection If one is aware of the channel filter ht one may want to redesign either the baseband filter at the transmitter or the baseband filter at the receiver so that there is no intersymbol interference between receiver samples and so that the noise on the samples is iid 1 Which filter should one redesign 2 Give an expression for the impulse response of the redesigned filter assume a carrier frequency fc 3 Draw a figure of the various filters at passband to show why your solution is correct We suggest you do this before answering the first two parts Exercise 29 Consider the twopath example in Section 214 with d 2 km and the receiver at 15 km from the transmitter moving at velocity 60 kmh away from the transmitter The carrier frequency is 900 MHz 1 Plot in MATLAB the magnitudes of the taps of the discretetime baseband channel at a fixed time t Give a few plots for several bandwidths W so as to exhibit both flat and frequencyselective fading 2 Plot the time variation of the phase and magnitude of a typical tap of the discretetime baseband channel for a bandwidth where the channel is approximately flat and for a bandwidth where the channel is frequencyselective How do the timevariations depend on the bandwidth Explain Exercise 210 For each tap of the discretetime channel response the Doppler spread is the range of Doppler shifts of the paths contributing to that tap Give an example of an environment ie location of reflectorsscatterers with respect to the location of the transmitter and the receiver in which the Doppler spread is the same for different taps and an environment in which they are different Exercise 211 Verify 239 and 240 Exercise 212 In this problem we consider generating passband orthogonal waveforms from baseband ones 1 Show that if the waveforms θt nTn form an orthogonal set then the waveforms ψn1 ψn2n also form an orthogonal set provided that θt is bandlimited to fc fc Here ψn1t θt nT cos 2π fc t ψn2t θt nT sin 2π fc t How should we normalize the energy of θt to make the ψt orthonormal 2 For a given fc find an example where the result in part 1 is false when the condition that θt is bandlimited to fc fc is violated Exercise 213 Verify 225 Does this equation contain any more information about the communication system in Figure 29 beyond what is in 224 Explain Exercise 214 Compute the probability density function of the magnitude X of a complex circular symmetric Gaussian random variable X with variance σ² Exercise 215 In the text we have discussed the various reasons why the channel tap gains hℓm vary in time as a function of m and how the various dynamics operate at different timescales The analysis is based on the assumption that communication takes place on a bandwidth W around a carrier frequency fc with fc W This assumption is not valid for ultrawideband UWB communication systems where the transmission bandwidth is from 31 GHz to 106 GHz as regulated by the FCC Redo the analysis for this system What is the main mechanism that causes the tap gains to vary at the fastest timescale and what is this fastest timescale determined by Exercise 216 In Section 242 we argue that the channel gain hℓm at a particular time m can be assumed to be circular symmetric Extend the argument to show that it is also reasonable to assume that the complex random vector h hℓm hℓm 1 hℓm nᵀ is circular symmetric for any n Exercise 217 In this question we will analyze in detail Clarkes onering model discussed at the end of the chapter Recall that the scatterers are assumed to be located in a ring around the receiver moving at speed v There are K paths coming in at angles θi 2πiK with respect to the direction of motion of the mobile i 0 K1 The path coming at angle θ has a delay of τθt and a timeinvariant gain aK not dependent on the angle and the inputoutput relationship is given by yt aK i0K1 xtτθit 271 1 Give an expression for the impulse response hτ t for this channel and give an expression for τθt in terms of τθ0 You can assume that the distance the mobile travelled in 0 t is small compared to the radius of the ring 2 Suppose communication takes place at carrier frequency fc and over a narrowband of bandwidth W such that the delay spread of the channel Td satisfies Td 1W Argue that the discretetime baseband model can be approximately represented by a single tap ym h0mxm wm 272 and give an approximate expression for that tap in terms of the aθs and τθts Hint Your answer should contain no sinc functions 3 Argue that it is reasonable to assume that the phase of the path from an angle θ at time 0 2πfcτθ0 mod 2π is uniformly distributed in 0 2π and that it is iid across θ 4 Based on the assumptions in part 3 for large K one can use the Central Limit Theorem to approximate h0m as a Gaussian process Verify that the limiting process is stationary and the autocorrelation function R0n is given by 258 5 Verify that the Doppler spectrum Sf is given by 260 Hint It is easier to show that the inverse Fourier transform of 260 is 258 6 Verify that Sfdf is indeed the received power from the paths that have Doppler shifts in f f df Is this surprising Exercise 218 Consider a onering model where there are K scatterers located at angles θi 2πiK i 0 K1 on a circle of radius 1 km around the receiver and the transmitter is 2 km away The angles are with respect to the line joining the transmitter and the receiver The transmit power is P The power attenuation along a path from the transmitter to a scatterer to the receiver is GK 1s² 1r² 273 where G is a constant and r and s are the distance from the transmitter to the scatterer and the distance from the scatterer to the receiver respectively Communication takes place at a carrier frequency fc 19 GHz and the bandwidth is W Hz You can assume that at any time the phases of each arriving path in the baseband representation of the channel are independent and uniformly distributed between 0 and 2π 1 What are the key differences and the similarities between this model and the Clarkes model in the text 2 Find approximate conditions on the bandwidth W for which one gets a flat fading channel 3 Suppose the bandwidth is such that the channel is frequency selective For large K find approximately the amount of power in tap ℓ of the discretetime baseband impulse response of the channel ie compute the powerdelay profile Make any simplifying assumptions but state them You can leave your answers in terms of integrals if you cannot evaluate them 4 Compute and sketch the powerdelay profile as the bandwidth becomes very large and K is large 5 Suppose now the receiver is moving at speed v towards the fixed transmitter What is the Doppler spread of tap ℓ Argue heuristically from physical considerations what the Doppler spectrum ie power spectral density of tap ℓ is for large K 6 We have made the assumptions that the scatterers are all on a circle of radius 1km around the receiver and the paths arrive with independent and uniform distributed phases at the receiver Mathematically are the two assumptions consistent If not do you think it matters in terms of the validity of your answers to the earlier parts of this question Exercise 219 Often in modeling multiple input multiple output MIMO fading channels the fading coefficients between different transmit and receive antennas are assumed to be independent random variables This problem explores whether this is a reasonable assumption based on Clarkes onering scattering model and the antenna separation 1 Antenna separation at the mobile Assume a mobile with velocity v moving away from the basestation with uniform scattering from the ring around it 48 The wireless channel a Compute the Doppler spread Ds for a carrier frequency fc and the correspond ing coherence time Tc b Assuming that fading states separated by Tc are approximately uncorrelated at what distance should we place a second antenna at the mobile to get an inde pendently faded signal Hint How much distance does the mobile travel in Tc 2 Antenna separation at the basestation Assume that the scattering ring has radius R and that the distance between the basestation and the mobile is d Further assume for the time being that the basestation is moving away from the mobile with velocity v Repeat the previous part to find the minimum antenna spacing at the basestation for uncorrelated fading Hint Is the scattering still uniform around the basestation 3 Typically the scatterers are local around the mobile near the ground and far away from the basestation high on a tower What is the implication of your result in part 2 for this scenario C H A P T E R 3 Pointtopoint communication detection diversity and channel uncertainty In this chapter we look at various basic issues that arise in communication over fading channels We start by analyzing uncoded transmission in a narrowband fading channel We study both coherent and noncoherent detection In both cases the error probability is much higher than in a nonfaded AWGN channel The reason is that there is a significant probability that the channel is in a deep fade This motivates us to investigate various diversity techniques that improve the performance The diversity techniques operate over time frequency or space but the basic idea is the same By sending signals that carry the same information through different paths multiple independently faded replicas of data symbols are obtained at the receiver end and more reliable detection can be achieved The simplest diversity schemes use repetition coding More sophisticated schemes exploit channel diversity and at the same time efficiently use the degrees of freedom in the channel Compared to repetition coding they provide coding gains in addition to diversity gains In space diversity we look at both transmit and receive diversity schemes In frequency diversity we look at three approaches singlecarrier with intersymbol interference equalization directsequence spreadspectrum orthogonal frequency division multiplexing Finally we study the impact of channel uncertainty on the performance of diversity combining schemes We will see that in some cases having too many diversity paths can have an adverse effect due to channel uncertainty To familiarize ourselves with the basic issues the emphasis of this chapter is on concrete techniques for communication over fading channels In Chapter 5 we take a more fundamental and systematic look and use information theory to derive the best performance one can achieve At that fundamental level we will see many of the issues discussed here recur The derivations in this chapter make repeated use of a few key results in vector detection under Gaussian noise We develop and summarize the basic results in Appendix A emphasizing the underlying geometry The reader is 49 encouraged to take a look at the appendix before proceeding with this chapter and to refer back to it often In particular a thorough understanding of the canonical detection problem in Summary A2 will be very useful 31 Detection in a Rayleigh fading channel 311 Noncoherent detection We start with a very simple detection problem in a fading channel For simplicity let us assume a flat fading model where the channel can be represented by a single discretetime complex filter tap h0m which we abbreviate as hm ym hmxm wm 31 where wm CN0 N0 We suppose Rayleigh fading ie hm CN0 1 where we normalize the variance to be 1 For the time being however we do not specify the dependence between the fading coefficients hm at different times m nor do we make any assumption on the prior knowledge the receiver might have of hm This latter assumption is sometimes called noncoherent communication First consider uncoded binary antipodal signaling or binary phaseshiftkeying BPSK with amplitude a ie xm a and the symbols xm are independent over time This signaling scheme fails completely even in the absence of noise since the phase of the received signal ym is uniformly distributed between 0 and 2π regardless of whether xm a or xm a is transmitted Further the received amplitude is independent of the transmitted symbol Binary antipodal signaling is binary phase modulation and it is easy to see that phase modulation in general is similarly flawed Thus signal structures are required in which either different signals have different magnitudes or coding between symbols is used Next we look at orthogonal signaling a special type of coding between symbols Consider the following simple orthogonal modulation scheme a form of binary pulseposition modulation For a pair of time samples transmit either xA x0 x1 a 0 32 or xB 0 a 33 We would like to perform detection based on y y0 y1 34 31 Detection in a Rayleigh fading channel This is a simple hypothesis testing problem and it is straightforward to derive the maximum likelihood ML rule Λy xA 0 xB 35 where Λy is the loglikelihood ratio Λy lnfyxA fyxB 36 It can be seen that if xA is transmitted y0 C N0 a2 N0 and y1 C N0 N0 and y0 y1 are independent Similarly if xB is transmitted y0 C N0 N0 and y1 C N0 a2 N0 Further y0 and y1 are independent Hence the loglikelihood ratio can be computed to be Λy y02 y12 a2 a2 N0 N0 37 The optimal rule is simply to decide xA is transmitted if y02 y12 and decide xB otherwise Note that the rule does not make use of the phases of the received signal since the random unknown phases of the channel gains h0 h1 render them useless for detection Geometrically we can interpret the detector as projecting the received vector y onto each of the two possible transmit vectors xA and xB and comparing the energies of the projections Figure 31 Thus this detector is also called an energy or a squarelaw detector It is somewhat surprising that the optimal detector does not depend on how h0 and h1 are correlated We can analyze the error probability of this detector By symmetry we can assume that xA is transmitted Under this hypothesis y0 and y1 are Figure 31 The noncoherent detector projects the received vector y onto each of the two orthogonal transmitted vectors xA and xB and compares the lengths of the projections independent circular symmetric complex Gaussian random variables with variances a2 N0 and N0 respectively See Section A13 in the appendices for a discussion on circular symmetric Gaussian random variables and vectors As shown there y02 y12 are exponentially distributed with mean a2 N0 and N0 respectively1 The probability of error can now be computed by direct integration pe Py12 y02xA 2 a2N01 38 We make the general definition SNR average received signal energy per complex symbol time noise energy per complex symbol time 39 which we use consistently throughout the book for any modulation scheme The noise energy per complex symbol time is N02 For the orthogonal modulation scheme here the average received energy per symbol time is a2 2 and so SNR a2 2N0 310 Substituting into 38 we can express the error probability of the orthogonal scheme in terms of SNR pe 1 21 SNR 311 This is a very discouraging result To get an error probability pe 103 one would require SNR 500 27 dB Stupendous amounts of power would be required for more reliable communication 312 Coherent detection Why is the performance of the noncoherent maximum likelihood ML receiver on a fading channel so bad It is instructive to compare its performance with detection in an AWGN channel without fading ym xm wm 312 1 Recall that a random variable U is exponentially distributed with mean μ if its pdf is fUu 1μ euμ 2 The orthogonal modulation scheme considered here uses only real symbols and hence transmits only on the I channel Hence it may seem more natural to define the SNR in terms of noise energy per real symbol ie N0 2 However later we will consider modulation schemes that use complex symbols and hence transmit on both the I and Q channels In order to be consistent throughout we choose to define SNR this way For antipodal signaling BPSK xm a a sufficient statistic is ℜym and the error probability is pe Qa N0 2 Q2 SNR 313 where SNR a2 N0 is the received signaltonoise ratio per symbol time and Q is the complementary cumulative distribution function of an N0 1 random variable This function decays exponentially with x2 more specifically Qx ex2 2 x 0 314 and Qx 1 2 π x 1 1x2 ex2 2 x 1 315 Thus the detection error probability decays exponentially in SNR in the AWGN channel while it decays only inversely with the SNR in the fading channel To get an error probability of 103 an SNR of only about 7 dB is needed in an AWGN channel as compared to 27 dB in the noncoherent fading channel Note that 2SNR is the separation between the two constellation points as a multiple of the standard deviation of the Gaussian noise the above observation says that when this separation is much larger than 1 the error probability is very small Compared to detection in the AWGN channel the detection problem considered in the previous section has two differences the channel gains hm are random and the receiver is assumed not to know them Suppose now that the channel gains are tracked at the receiver so that they are known at the receiver but still random In practice this is done either by sending a known sequence called a pilot or training sequence or in a decision directed manner estimating the channel using symbols detected earlier The accuracy of the tracking depends of course on how fast the channel varies For example in a narrowband 30kHz channel such as that used in the North American TDMA cellular standard IS136 with a Doppler spread of 100 Hz the coherence time Tc is roughly 80 symbols and in this case the channel can be estimated with minimal overhead expended in the pilot3 For our current purpose let us suppose that the channel estimates are perfect Knowing the channel gains coherent detection of BPSK can now be performed on a symbol by symbol basis We can focus on one symbol time and drop the time index y h x w 316 3 The channel estimation problem for a broadband channel with many taps in the impulse response is more difficult we will get to this in Section 35 Detection of x from y can be done in a way similar to that in the AWGN case the decision is now based on the sign of the real sufficient statistic r Rehh y hx z 317 where z sim mathcalN0 N02 If the transmitted symbol is x pm a then for a given value of h the error probability of detecting x is QleftfracahsqrtN02right Qleftsqrt2h2 extSNRright 318 where SNR a2N0 is the average received signaltonoise ratio per symbol time Recall that we normalized the channel gain such that Eh2 1 We average over the random gain h to find the overall error probability For Rayleigh fading when h sim mathcalCN01 direct integration yields pe EleftQleftsqrt2h2 extSNRrightright frac12 left1 sqrtfrac extSNR1 extSNRright 319 See Exercise 31 Figure 32 compares the error probabilities of coherent BPSK and noncoherent orthogonal signaling over the Rayleigh fading channel as well as BPSK over the AWGN channel We see that while the error probability for BPSK over the AWGN channel decays very fast with the SNR the error probabilities for the Rayleigh fading channel are much worse Figure 32 Performance of coherent BPSK vs noncoherent orthogonal signaling over Rayleigh fading channel vs BPSK over AWGN schannel whether the detection is coherent or noncoherent At high SNR Taylor series expansion yields sqrtfrac extSNR1 extSNR 1 frac12 extSNR Oleftfrac1 extSNR2right 320 Substituting into 319 we get the approximation pe approx frac14 extSNR 321 which decays inversely proportional to the SNR just as in the noncoherent orthogonal signaling scheme cf 311 There is only a 3 dB difference in the required SNR between the coherent and noncoherent schemes in contrast at an error probability of 103 there is a 17 dB difference between the performance on the AWGN channel and coherent detection on the Rayleigh fading channel4 We see that the main reason why detection in the fading channel has poor performance is not because of the lack of knowledge of the channel at the receiver It is due to the fact that the channel gain is random and there is a significant probability that the channel is in a deep fade At high SNR we can in fact be more precise about what a deep fade means by inspecting 318 The quantity h2 SNR is the instantaneous received SNR Under typical channel conditions ie h2 SNR gg 1 the conditional error probability is very small since the tail of the Qfunction decays very rapidly In this regime the separation between the constellation points is much larger than the standard deviation of the Gaussian noise On the other hand when h2 SNR is of the order of 1 or less the separation is of the same order as the standard deviation of the noise and the error probability becomes significant The probability of this event is Ph2 extSNR 1 int01 extSNR ex dx 322 frac1 extSNR Oleftfrac1 extSNR2right 323 This probability has the same order of magnitude as the error probability itself cf 321 Thus we can define a deep fade via an orderofmagnitude approximation Deep fade event h2 frac1 extSNR Pdeep fade approx frac1 extSNR 4 Communication engineers often compare schemes based on the difference in the required SNR to attain the same error probability This corresponds to the horizontal gap between the error probability versus SNR curves of the two schemes We conclude that highSNR error events most often occur because the channel is in deep fade and not as a result of the additive noise being large In contrast in the AWGN channel the only possible error mechanism is for the additive noise to be large Thus the error probability performance over the AWGN channel is much better We have used the explicit error probability expression 319 to help identify the typical error event at high SNR We can in fact turn the table around and use it as a basis for an approximate analysis of the highSNR performance Exercises 32 and 33 Even though the error probability pe can be directly computed in this case the approximate analysis provides much insight as to how typical errors occur Understanding typical error events in a communication system often suggests how to improve it Moreover the approximate analysis gives some hints as to how robust the conclusion is to the Rayleigh fading model In fact the only aspect of the Rayleigh fading model that is important to the conclusion is the fact that Ph2 epsilon is proportional to epsilon for epsilon small This holds whenever the pdf of h2 is positive and continuous at 0 313 From BPSK to QPSK exploiting the degrees of freedom In Section 312 we have considered BPSK modulation xm pm a This uses only the real dimension the I channel while in practice both the I and Q channels are used simultaneously in coherent communication increasing spectral efficiency Indeed an extra bit can be transmitted by instead using QPSK quadrature phaseshiftkeying modulation ie the constellation is a1j a1j a1j a1j 324 in effect a BPSK symbol is transmitted on each of the I and Q channels simultaneously Since the noise is independent across the I and Q channels the bits can be detected separately and the bit error probability on the AWGN channel cf 312 is Qsqrt2a2overN0 325 the same as BPSK cf 313 For BPSK the SNR as defined in 39 is given by SNR fraca2N0 326 while for QPSK SNR frac2a2N0 327 is twice that of BPSK since both the I and Q channels are used Equivalently for a given SNR the bit error probability of BPSK is Q2SNR cf 313 and that of QPSK is QSNR The error probability of QPSK under Rayleigh fading can be similarly obtained by replacing SNR by SNR2 in the corresponding expression 319 for BPSK to yield pe 12 1 SNR2SNR 12SNR 328 at high SNR For expositional simplicity we will consider BPSK modulation in many of the discussions in this chapter but the results can be directly mapped to QPSK modulation One important point worth noting is that it is much more energyefficient to use both the I and Q channels rather than just one of them For example if we had to send the two bits carried by the QPSK symbol on the I channel alone then we would have to transmit a 4PAM symbol The constellation is 3b b b 3b and the average error probability on the AWGN channel is 32 Q2b2N0 329 To achieve approximately the same error probability as QPSK the argument inside the Qfunction should be the same as that in 325 and hence b should be the same as a ie the same minimum separation between points in the two constellations Figure 33 But QPSK requires a transmit energy of 2a2 per symbol while 4PAM requires a transmit energy of 5b2 per symbol Hence for the same error probability approximately 25 times more transmit energy is needed a 4 dB worse performance Exercise 34 shows that this loss is even more significant for larger constellations The loss is due to the fact that it is more energy efficient to pack for a desired minimum distance separation a given number of constellation points in a higherdimensional space than in a lowerdimensional space We have thus arrived at a general design principle cf Discussion 21 A good communication scheme exploits all the available degrees of freedom in the channel This important principle will recur throughout the book and in fact will be shown to be of a fundamental nature as we talk about channel capacity in Chapter 5 Here the choice is between using just the I channel and using both the I and Q channels but the same principle applies to many other situations As another example the noncoherent orthogonal signaling scheme discussed in Section 311 conveys one bit of information and uses one real dimension per two symbol times Figure 34 This scheme does not assume any relationship between consecutive channel gains but if we assume that they do not change much from symbol to symbol an alternative scheme is differential BPSK which conveys information in the relative phases of consecutive transmitted symbols That is if the BPSK information symbol is um at time m um 1 the transmitted symbol at time m is given by xm um xm 1 330 Exercise 35 shows that differential BPSK can be demodulated noncoherently at the expense of a 3dB loss in performance compared to coherent BPSK at high SNR But since noncoherent orthogonal modulation also has a 3dB worse performance compared to coherent BPSK this implies that differential BPSK and noncoherent orthogonal modulation have the same error probability performance On the other hand differential BPSK conveys one 59 31 Detection in a Rayleigh fading channel bit of information and uses one real dimension per single symbol time and therefore has twice the spectral efficiency of orthogonal modulation Better performance is achieved because differential BPSK uses more efficiently the available degrees of freedom 314 Diversity The performance of the various schemes considered so far for fading channels is summarized in Table 31 Some schemes are spectrally more efficient than others but from a practical point of view they are all bad the error proba bilities all decay very slowly like 1SNR From Section 312 it can be seen that the root cause of this poor performance is that reliable communication depends on the strength of a single signal path There is a significant proba bility that this path will be in a deep fade When the path is in a deep fade any communication scheme will likely suffer from errors A natural solution to improve the performance is to ensure that the information symbols pass through multiple signal paths each of which fades independently making sure that reliable communication is possible as long as one of the paths is strong This technique is called diversity and it can dramatically improve the performance over fading channels There are many ways to obtain diversity Diversity over time can be obtained via coding and interleaving information is coded and the coded sym bols are dispersed over time in different coherence periods so that different parts of the codewords experience independent fades Analogously one can also exploit diversity over frequency if the channel is frequencyselective In a channel with multiple transmit or receive antennas spaced sufficiently diversity can be obtained over space as well In a cellular network macro diversity can be exploited by the fact that the signal from a mobile can be received at two basestations Since diversity is such an important resource a wireless system typically uses several types of diversity In the next few sections we will discuss diversity techniques in time frequency and space In each case we start with a simple scheme based on repetition coding the same information symbol is transmitted over several signal paths While repetition coding achieves the maximal diversity gain it is usually quite wasteful of the degrees of freedom of the channel More sophisticated schemes can increase the data rate and achieve a coding gain along with the diversity gain To keep the discussion simple we begin by focusing on the coherent scenario the receiver has perfect knowledge of the channel gains and can coherently combine the received signals in the diversity paths As discussed in the previous section this knowledge is learnt via training pilot symbols and the accuracy depends on the coherence time of the channel and the received power of the transmitted signal We discuss the impact of channel measurement error and noncoherent diversity combining in Section 35 Table 31 Performance of coherent and noncoherent schemes under Rayleigh fading The data rates are in bitssHz which is the same as bits per complex symbol time The performance of differential QPSK is derived in Exercise 35 It is also 3dB worse than coherent QPSK Scheme Bit error prob High SNR Data rate bitssHz Coherent BPSK 14SNR 1 Coherent QPSK 12SNR 2 Coherent 4PAM 54SNR 2 Coherent 16QAM 52SNR 4 Noncoherent orth mod 12SNR 12 Differential BPSK 12SNR 1 Differential QPSK 1SNR 2 32 Time diversity Time diversity is achieved by averaging the fading of the channel over time Typically the channel coherence time is of the order of tens to hundreds of symbols and therefore the channel is highly correlated across consecutive symbols To ensure that the coded symbols are transmitted through independent or nearly independent fading gains interleaving of codewords is required Figure 35 For simplicity let us consider a flat fading channel We transmit a codeword x x1 xLt of length L symbols and the received signal is given by yℓ hℓ xℓ wℓ ℓ 1 L 331 Assuming ideal interleaving so that consecutive symbols xℓ are transmitted sufficiently far apart in time we can assume that the hℓ are independent The parameter L is commonly called the number of diversity branches The additive noises w1 wL are iid N0 N0 random variables 321 Repetition coding The simplest code is a repetition code in which xℓ x1 for ℓ 1 L In vector form the overall channel becomes y h x1 w 332 where y y1 yLt h h1 hLt and w w1 wLt Figure 35 The codewords are transmitted over consecutive symbols top and interleaved bottom A deep fade will wipe out the entire codeword in the former case but only one coded symbol from each codeword in the latter In the latter case each codeword can still be recovered from the other three unfaded symbols Consider now coherent detection of x₁ ie the channel gains are known to the receiver This is the canonical vector Gaussian detection problem in Summary A2 of Appendix A The scalar hh y h x₁ hh w is a sufficient statistic Thus we have an equivalent scalar detection problem with noise hh w CN0 N₀ The receiver structure is a matched filter and is also called a maximal ratio combiner it weighs the received signal in each branch in proportion to the signal strength and also aligns the phases of the signals in the summation to maximize the output SNR This receiver structure is also called coherent combining Consider BPSK modulation with x₁ a The error probability conditional on h can be derived exactly as in 318 Q2h² SNR where as before SNR a²N₀ is the average received signaltonoise ratio per complex symbol time and h² SNR is the received SNR for a given channel vector h We average over h² to find the overall error probability Under Rayleigh fading with each gain hℓ iid CN01 h² ℓ1 to L hℓ² is a sum of the squares of 2L independent real Gaussian random variables each term hℓ² being the sum of the squares of the real and imaginary parts of hℓ It is Chisquare distributed with 2L degrees of freedom and the density is given by fx 1L1 xL1 ex x 0 The average error probability can be explicitly computed to be cf Exercise 36 pₑ ₀ Q2xSNRfxdx 1μ2 L ℓ0 to L1 L1ℓℓ 1μ2 ℓ where μ SNR1SNR The error probability as a function of the SNR for different numbers of diversity branches L is plotted in Figure 36 Increasing L dramatically decreases the error probability At high SNR we can see the role of L analytically consider the leading term in the Taylor series expansion in 1SNR to arrive at the approximations 1μ2 1 and 1μ2 14SNR Figure 36 Error probability as a function of SNR for different numbers of diversity branches L Furthermore ℓ0 to L1 L1ℓℓ 2L1L Hence pₑ 2L1L 14SNRL at high SNR In particular the error probability decreases as the Lth power of SNR corresponding to a slope of L in the error probability curve in dBdB scale To understand this better we examine the probability of the deep fade event as in our analysis in Section 312 The typical error event at high SNR is when the overall channel gain is small This happens with probability Ph² 1SNR Figure 37 plots the distribution of h² for different values of L clearly the tail of the distribution near zero becomes lighter for larger L For small x the probability density function of h² is approximately fx 1L1 xL1 and so Ph² 1SNR ₀1SNR 1L1 xL1 dx 1L 1SNRL Figure 37 The probability density function of h² for different values of L The larger the L the faster the probability density function drops off around 0 This analysis is too crude to get the correct constant before the 1SNRL term in 341 but does get the correct exponent L Basically an error occurs when suml1 to L hl2 is of the order of or smaller than 1SNR and this happens when all the magnitudes of the gains hl2 are small of the order of 1SNR Since the probability that each hl2 is less than 1SNR is approximately 1SNR and the gains are independent the probability of the overall gain being small is of the order 1SNRL Typically L is called the diversity gain of the system 322 Beyond repetition coding The repetition code is the simplest possible code Although it achieves a diversity gain it does not exploit the degrees of freedom available in the channel effectively because it simply repeats the same symbol over the L symbol times By using more sophisticated codes a coding gain can also be obtained beyond the diversity gain There are many possible codes that one can use We first focus on the example of a rotation code to explain some of the issues in code design for fading channels Consider the case L 2 A repetition code which repeats a BPSK symbol u a twice obtains a diversity gain of 2 but would only transmit one bit of information over the two symbol times Transmitting two independent BPSK symbols u1 u2 over the two times would use the available degrees of freedom more efficiently but of course offers no diversity gain an error would be made whenever one of the two channel gains h1 h2 is in deep fade To get both benefits consider instead a scheme that transmits the vector x R u1 u2T 345 over the two symbol times where R cos theta sin theta sin theta cos theta 346 is a rotation matrix for some theta 0 2pi This is a code with four codewords xA R a aT xB R a aT xC R a aT xD R a aT 347 they are shown in Figure 38a The received signal is given by yl hl xl wl l 1 2 348 Here communication is over the real I channel since both x1 and x2 are real but as in Section 313 the spectral efficiency can be doubled by using both the I and the Q channels Since the two channels are orthogonal one can apply the same code separately to the symbols transmitted in the two channels to get the same performance gain Figure 38 a Codewords of rotation code b Codewords of repetition code It is difficult to obtain an explicit expression for the exact error probability So we will proceed by looking at the union bound Due to the symmetry of the code without loss of generality we can assume xA is transmitted The union bound says that pe PxA xB PxA xC PxA xD 349 where PxA xB is the pairwise error probability of confusing xA with xB when xA is transmitted and when these are the only two hypotheses Conditioned on the channel gains h1 and h2 this is just the binary detection problem in Summary A2 of Appendix A with uA h1 xA1 h2 xA2T and uB h1 xB1 h2 xB2T 350 Hence PxA xB h1 h2 Q uA uB 2 sqrtN02 QsqrtSNRh12 d12 h22 d22 2 351 where SNR a2 N0 and d 1axA xB 2 cos theta 2 sin thetaT 352 is the normalized difference between the codewords normalized such that the transmit energy is 1 per symbol time We use the upper bound Qx ex22 for x 0 in 351 to get PxA xB h1 h2 exp SNR h12 d12 h22 d22 4 353 Averaging with respect to h1 and h2 under the independent Rayleigh fading assumption we get PxA xB Eh1 h2 exp SNR h12 d12 h22 d22 4 1 1 SNR d12 4 1 1 SNR d22 4 354 Here we have used the fact that the moment generating function for a unit mean exponential random variable X is EesX 11 s for s 1 While it is possible to get an exact expression for the pairwise error probability this upper bound is more explicit moreover it is asymptotically tight at high SNR Exercise 37 We first observe that if d1 0 or d2 0 then the diversity gain of the code is only 1 If they are both nonzero then at high SNR the above bound on the pairwise error probability becomes PxA xB 16 d1 d22 SNR2 355 Call δAB d1 d22 356 the squared product distance between xA and xB when the average energy of the code is normalized to be 1 per symbol time cf 352 This determines the pairwise error probability between the two codewords Similarly we can define δij to be the squared product distance between xi and xj i j A B C D Combining 355 with 349 yields a bound on the overall error probability pe 16 1 δAB 1 δAC 1 δAD SNR2 48 minjBCD δAj SNR2 357 We see that as long as δij 0 for all i j we get a diversity gain of 2 The minimum squared product distance minjBCD δAj then determines the coding gain of the scheme beyond the diversity gain This parameter depends on θ and we can optimize over θ to maximize the coding gain Here δAB δAD 4 sin2 2θ and δAC 16 cos2 2θ 358 The angle θ that maximizes the minimum squared product distance makes δAB equal δAC yielding θ 12 tan1 2 and min δij 165 The bound in 357 now becomes pe 15 SNR2 359 To get more insight into why the product distance is important we see from 351 that the typical way for xA to be confused with xB is for the squared Euclidean distance h12d12 h22d22 between the received codewords to be of the order of 1SNR This event holds roughly when both h12d12 and h22d22 are of the order of 1SNR and this happens with probability approximately 1d12 SNR 1d22 SNR 1d12d22 SNR2 360 Thus it is important that both d12 and d22 are large to ensure diversity against fading in both components It is interesting to see how this code compares to the repetition scheme To keep the bit rate the same 2 bits over 2 realvalued symbols the repetition scheme would be using 4PAM modulation 3b b b 3b The codewords of the repetition scheme are shown in Figure 38b From 351 the pairwise error probability between two adjacent codewords say xA and xB is PxA xB EQSNR2 h12d12 h22d22 361 But now SNR 5b2N0 is the average SNR per symbol time for the 4PAM constellation and d1 d2 25 are the normalized component differences between the adjacent codewords The minimum squared product distance for the repetition code is therefore 1625 and we can compare this to the minimum squared product distance of 165 for the previous rotation code Since the error probability is proportional to SNR2 in both cases we conclude that the rotation code has an improved coding gain over the repetition code in terms of a saving in transmit power by a factor of 5 35 dB for the same product distance This improvement comes from increasing the overall product distance and this is in turn due to spreading the codewords in the twodimensional space rather than packing them on a singledimensional line as in the repetition code This is the same reason that QPSK is more efficient than BPSK as we have discussed in Section 313 We summarize and generalize the above development to any time diversity code ⁶ As we have seen earlier the 4PAM constellation requires five times more energy than BPSK for the same separation between the constellation points Summary 31 Time diversity code design criterion Ideal timeinterleaved channel yℓ hℓ xℓ wℓ ℓ 1 L 362 where hℓ are iid CN0 1 Rayleigh faded channel gains x1 xM are the codewords of a time diversity code with block length L normalized such that 1ML Σ from i1 to M xi2 1 363 Union bound on overall probability of error pe 1M Σij Pxi xj 364 Bound on pairwise error probability Pxi xj Π from ℓ1 to L 11 SNRxiℓ xjℓ24 365 where xiℓ is the ℓth component of codeword xi and SNR 1N0 Let Lij be the number of components on which the codewords xi and xj differ Diversity gain of the code is min over ij Lij 366 If Lij L for all i j then the code achieves the full diversity L of the channel and pe 4LM Σij 1δij SNRL 4LM1min over ij δij SNRL 367 where δij Π from ℓ1 to L xiℓ xjℓ2 368 is the squared product distance between xi and xj The rotation code discussed above is specifically designed to exploit time diversity in fading channels In the AWGN channel however rotation of the constellation does not affect performance since the iid Gaussian noise is invariant to rotations On the other hand codes that are designed for the AWGN channel such as linear block codes or convolutional codes can be used to extract time diversity in fading channels when combined with interleaving Their performance can be analyzed using the general framework above For example the diversity gain of a binary linear block code where the coded symbols are ideally interleaved is simply the minimum Hamming distance between the codewords or equivalently the minimum weight of a codeword the diversity gain of a binary convolutional code is given by the free distance of the code which is the minimum weight of the coded sequence of the convolutional code The performance analysis of these codes and various decoding techniques is further pursued in Exercise 311 It should also be noted that the above code design criterion is derived assuming iid Rayleigh fading across the symbols This can be generalized to the case when the coded symbols pass through correlated fades of the channel see Exercise 312 Generalization to the case when the fading is Rician is also possible and is studied in Exercise 318 Nevertheless these code design criteria all depend on the specific channel statistics assumed Motivated by information theoretic considerations we take a completely different approach in Chapter 9 where we seek a universal criterion which works for all channel statistics We will also be able to define what it means for a timevarying code to be optimal Example 31 Time diversity in GSM Global System for Mobile GSM is a digital cellular standard developed in Europe in the 1980s GSM is a frequency division duplex FDD system and uses two 25MHz bands one for the uplink mobiles to basestation and one for the downlink basestation to mobiles The original bands set aside for GSM are the 890915 MHz band uplink and the 935960 MHz band downlink The bands are further divided into 200kHz subchannels and each subchannel is shared by eight users in a timedivision fashion timedivision multiple access TDMA The data of each user are sent over time slots of length 577 microseconds μs and the time slots of the eight users together form a frame of length 4615 ms Figure 39 Voice is the main application for GSM Voice is coded by a speech encoder into speech frames each of length 20 ms The bits in each speech frame are encoded by a convolutional code of rate 12 with the two generator polynomials D4 D3 1 and D4 D3 D 1 The number of coded bits for each speech frame is 456 To achieve time diversity these coded bits are interleaved across eight consecutive time slots assigned to that specific user the 0th 8th 448th bits are put into the first time slot the 1st 9th 449th bits are put into the second time slot etc 102 Pointtopoint communication Compared to MLSD complexity of the Rake receiver is much lower ISI is avoided because of the very low spectral efficiency per user but the spectrum is typically shared between many interfering users Complexity is thus shifted to the problem of interference management 3 Orthogonal frequency division multiplexing Information is modulated on noninterfering subcarriers in the frequency domain The transformation between the time and frequency domains is done by means of addingsubtracting a cyclic prefix and IDFTDFT operations This incurs an overhead in terms of time and power Frequency diversity is attained by coding over independently faded sub carriers This coding problem is identical to that for time diversity Complexity is shared between the transmitter and the receiver in perform ing the IDFT and DFT operations the complexity of these operations is insensitive to the number of taps scales moderately with the number of subcarriers Nc and is very manageable with current implementation technology Complexity of diversity coding across subcarriers can be traded off with the amount of diversity desired 35 Impact of channel uncertainty In the past few sections we assumed perfect channel knowledge so that coherent combining can be performed at the receiver In fast varying channels it may not be easy to estimate accurately the phases and magnitudes of the tap gains before they change In this case one has to understand the impact of estimation errors on performance In some situations noncoherent detection which does not require an estimate of the channel may be the preferred route In Section 311 we have already come across a simple noncoherent detector for fading channels without diversity In this section we will extend this to channels with diversity When we compared coherent and noncoherent detection for channels with out diversity the difference was seen to be relatively small cf Figure 32 An important question is what happens to that difference as the number of diversity paths L increases The answer depends on the specific diversity scenario We first focus on the situation where channel uncertainty has the most impact DS spreadspectrum over channels with frequency diversity Once we understand this case it is easy to extend the insights to other scenarios 108 Pointtopoint communication measurement problem is the same as with a single receive antenna and does not become harder The situation is similar in the time diversity scenario In antenna diversity with L transmit antennas the received energy per diversity path does decrease with the number of antennas used but certainly we can restrict the number L to be the number of different channels that can be reliably learnt by the receiver How about in OFDM systems with frequency diversity Here the designer has control over how many subcarriers to spread the signal energy over Thus while the number of available diversity branches L may increase with the bandwidth the signal energy can be restricted to a fixed number of sub carriers L L over any one OFDM time block Such communication can be restricted to concentrated timefrequency blocks and Figure 325 visualizes one such scheme for L 2 where the choice of the L subcarriers is different for different OFDM blocks and is hopped over the entire bandwidth Since the energy in each OFDM block is concentrated within a fixed number of subcarriers at any one time coherent reception is possible On the other hand the maximum diversity gain of L can still be achieved by coding across the subcarriers within one OFDM block as well as across different blocks One possible drawback is that since the total power is only concentrated within a subset of subcarriers the total degrees of freedom available in the system are not utilized This is certainly the case in the context of pointto point communication in a system with other users sharing the same band width however the other degrees of freedom can be utilized by the other users and need not go wasted In fact one key advantage of OFDM over DS spreadspectrum is the ability to maintain orthogonality across multiple users in a multiple access scenario We will return to this point in Chapter 4 Figure 325 An illustration of a scheme that uses only a fixed part of the bandwidth at every time Here one small square denotes a single subcarrier within one OFDM block The timeaxis indexes the different OFDM blocks the frequencyaxis indexes the different subcarriers Time Frequency 109 35 Impact of channel uncertainty Chapter 3 The main plot Baseline We first looked at detection on a narrowband slow fading Rayleigh channel Under both coherent and noncoherent detection the error probability behaves like pe SNR1 3157 at high SNR In contrast the error probability decreases exponentially with the SNR in the AWGN channel The typical error event for the fading channel is due to the channel being in deep fade rather than the Gaussian noise being large Diversity Diversity was presented as an effective approach to improve performance drastically by providing redundancy across independently faded branches Three modes of diversity were considered time the interleaving of coded symbols over different coherence time periods space the use of multiple transmit andor receive antennas frequency the use of a bandwidth greater than the coherence bandwidth of the channel In all cases a simple scheme that repeats the information symbol across the multiple branches achieves full diversity With L iid Rayleigh branches of diversity the error probability behaves like pe c SNRL 3158 at high SNR Examples of repetition schemes repeating the same symbol over different coherence periods repeating the same symbol over different transmit antennas one at a time repeating the same symbol across OFDM subcarriers in different coher ence bands transmitting a symbol once every delay spread in a frequencyselective channel so that multiple delayed replicas of the symbol are received without interference Code design and degrees of freedom More sophisticated schemes cannot achieve higher diversity gain but can provide a coding gain by improving the constant c in 3158 This is 110 Pointtopoint communication achieved by utilizing the available degrees of freedom better than in the repetition schemes Examples rotation and permutation codes for time diversity and for frequency diversity in OFDM Alamouti scheme for transmit diversity uncoded transmission at symbol rate in a frequencyselective channel with ISI equalization Criteria to design schemes with good coding gain were derived for the different scenarios by using the union bound based on pairwise error probabilities on the actual error probability product distance between codewords for time diversity determinant criterion for spacetime codes Channel uncertainty The impact of channel uncertainty is significant in scenarios where there are many diversity branches but only a small fraction of signal energy is received along each branch Directsequence spreadspectrum is a prime example The gap between coherent and noncoherent schemes is very significant in this regime Noncoherent schemes do not work well as they cannot combine the signals along each branch effectively Accurate channel estimation is crucial Given the amount of transmit power devoted to channel estimation the efficacy of detection performance depends on the key parameter SNRest the received SNR per coherence time per diversity branch If SNRest 0dB then detection performance is near coherent If SNRest 0dB then effective combining is impossible Impact of channel uncertainty can be ameliorated in some schemes where the transmit energy can be focused on smaller number of diversity branches Effectively SNRest is increased OFDM is an example 36 Bibliographical notes Reliable communication over fading channels has been studied since the 1960s Improving the performance via diversity is also an old topic Standard digital commu nication texts contain many formulas for the performance of coherent and noncoherent diversity combiners which we have used liberally in this chapter see Chapter 14 of Proakis 96 for example Early works recognizing the importance of the product distance criterion for improv ing the coding gain under Rayleigh fading are Wilson and Leung 144 and Divsalar 117 37 Exercises L 2 In particular Theorem 552 of 117 constructs orthogonal designs for all L and n 2L This does not preclude the existence of orthogonal designs with rate larger than 05 A reading exercise is to study 117 where orthogonal designs with rate larger than 05 are constructed Exercise 318 The pairwise error probability analysis for the iid Rayleigh fading channel has led us to the product distance for time diversity and generalized product distance for transmit diversity code design criteria Extend this analysis for the iid Rician fading channel 1 Does the diversity order change for repetition coding over a time diversity channel with the L branches iid Rician distributed 2 What is the new code design criterion analogous to product distance based on the pairwise error probability analysis Exercise 319 In this exercise we study the performance of spacetime codes the subject of Section 332 in the presence of multiple receive antennas 1 Derive as an extension of 383 the pairwise error probability for spacetime codes with nr receive antennas 2 Assuming that the channel matrix has iid Rayleigh components derive as an extension of 386 a simple upper bound for the pairwise error probability 3 Conclude that the code design criterion remains unchanged with multiple receive antennas Exercise 320 We have studied the performance of the Alamouti scheme in a channel with two transmit and one receive antenna Suppose now we have an additional receive antenna Derive the ML detector for the symbols based on the received signals at both receive antennas Show that the scheme effectively provides two independent scalar channels What is the gain of each of the channels Exercise 321 In this exercise we study some expressions for error probabilities that arise in Section 333 1 Verify Eqs 393 and 394 In which SNR range is 393 smaller than 394 2 Repeat the derivation of 393 and 394 for a general target rate of R bitssHz suppose that R is an integer How does the SNR range in which the spatial multiplexing scheme performs better depend on R Exercise 322 In Section 333 the performance comparison between the spatial multiplexing scheme and the Alamouti scheme is done for PAM symbols Extend the comparison to QAM symbols with the target data rate R bitssHz suppose that R 4 is an even integer Exercise 323 In the text we have developed code design criteria for pure time diversity and pure spatial diversity scenarios In some wireless systems one can get both time and spatial diversity simultaneously and we want to develop a code design criterion for that More specifically consider a channel with L transmit antennas and 1 receive antenna The channel remains constant over blocks of k symbol times but changes to an independent realization every k symbols as a result of interleaving say The channel is assumed to be independent across antennas All channel gains are Rayleigh distributed C H A P T E R 4 Cellular systems multiple access and interference management 41 Introduction In Chapter 3 our focus was on pointtopoint communication ie the sce nario of a single transmitter and a single receiver In this chapter we turn to a network of many mobile users interested in communicating with a common wireline network infrastructure1 This form of wireless communication is dif ferent from radio or TV in two important respects first users are interested in messages specific to them as opposed to the common message that is broad cast in radio and TV Second there is twoway communication between the users and the network In particular this allows feedback from the receiver to the transmitter which is missing in radio and TV This form of communica tion is also different from the allwireless walkietalkie communication since an access to a wireline network infrastructure is demanded Cellular systems address such a multiuser communication scenario and form the focus of this chapter Broadly speaking two types of spectra are available for commercial cel lular systems The first is licensed typically nationwide and over a period of a few years from the spectrum regulatory agency FCC in the United States The second is unlicensed spectrum made available for experimental systems and to aid development of new wireless technologies While licens ing spectrum provides immunity from any kind of interference outside of the system itself bandwidth is very expensive This skews the engineering design of the wireless system to be as spectrally efficient as possible There are no hard constraints on the power transmitted within the licensed spectrum but the power is expected to decay rapidly outside On the other hand unli censed spectrum is very cheap to transmit on and correspondingly larger 1 A common example of such a network wireline albeit is the public switched telephone network 120 121 41 Introduction than licensed spectrum but there is a maximum power constraint over the entire spectrum as well as interference to deal with The emphasis thus is less on spectral efficiency The engineering design can thus be very different depending on whether the spectrum is licensed or not In this chapter we focus on cellular systems that are designed to work on licensed spectrum Such cellular systems have been deployed nationwide and one of the driving factors for the use of licensed spectrum for such networks is the risk of huge capital investment if one has to deal with malicious interference as would be the case in unlicensed bands A cellular network consists of a number of fixed basestations one for each cell The total coverage area is divided into cells and a mobile communicates with the basestations close to it See Figure 12 At the physical and medium access layers there are two main issues in cellular communication multiple access and interference management The first issue addresses how the overall resource time frequency and space of the system is shared by the users in the same cell intracell and the second issue addresses the interference caused by simultaneous signal transmissions in different cells intercell At the network layer an important issue is that of seamless connectivity to the mobile as it moves from one cell to the other and thus switching communication from one basestation to the other an operation known as handoff In this chapter we will focus primarily on the physical layer issues of multiple access and interference management although we will see that in some instances these issues are also coupled with how handoff is done In addition to resource sharing between different users there is also an issue of how the resource is allocated between the uplink the communication from the mobile users to the basestation also called the reverse link and the downlink the communication from the basestation to the mobile users also called the forward link There are two natural strategies for separating resources between the uplink and the downlink time division duplex TDD separates the transmissions in time and frequency division duplex FDD achieves the separation in frequency Most commercial cellular systems are based on FDD Since the powers of the transmitted and received signals typically differ by more than 100 dB at the transmitter the signals in each direction occupy bands that are separated far apart tens of MHz and a Sector 3 Sector 1 Sector 2 Figure 41 A hexagonal cell with three sectors device called a duplexer is required to filter out any interference between the two bands A cellular network provides coverage of the entire area by dividing it into cells We can carry this idea further by dividing each cell spatially This is called sectorization and involves dividing the cell into say three sectors Figure 41 shows such a division of a hexagonal cell One way to think about sectors is to consider them as separate cells except that the basestation corresponding to the sectors is at the same location Sectorization is achieved by having a directional antenna at the basestation that focuses transmissions 122 Cellular systems into the sector of interest and is designed to have a null in the other sectors The ideal end result is an effective creation of new cells without the added burden of new basestations and network infrastructure Sectorization is most effective when the basestation is quite tall with few obstacles surrounding it Even in this ideal situation there is intersector interference On the other hand if there is substantial local scattering around the basestation as is the case when the basestations are lowlying such as on the top of lamp posts sectorization is far less effective because the scattering and reflection would transfer energy to sectors other than the one intended We will discuss the impact of sectorization on the choice of the system design In this chapter we study three cellular system designs as case studies to illustrate several different approaches to multiple access and interference management Both the uplink and the downlink designs will be studied In the first system which can be termed a narrowband system user transmissions within a cell are restricted to separate narrowband channels Further neigh boring cells use different narrowband channels for user transmissions This requires that the total bandwidth be split and reduces the frequency reuse in the network However the network can now be simplified and approximated by a collection of pointtopoint noninterfering links and the physicallayer issues are essentially pointtopoint ones The IS136 and GSM standards are prime examples of this system Since the level of interference is kept minimal the pointtopoint links typically have high signaltointerferenceplusnoise ratios SINRs2 The second and third system designs propose a contrasting strategy all transmissions are spread to the entire bandwidth and are hence wideband The key feature of these systems is universal frequency reuse the same spectrum is used in every cell However simultaneous transmissions can now interfere with each other and links typically operate at low SINRs The two system designs differ in how the users signals are spread The code division multiple access CDMA system is based on directsequence spreadspectrum Here users information bits are coded at a very low rate and modulated by pseudonoise sequences In this system the simultaneous transmissions intra cell and intercell cause interference The IS95 standard is the main example to highlight the design features of this system In the orthogonal frequency division multiplexing OFDM system on the other hand users information is spread by hopping in the timefrequency grid Here the transmissions within a cell can be kept orthogonal but adjacent cells share the same bandwidth and intercell interference still exists This system has the advantage of the full frequency reuse of CDMA while retaining the benefits of the narrowband system where there is no intracell interference 2 Since interference plays an important role in multiuser systems SINR takes the place of the parameter SNR we used in Chapter 3 when we only talked about pointtopoint communication 123 42 Narrowband cellular systems We also study the power profiles of the signals transmitted in these systems This study will be conducted for both the downlink and the uplink to obtain an understanding of the peak and average power profile of the transmissions We conclude by detailing the impact on power amplifier settings and overall power consumption in the three systems Towards implementing the multiple access design there is an overhead in terms of communicating certain parameters from the basestation to the mobiles and vice versa They include authentication of the mobile by the network allocation of traffic channels training data for channel measurement transmit power level and acknowledgement of correct reception of data Some of these parameters are onetime communication for a mobile others continue in time The amount of overhead this constitutes depends to some extent on the design of the system itself Our discussions include this topic only when a significant overhead is caused by a specific design choice The table at the end of the chapter summarizes the key properties of the three systems 42 Narrowband cellular systems In this section we discuss a cellular system design that uses naturally the ideas of reliable pointtopoint wireless communication towards constructing a wireless network The basic idea is to schedule all transmissions so that no two simultaneous transmissions interfere with each other for the most part We describe an identical uplink and downlink design of multiple access and interference management that can be termed narrowband to signify that the user transmissions are restricted to a narrow frequency band and the main design goal is to minimize all interference Our description of the narrowband system is the same for the uplink and the downlink The uplink and downlink transmissions are separated either in time or frequency For concreteness let us consider the separation to be in frequency implemented by adopting an FDD scheme which uses widely separated frequency bands for the two types of transmissions A bandwidth of W Hz is allocated for the uplink as well as for the downlink Transmissions of different users are scheduled to be nonoverlapping in time and frequency thus eliminating intracell interference Depending on how the overall resource time and bandwidth is split among transmissions to the users the system performance and design implications of the receivers are affected We first divide the bandwidth into N narrowband chunks also denoted as channels Each narrowband channel has width WN Hz Each cell is allotted some n of these N channels These n channels are not necessarily contigu ous The idea behind this allocation is that all transmissions within this cell in both the uplink and the downlink are restricted to those n channels To prevent interference between simultaneous transmissions in neighboring 128 Cellular systems which it could originate and the channel variations But due to the fact that the interfering user can be at a wide range of locations the variance of I is quite high We see that the SINR is a random parameter leading to an undesirably poor performance There is an appreciably high probability of unreliable trans mission of even a small and fixed data rate in the frame In Chapter 3 we focused on techniques that impart channel diversity to the system for exam ple antenna diversity techniques make the channel less variable improving performance However there is an important distinction in the variability of the SINR here that cannot be improved by the diversity techniques of Chapter 3 The randomness in the interference I due to the interferers loca tion is inherent in this system and remains Due to this we can conclude that narrowband systems are unsuitable for universal frequency reuse To reduce the randomness in the SINR we would really like the interference to be averaged over several simultaneous lowerpowered transmissions from the neighboring cell instead of coming from one user only This is one of the important underlying themes in the design of the next two systems that have universal frequency reuse Summary 41 Narrowband systems Orthogonal narrowband channels are assigned to users within a cell Users in adjacent cells cannot be assigned the same channel due to the lack of interference averaging across users This reduces the frequency reuse factor and leads to inefficient use of the total bandwidth The network is decomposed into a set of high SINR pointtopoint links simplifying the physicallayer design Frequency planning is complex particularly when new cells have to be added 43 Wideband systems CDMA In narrowband systems users are assigned disjoint timefrequency slots within the cell and users in adjacent cells are assigned different frequency bands The network is decomposed into a set of pointtopoint noninterfering links In a code division multiple access CDMA system design the multiple access and interference management strategies are different Using the direct sequence spreadspectrum technique briefly mentioned in Section 343 each user spreads its signal over the entire bandwidth such that when demodulating any particular users data other users signals appear as pseudo white noise 129 43 Wideband systems CDMA Thus not only all users in the same cell share all the timefrequency degrees of freedom so do the users in different cells Universal frequency reuse is a key property of CDMA systems Roughly the design philosophy of CDMA systems can be broken down into two design goals First the interference seen by any user is made as similar to white Gaussian noise as possible and the power of that interference is kept to a minimum level and as consistent as possible This is achieved by Making the received signal of every user as random looking as possible via modulating the coded bits onto a long pseudonoise sequence Tight power control among users within the same cell to ensure that the received power of each user is no more than the minimum level needed for demodulation This is so that the interference from users closer to the basestation will not overwhelm users further away the socalled nearfar problem Averaging the interference of many geographically distributed users in nearby cells This averaging not only makes the aggregate interference look Gaussian but more importantly reduces the randomness of the inter ference level due to varying locations of the interferers thus increasing link reliability This is the key reason why universal frequency reuse is possible in a wideband system but impossible in a narrowband system Assuming the first design goal is met each user sees a pointtopoint wideband fading channel with additive Gaussian noise Diversity techniques introduced in Chapter 3 such as coding timeinterleaving Rake combining and antenna diversity can be employed to improve the reliability of these pointtopoint links Thus CDMA is different from narrowband system design in the sense that all users share all degrees of freedom and therefore interfere with each other the system is interferencelimited rather than degreeoffreedomlimited On the other hand it is similar in the sense that the design philosophy is still to decompose the network problem into a set of independent pointtopoint links only now each link sees both interference as well as the background thermal noise We do not question this design philosophy here but we will see that there are alternative approaches in later chapters In this section we confine ourselves to discussing the various components of a CDMA system in the quest to meet the two design goals We use the IS95 standard to discuss concretely the translation of the design goals into a real system Compared to the narrowband systems described in the previous section CDMA has several potential benefits Universal frequency reuse means that users in all cells get the full band width or degrees of freedom of the system In narrowband systems the number of degrees of freedom per user is reduced by both the number of users sharing the resources within a cell as well as by the frequencyreuse 130 Cellular systems factor This increase in degrees of freedom per user of a CDMA system however comes at the expense of a lower signaltointerferenceplusnoise ratio SINR per degree of freedom of the individual links Because the performance of a user depends only on the aggregate inter ference level the CDMA approach automatically takes advantage of the source variability of users if a user stops transmitting data the total inter ference level automatically goes down and benefits all the other users Assuming that users activities are independent of each other this provides a statistical multiplexing effect to enable the system to accommodate more users than would be possible if every user were transmitting continuously Unlike narrowband systems no explicit reassignment of time or frequency slots is required In a narrowband system new users cannot be admitted into a network once the timefrequency slots run out This imposes a hard capacity limit on the system In contrast increasing the number of users in a CDMA system increases the total level of interference This allows a more graceful degradation on the performance of a system and provides a soft capacity limit on the system Since all cells share a common spectrum a user on the edge of a cell can receive or transmit signals to two or more basestations to improve recep tion This is called soft handoff and is yet another diversity technique but at the network level sometimes called macrodiversity It is an important mechanism to increase the capacity of CDMA systems In addition to these network benefits there is a further linklevel advantage over narrowband systems every user in a CDMA experiences a wideband fading channel and can therefore exploit the inherent frequency diversity in the system This is particularly important in a slow fading environment where there is a lack of time diversity It significantly reduces the fade margin of the system the increased SINR required to achieve the same error probability as in an AWGN channel On the cons side it should be noted that the performance of CDMA sys tems depends crucially on accurate power control as the channel attenuation of nearby and cell edge users can differ by many tens of dBs This requires frequent feedback of power control information and incurs a significant over head per active user In contrast tight power control is not necessary in narrowband systems and power control is exercised mainly for reducing bat tery consumption rather than managing interference Also it is important in a CDMA system that there be sufficient averaging of outofcell interference While this assumption is rather reasonable in the uplink because the interfer ence comes from many weak users it is more questionable in the downlink where the interference comes from a few strong adjacent basestations3 3 In fact the downlink of IS95 is the capacity limiting link 139 43 Wideband systems CDMA Channel 1dB Transmitted power Measured error probability or target rate Measured SINR or β Measured SINR Inner loop Closed loop Outer loop Open loop Update β Received signal Frame decoder Estimate uplink power required Initial downlink power measurement adjusts the SINR threshold as a function of frame error rates Figure 45 Figure 45 Inner and outer loops of power control An important point however is that even though feedback occurs at a high rate 800 Hz because of the limited resolution of 1 bit per feedback power control does not track the fast multipath fading of the users when they are at vehicular speeds It only tracks the slower shadow fading and varying path loss The multipath fading is dealt with primarily by the diversity techniques discussed earlier Soft handoff Handoff from one cell to the other is an important mechanism in cellular systems Traditionally handoffs are hard users are either assigned to one cell or the other but not both In CDMA systems since all the cells share the same spectrum soft handoffs are possible multiple basestations can simultaneously decode the mobiles data with the switching center choosing Figure 46 Soft handoff Switching center Basestation 1 Basestation 2 Mobile 1 dB Power control bits 1 dB 140 Cellular systems the best reception among them Figure 46 Soft handoffs provide another level of diversity to the users The soft handoff process is mobileinitiated and works like this While a user is tracking the downlink pilot of the cell it is currently in it can be searching for pilots of adjacent cells these pilots are known pseudonoise sequences shifted by known offsets In general this involves timing acqui sition of the adjacent cell as well However we have observed that timing acquisition is a computationally very expensive step Thus a practical alter native is for the basestation clocks to be synchronized so that the mobile only has to acquire timing once Once a pilot is detected and found to have sufficient signal strength relative to the first pilot the mobile will signal the event to its original basestation The original basestation will in turn notify the switching center which enables the second cells basestation to both send and receive the same traffic to and from the mobile In the uplink each basestation demodulates and decodes the frame or packet independently and it is up to the switching center to arbitrate Normally the better cells decision will be used If we view the basestations as multiple receive antennas soft handoff is providing a form of receive diversity We know from Section 331 that the optimal processing of signals from the multiple antennas is maximal ratio combining this is however difficult to do in the handoff scenario as the antennas are geographically apart Instead what soft handoff achieves is selection combining cf Exercise 313 In IS95 there is another form of handoff called softer handoff which takes place between sectors of the same cell In this case since the signal from the mobile is received at the sectored antennas which are colocated at the same basestation maximal ratio combining can be performed How does power control work in conjunction with soft handoff Soft handoff essentially allows users to choose among several cell sites In the power control formulation discussed in the previous section each user is assumed to be assigned to a particular cell but cell site selection can be easily incorporated in the framework Suppose user k has an active set Sk of cells among which it is performing soft handoff Then the transmit powers Pk and the cell site assignments ck Sk should be chosen such that the SINR requirements 410 are simultaneously met Again if there is a feasible solution it can be shown that there is a componentwise minimal solution for the transmit powers Exercise 45 Moreover there is an analogous distributed asynchronous algorithm that will converge to the optimal solution at each step each user is assigned the cell site that will minimize the transmit power required to meet its SINR requirement given the current interference levels at the basestations Its transmit power is set accordingly Exercise 48 Put it another way the transmit power is set in such a way that the SINR requirement is just met at the cell with the best reception This is implemented in the IS95 system as follows all the basestations in the soft handoff set will feedback 145 43 Wideband systems CDMA of fluctuation of the aggregate interference level Further randomness arises due to imperfect power control The same principle of interference averaging applies to these settings as well allowing CDMA systems to benefit from an increase in the system size These settings are analyzed in Exercises 411 and 412 To conclude our discussion we note that we have made an implicit assump tion of separation of timescales in our analysis of the effect of interference in CDMA systems At a faster timescale we average over the pseudoran dom characteristics of the signal and the fast multipath fading to compute the statistics of the interference which determine the bit error rates of the point topoint demodulators At a slower timescale we consider the burstiness of user traffic and the largescale motion of the users to determine the outage probability ie the probability that the target bit error rate performance of users cannot be met Since these error events occur at completely different timescales and have very different ramifications from a systemlevel per spective this way of measuring the performance of the system makes more sense than computing an overall average performance 432 CDMA downlink The design of the onetomany downlink uses the same basic principles of pseudorandom spreading diversity techniques power control and soft handoff we already discussed for the uplink However there are several important differences The nearfar problem does not exist for the downlink since all the signals transmitted from a basestation go through the same channel to reach any given user Thus power control is less crucial in the downlink than in the uplink Rather the problem becomes that of allocating different powers to different users as a function of primarily the amount of outofcell interference they see However the theoretical formulation of this power allocation problem has the same structure as the uplink power control problem See Exercise 413 Since signals for the different users in the cell are all transmitted at the base station it is possible to make the users orthogonal to each other something that is more difficult to do in the uplink as it requires chiplevel syn chronization between distributed users This reduces but does not remove intracell interference since the transmitted signal goes through multipath channels and signals with different delays from different users still interfere with each other Still if there is a strong lineof sight component this tech nique can significantly reduce the intracell interference since then most of the energy is in the first tap of the channel On the other hand intercell interference is more poorly behaved in the downlink than in the uplink In the uplink there are many distributed 146 Cellular systems 96 kbps Downlink data 48 kbps 24 kbps 12 kbps Symbol cover Block interleaver 12288 Msyms PN code generator for I channel PN code generator for Q channel Baseband shaping filter Baseband shaping filter Hadamard Walsh sequence 90 Carrier generator 12288 Mchipss 12288 Mchips s 192 ksym s Rate 05 K 9 Convolutional encoder Output CDMA signal users transmitting with small power and significant interference averaging Figure 48 The IS95 downlink occurs In the downlink in contrast there are only a few neighboring base stations but each transmits at high power There is much less interference averaging and the downlink capacity takes a significant hit compared to the uplink In the uplink soft handoff is accomplished by multiple basestations lis tening to the transmitted signal from the mobile No extra system resource needs to be allocated for this task In the downlink however multiple base stations have to simultaneously transmit to a mobile in soft handoff Since each cell has a fixed number of orthogonal codes for the users this means that a user in soft handoff is consuming double or more system resources See Exercise 413 for a precise formulation of the downlink soft handoff problem It is common to use a strong pilot and perform coherent demodulation in the downlink since the common pilot can be shared by all the users With the knowledge of the channels from each basestation a user in soft handoff can also coherently combine the signals from the different basestations Synchronization tasks are also made easier in the presence of a strong pilot As an example the IS95 downlink is shown in Figure 48 Note the different roles of the Hadamard sequences in the uplink and in the downlink In the uplink the Hadamard sequences serve as an orthogonal modulation for each individual user so that noncoherent demodulation can be performed In the downlink in contrast each user in the cell is assigned a different Hadamard sequence to keep them orthogonal at the transmitter 147 43 Wideband systems CDMA 433 System issues Signal characteristics Consider the baseband uplink signal of a user given in 41 Due to the abrupt transitions from 1 to 1 and vice versa of the pseudonoise sequences sn the bandwidth occupied by this signal is very large On the other hand the signal has to occupy an allotted bandwidth As an example we see that the IS 95 system uses a bandwidth of 12288 MHz and a steep fall off after 167 MHz To fit this allotted bandwidth the signal in 41 is passed through a pulse shaping filter and then modulated on to the carrier Thus though the signal in 41 has a perfect PAPR equal to 1 the resulting transmit signal has a larger PAPR The overall signal transmitted from the basestation is the superposition of all the user signals and this aggregate signal has PAPR performance similar to that of the narrowband system described in the previous section Sectorization In the narrowband system we saw that all users can maintain high SINR due to the nature of the allocations In fact this was the benefit gained by paying the price of poor reuse of the spectrum In the CDMA system however due to the intra and intercell interferences the values of SINR possible are very small Now consider sectorization with universal frequency reuse among the sectors Ideally with full isolation among the sectors this allows us to increase the system capacity by a factor equal to the number of sectors However in practice each sector now has to contend with intersector interference as well Since intrasector and intercell interference dominate the noise faced by the user signals the additional interference caused due to sectorization does not cause a further degradation in SINR Thus sectors of the same cell reuse the frequency without much of an impact on the performance Network issues We have observed that timing acquisition at a chip level accuracy by a mobile is a computationally intensive step Thus we would like to have this step repeated as infrequently as possible On the other hand to achieve soft handoff this acquisition has to be done synchronously for all basestations with which the mobile communicates To facilitate this step and the eventual handoff implementations of the IS95 system use high precision clocks about 1 ppm parts per million and further synchronize the clocks at the base stations through a proprietary wireline network that connects the basestations This networking cost is the price paid in the design to ease the handoff process Summary 42 CDMA Universal frequency reuse all users both within a cell and across different cells transmit and receive on the entire bandwidth 148 Cellular systems The signal of each user is modulated onto a pseudonoise sequence so that it appears as white noise to others Interference management is crucial for allowing universal frequency reuse Intracell interference is managed via power control Accurate closed loop power control is particularly important for combating the nearfar problem in the uplink Intercell interference is managed via averaging of the effects of multiple interferers It is more effective in the uplink than in the downlink Interference averaging also allows statistical multiplexing of bursty users thus increasing system capacity Diversity of the pointtopoint links is achieved by a combination of lowrate coding timeinterleaving and Rake combining Soft handoff provides a further level of macrodiversity allowing users to communicate with multiple basestations simultaneously 44 Wideband systems OFDM The narrowband system design of making transmissions interferencefree simplified several aspects of network design One such aspect was that the performance of a user is insensitive to the received powers of other users In contrast to the CDMA approach the requirement for accurate power control is much less stringent in systems where user transmissions in the same cell are kept orthogonal This is particularly important in systems designed to accom modate many users each with very low average data rate the fixed overhead needed to perform tight power control for each user may be too expensive for such systems On the other hand there is a penalty of poor spectral reuse in narrowband systems compared to the CDMA system Basically narrowband systems are ill suited for universal frequency reuse since they do not average interference In this section we describe a system that combines the desirable features of both these systems maintaining orthogonality of transmissions within the cell and having universal frequency reuse across cells Again the latter feature is made possible through interference averaging 441 Allocation design principles The first step in the design is to decide on the user signals that ensure orthogonality after passing through the wireless channel Recall from the discussion of the downlink signaling in the CDMA system that though the transmit signals of the users are orthogonal they interfere with each other at the receiver after passing through the multipath channel Thus any orthogonal 149 44 Wideband systems OFDM set of signals will not suffice If we model the wireless channel as a linear time invariant multipath channel then the only eigenfunctions are the sinusoids Thus sinusoid inputs remain orthogonal at the receiver no matter what the multipath channel is However due to the channel variations in time we want to restrict the notion of orthogonality to no more than a coherence time interval In this context sinusoids are no longer orthogonal but the sub carriers of the OFDM scheme of Section 344 with the cyclic prefix for the multipath channel provide a set of orthogonal signals over an OFDM block length We describe an allocation of sets of OFDM subcarriers as the user signals this description is identical for both the downlink and the uplink As in Section 344 the bandwidth W is divided into Nc subcarriers The number of subcarriers Nc is chosen to be as large as possible As we discussed earlier Nc is limited by the coherence time ie the OFDM symbol period NcW Tc In each cell we would like to distribute these Nc subcarriers to the users in it with say n subcarriers per user The n subcarriers should be spread out in frequency to take advantage of frequency diversity There is no interference among user transmissions within a cell by this allocation With universal frequency reuse there is however intercell interference To be specific let us focus on the uplink Two users in neighboring cells sharing the same subcarrier in any OFDM symbol time interfere with each other directly If the two users are close to each other the interference can be very severe and we would like to minimize such overlaps However due to full spectral reuse there is such an overlap at every OFDM symbol time in a fully loaded system Thus the best one can do is to ensure that the interference does not come solely from one user or a small set of users and the interference seen over a coded sequence of OFDM symbols forming a frame can be attributed to most of the user transmissions in the neighboring cell Then the overall interference seen over a frame is a function of the average received power of all the users in the neighboring cells This is yet another example of the interference diversity concept we already saw in Section 43 How are the designs of the previous two systems geared towards harvesting interference diversity The CDMA design fully exploits interferer diversity by interference averaging This is achieved by every user spreading its signals over the entire spectrum On the other hand the orthogonal allocation of channels in the GSM system is poorly suited from the point of view of interferer diversity As we saw in Section 42 users in neighboring cells that are close to each other and transmitting on the same channel over the same slot cause severe interference to each other This leads to a very degraded performance and the reason for it is clear interference seen by a user comes solely from one interferer and there is no scope to see an average interference from all the users over a slot If there were no hopping and coding across the subcarriers the OFDM system would behave exactly like a narrowband system and suffer the same fate 152 Cellular systems interleaving is permitted then the time diversity in the system can also be obtained To implement these design goals in a cellular system successfully the users within the cell must be synchronized to their corresponding basestation This way the simultaneous uplink transmissions are still orthogonal at the base station Further the transmissions of neighboring basestations also have to be synchronized This way the design of the hopping patterns to average the interference is fully utilized Observe that the synchronization needs to be done only at the level of OFDM symbols which is much coarser than at the level of chips 443 Signal characteristics and receiver design Let us consider the signal transmission corresponding to a particular user either in the uplink or the downlink The signal consists of n virtual chan nels which over a slot constitute a set of n OFDM subcarriers that are hopped over OFDM symbol times Thus though the signal information con tent can be narrow for small ratios nNc the signal bandwidth itself is wide Further since the bandwidth range occupied varies from symbol to symbol each mobile receiver has to be wideband That is the sam pling rate is proportional to 1W Thus this signal constitutes a frequency hopped spreadspectrum signal just as the CDMA signal is the ratio of data rate to bandwidth occupied by the signal is small However unlike the CDMA signal which spreads the energy over the entire bandwidth here the energy of the signal is only in certain subcarriers n of a total Nc As discussed in Chapter 3 fewer channel parameters have to be measured and channel estimation with this signal is superior to that with the CDMA signal The major advantages of the third system design are the frequency and interferer diversity features There are a few engineering drawbacks to this choice The first is that the mobile sampling rate is quite high same as that of the CDMA system design but much higher than that of the first system All signal processing operations such as the FFT and IFFT are driven off this basic rate and this dictates the processing power required at the mobile receiver The second drawback is with respect to the transmit signal on the uplink In Exercise 415 we calculate the PAPR of a canoni cal transmit signal in this design and observe that it is significantly high as compared to the signal in the GSM and CDMA systems As we discussed in the first system earlier this higher PAPR translates into a larger bias in the power amplifier settings and a correspondingly lower average efficiency Several engineering solutions have been proposed to this essentially engineer ing problem as opposed to the more central communication problem which deals with the uncertainties in the channel and we review some of these in Exercise 416 154 Cellular systems QPSK or 16QAM is used to convert the raw information bits into the 672 OFDM symbols The different levels of granularity of the traffic channels are ideally suited to carry bursty traffic Indeed FlashOFDM is designed to act in a data network where it harnesses the statistical multiplexing gains of the users bursty data traffic by its packetswitching operation The mobiles are in three different states in the network When they are inactive they go to a sleep mode monitoring the basestation signal every once in a while this mode saves power by turning off most of the mobile device functionalities On the other hand when the mobile is actively receiv ing andor sending data it is in the ON mode this mode requires the net work to assign resources to the mobile to perform periodic power control updates and timing and frequency synchronization Apart from these two states there is an inbetween HOLD mode here mobiles that have been recently active are placed without power control updates but still maintain ing timing and frequency synchronization with the basestation Since the intracell users are orthogonal and the accuracy of power control can be coarse users in a HOLD state can be quickly moved to an ON state when there is a need to send or receive data FlashOFDM has the ability to hold approximately 30 130 and 1000 mobiles in the ON HOLD and sleep modes Formanydataapplicationsitisimportanttobeabletokeepalargenumber of users in the HOLD state since each user may send traffic only once in a while and in short bursts requests for http transfers acknowledgements etc but when they do want to send they require short latency and quick access to the wireless resource It is difficult to support this HOLD state in a CDMA system Since accurate power control is crucial because of the nearfar problem a user who is not currently powercontrolled is required to slowly ramp up its power before it can send traffic This incurs a very significant delay12 On the other hand it is very expensive to power control a large number of users who only transmit infrequently In an orthogonal system like OFDM this overhead can be largely avoided The issue does not ariseinavoicesystemsinceeachusersendsconstantlyandthepowercontrol overhead is only a small percentage of the payload about 10 in IS95 Chapter 4 The main plot The focus of this chapter is on multiple access interference management and the system issues in the design of cellular networks To highlight the 12 Readers from the San Francisco Bay area may be familiar with the notorious Fast Track lanes for the Bay Bridge Once a car gets on one of these lanes it can cross the toll plaza very quickly But the problem is that most of the delay is in getting to them through the traffic jam 155 46 Exercises issues we looked at three different system designs Their key characteris tics are compared and contrasted in the table below Narrowband system Wideband CDMA Wideband OFDM Signal Narrowband Wideband Wideband Intracell BW allocation Orthogonal Pseudorandom Orthogonal Intracell interference None Significant None Intercell BW allocation Partial reuse Universal reuse Universal reuse Intercell uplink interference Bursty Averaged Averaged Accuracy of power control Low High Low Operating SINR High Low Range low to high PAPR of uplink signal Low Medium High Example system GSM IS95 FlashOFDM 45 Bibliographical notes The two important aspects that have to be addressed by a wireless system designer are how resource is allocated within a cell among the users and how interference both intra and intercell is handled Three topical wireless technologies have been used as case studies to bring forth the tradeoffs the designer has to make The standards IS136 60 and GSM 99 have been the substrate on which the discussion of the narrowband system design is built The wideband CDMA design is based on the widely implemented secondgenerational technology IS95 61 A succinct description of the the technical underpinnings of the IS95 design has been done by Viterbi 140 with emphasis on a system view and our discussion here has been influenced by it The frequency hopping OFDM system based on Latin squares was first suggested by Wyner 150 and Pottie and Calderbank 94 This basic physicallayer construct has been built into a technology FlashOFDM 38 46 Exercises Exercise 41 In Figure 42 we set a specific reuse pattern A channel used in a cell precludes its use in all the neighboring cells With this allocation policy the reuse factor is at least 17 This is a rather ad hoc allocation of channels to the cells and the reuse ratio can be improved for example the fourcolor theorem 102 asserts that a planar graph can be colored with four colors with no two vertices joined by an edge C H A P T E R 5 Capacity of wireless channels In the previous two chapters we studied specific techniques for communi cation over wireless channels In particular Chapter 3 is centered on the pointtopoint communication scenario and there the focus is on diversity as a way to mitigate the adverse effect of fading Chapter 4 looks at cellular wireless networks as a whole and introduces several multiple access and interference management techniques The present chapter takes a more fundamental look at the problem of communication over wireless fading channels We ask what is the optimal performance achievable on a given channel and what are the techniques to achieve such optimal performance We focus on the pointtopoint scenario in this chapter and defer the multiuser case until Chapter 6 The material covered in this chapter lays down the theoretical basis of the modern development in wireless communication to be covered in the rest of the book The framework for studying performance limits in communication is infor mation theory The basic measure of performance is the capacity of a chan nel the maximum rate of communication for which arbitrarily small error probability can be achieved Section 51 starts with the important exam ple of the AWGN additive white Gaussian noise channel and introduces the notion of capacity through a heuristic argument The AWGN chan nel is then used as a building block to study the capacity of wireless fading channels Unlike the AWGN channel there is no single definition of capacity for fading channels that is applicable in all scenarios Sev eral notions of capacity are developed and together they form a system atic study of performance limits of fading channels The various capacity measures allow us to see clearly the different types of resources available in fading channels power diversity and degrees of freedom We will see how the diversity techniques studied in Chapter 3 fit into this big pic ture More importantly the capacity results suggest an alternative technique opportunistic communication which will be explored further in the later chapters 166 209 54 Capacity of fading channels channel is near its peak In a nonfading AWGN channel the channel stays constant at the average level and there are no peaks to take advantage of For models like Rayleigh fading the channel gain is actually unbounded Hence theoretically the gain of the fading channel waterfilling capacity over the AWGN channel capacity is also unbounded See Figure 523 However to get very large relative gains one has to operate at very low SNR In this regime it may be difficult for the receiver to track and feed back the channel state to the transmitter to implement the waterfilling strategy Overall the performance gain from full CSI is not that large compared to CSIR unless the SNR is very low On the other hand full CSI potentially simplifies the code design problem as no coding across channel states is necessary In contrast one has to interleave and code across many channel states with CSIR Waterfilling versus channel inversion The capacity of the fading channel with full CSI by using the waterfill ing power allocation should be interpreted as a longterm average rate of flow of information averaged over the fluctuations of the channel While the waterfilling strategy increases the longterm throughput of the system by transmitting when the channel is good an important issue is the delay entailed In this regard it is interesting to contrast the waterfilling power allo cation strategy with the channel inversion strategy Compared to waterfilling channel inversion is much less powerefficient as a huge amount of power is consumed to invert the channel when it is bad On the other hand the rate of flow of information is now the same in all fading states and so the associ ated delay is independent of the timescale of channel variations Thus one can view the channel inversion strategy as a delaylimited power allocation strategy Given an average power constraint the maximum achievable rate by this strategy can be thought of as a delaylimited capacity For applications with very tight delay constraints this delaylimited capacity may be a more appropriate measure of performance than the waterfilling capacity Without diversity the delaylimited capacity is typically very small With increased diversity the probability of encountering a bad channel is reduced and the average power consumption required to support a target delaylimited rate is reduced Put another way a larger delaylimited capacity is achieved for a given average power constraint Exercise 524 Example 53 Rate adaptation in IS856 IS856 downlink IS856 also called CDMA 2000 1 EVDO Enhanced Version Data Opti mized is a cellular data standard operating on the 125MHz bandwidth 210 Capacity of wireless channels Fixed transmit power User 2 User 1 Base station Data Measure channel request rate Figure 525 Downlink of IS856 CDMA 2000 1 EVDO Users measure their channels based on the downlink pilot and feed back requested rates to the basestation The basestation schedules users in a timedivision manner The uplink is CDMAbased not too different from IS95 but the downlink is quite different Figure 525 Multiple access is TDMA with one user transmission at a time The finest granularity for scheduling the user transmissions is a slot of duration 167 ms Each user is ratecontrolled rather than power controlled The transmit power at the basestation is fixed at all times and the rate of transmission to a user is adapted based on the current channel condition In contrast the uplink of IS95 cf Section 432 is CDMAbased with the total power dynamically allocated among the users to meet their individual SIR requirements The multiple access and scheduling aspects of IS856 are discussed in Chapter 6 here the focus is only on rate adaptation Rate versus power control The contrast between power control in IS95 and rate control in IS856 is roughly analogous to that between the channel inversion and the waterfilling strategies discussed above In the former power is allocated dynamically to a user to maintain a constant target rate at all times this is suitable for voice whichhasastringentdelayrequirementandrequiresaconsistentthroughput In the latter rate is adapted to transmit more information when the channel is strong this is suitable for data which have a laxer delay requirement and can take better advantage of a variable transmission rate The main difference betweenIS856andthewaterfillingstrategyisthatthereisnodynamicpower adaptation in IS856 only rate adaption Rate control in IS856 Like IS95 IS856 is an FDD system Hence rate control has to be performed based on channel state feedback from the mobile to the base station The mobile measures its own channel based on a common strong pilot broadcast by the basestation Using the measured values the mobile predicts the SINR for the next time slot and uses that to predict the rate the basestation can send information to it This requested rate is fed back to the basestation on the uplink The transmitter then sends a packet at 213 54 Capacity of fading channels To reduce the loss in performance due to the conservativeness of the channel prediction IS856 employs an incremental ARQ or hybrid ARQ mechanism for the repetitioncoded multiple slot packets Instead of waiting until the end of the transmission of all slots before decoding the mobile will attempt to decode the information incrementally as it receives the repeated copies over the time slots When it succeeds in decoding it will send an acknowledgement back to the basestation so that it can stop the transmission of the remaining slots This way a rate higher than the requested rate can be achieved if the actual SINR is higher than the predicted SINR 547 Frequencyselective fading channels So far we have considered flat fading channels cf 553 In Section 533 the capacity of the timeinvariant frequencyselective channel 532 was also analyzed It is simple to extend the understanding to underspread timevarying frequencyselective fading channels these are channels with the coherence time much larger than the delay spread We model the channel as a time invariant Ltap channel as in 532 over each coherence time interval and view it as Nc parallel subchannels in frequency For underspread chan nels Nc can be chosen large so that the cyclic prefix loss is negligible This model is a generalization of the flat fading channel in 553 here there are Nc frequency subchannels over each coherence time interval and multiple time subchannels over the different coherence time inter vals Overall it is still a parallel channel We can extend the capacity results from Sections 545 and 546 to the frequencyselective fading channel In particular the fast fading capacity with full CSI cf Section 546 can be generalized here to a combination of waterfilling over time and frequency the coherence time intervals provide subchannels in time and each coher ence time interval provides subchannels in frequency This is carried out in Exercise 530 548 Summary a shift in point of view Let us summarize our investigation on the performance limits of fading channels In the slow fading scenario without transmitter channel knowledge the amount of information that is allowed through the channel is random and no positive rate of communication can be reliably supported in the sense of arbitrarily small error probability The outage probability is the main performance measure and it behaves like 1SNR at high SNR This is due to a lack of diversity and equivalently the outage capacity is very small With L branches of diversity either over space time or frequency the outage C H A P T E R 6 Multiuser capacity and opportunistic communication In Chapter 4 we studied several specific multiple access techniques TDMAFDMA CDMA OFDM designed to share the channel among sev eral users A natural question is what are the optimal multiple access schemes To address this question one must now step back and take a fun damental look at the multiuser channels themselves Information theory can be generalized from the pointtopoint scenario considered in Chapter 5 to the multiuser ones providing limits to multiuser communications and suggesting optimal multiple access strategies New techniques and concepts such as successive cancellation superposition coding and multiuser diversity emerge The first part of the chapter focuses on the uplink manytoone and downlink onetomany AWGN channel without fading For the uplink an optimal multiple access strategy is for all users to spread their signal across the entire bandwidth much like in the CDMA system in Chapter 4 However rather than decoding every user treating the interference from other users as noise a successive interference cancellation SIC receiver is needed to achieve capacity That is after one user is decoded its signal is stripped away from the aggregate received signal before the next user is decoded A similar strategy is optimal for the downlink with signals for the users superimposed on top of each other and SIC done at the mobiles each user decodes the information intended for all of the weaker users and strips them off before decoding its own It is shown that in situations where users have very disparate channels to the basestation CDMA together with successive cancellation can offer significant gains over the conventional multiple access techniques discussed in Chapter 4 In the second part of the chapter we shift our focus to multiuser fading channels One of the main insights learnt in Chapter 5 is that for fast fading channels the ability to track the channel at the transmitter can increase point topoint capacity by opportunistic communication transmitting at high rates when the channel is good and at low rates or not at all when the channel is poor We extend this insight to the multiuser setting both for the uplink 228 232 Multiuser capacity and opportunistic communication much larger than the other In this case consider operating at the corner point in which the strong user is decoded first now the weak user gets the best possible rate3 In the case when the weak user is the one further away from the basestation it is shown in Exercise 610 that this decoding order has the property of minimizing the total transmit power to meet given target rates for the two users Not only does this lead to savings in the battery power of the users it also translates to an increase in the system capacity of an interferencelimited cellular system Exercise 611 612 Comparison with conventional CDMA There is a certain similarity between the multiple access technique that achieves points A and B and the CDMA technique discussed in Chapter 4 The only difference is that in the CDMA system described there every user is decoded treating the other users as interference This is sometimes called a conventional or a singleuser CDMA receiver In contrast the SIC receiver is a multiuser receiver one of the users say user 1 is decoded treating user 2 as interference but user 2 is decoded with the benefit of the signal of user 1 already removed Thus we can immediately conclude that the performance of the conventional CDMA receiver is suboptimal in Figure 62 it achieves the point C which is strictly in the interior of the capacity region The benefit of SIC over the conventional CDMA receiver is particularly significant when the received power of one user is much larger than that of the other by decoding and subtracting the signal of the strong user first the weaker user can get a much higher data rate than when it has to contend with the interference of the strong user Figure 63 In the context of a cellular system this means that rather than having to keep the received powers of all users equal by transmit power control users closer to the basestation can be allowed to take advantage of the stronger channel and transmit at a higher rate while not degrading the performance of the users in the edge of the cell With a conventional receiver this is not possible due to the nearfar problem With the SIC we are turning the nearfar problem into a nearfar advantage This advantage is less apparent in providing voice service where the required data rate of a user is constant over time but it can be important for providing data services where users can take advantage of the higher data rates when they are closer to the basestation 613 Comparison with orthogonal multiple access How about orthogonal multiple access techniques Can they be information theoretically optimal Consider an orthogonal scheme that allocates a fraction 3 This operating point is said to be maxmin fair 254 Multiuser capacity and opportunistic communication Figure 611 Sum capacity of the uplink Rayleigh fading channel plotted as a function of SNR KPN0 2 4 6 5 5 10 15 20 10 15 20 8 AWGN CSIR Full CSI Csumbits s Hz SNR dB K 16 K 2 K 4 K 1 AWGN Figure 612 Sum capacity of the uplink Rayleigh fading channel plotted as a function of SNR KPN0 in the low SNR regime Everything is plotted as a fraction of the AWGN channel capacity 1 5 5 15 20 25 30 10 2 3 4 5 6 7 CSIR Full CSI SNR dB Csum CAWGN K 16 K 4 K 2 K 1 10 Several observations can be made from the plots The sum capacity without transmitter CSI increases with the number of the users but not significantly This is due to the multiuser averaging effect explained in the last section This sum capacity is always bounded by the capacity of the AWGN channel The sum capacity with full CSI increases significantly with the number of users In fact with even two users this sum capacity already exceeds that 257 67 Multiuser diversity system aspects the ability of the basestation to schedule transmissions among the users as well as to adapt the data rate as a function of the instantaneous channel quality These features are already present in the designs of many thirdgeneration systems Nevertheless in practice there are several considerations to take into account before realizing such gains In this section we study three main hurdles towards a system implementation of the multiuser diversity idea and some prominent ways of addressing these issues 1 Fairness and delay To implement the idea of multiuser diversity in a real system one is immediately confronted with two issues fairness and delay In the ideal situation when users fading statistics are the same the strategy of communicating with the user having the best channel maximizes not only the total throughput of the system but also that of individual users In reality the statistics are not symmetric there are users who are closer to the basestation with a better average SNR there are users who are stationary and some that are moving there are users who are in a rich scattering environment and some with no scatterers around them More over the strategy is only concerned with maximizing longterm average throughputs in practice there are latency requirements in which case the average throughput over the delay timescale is the performance metric of interest The challenge is to address these issues while at the same time exploiting the multiuser diversity gain inherent in a system with users hav ing independent fluctuating channel conditions As a case study we will look at one particular scheduler that harnesses multiuser diversity while addressing the realworld fairness and delay issues 2 Channel measurement and feedback One of the key system requirements to harness multiuser diversity is to have scheduling decisions by the base station be made as a function of the channel states of the users In the uplink the basestation has access to the user transmissions over trickle channels which are used to convey control information and has an estimate of the user channels In the downlink the users have access to their channel states but need to feedback these values to the basestation Both the error in channel state measurement and the delay in feeding it back constitute a significant bottleneck in extracting the multiuser diversity gains 3 Slow and limited fluctuations We have observed that the multiuser diver sity gains depend on the distribution of channel fluctuations In particular larger and faster variations in a channel are preferred over slow ones However there may be a lineofsight path and little scattering in the environment and hence the dynamic range of channel fluctuations may be small Further the channel may fade very slowly compared to the delay constraints of the application so that transmissions cannot wait until the channel reaches its peak Effectively the dynamic range of channel fluctuations is small within the timescale of interest Both are important 259 67 Multiuser diversity system aspects Figure 614 For symmetric channel statistics of users the scheduling algorithm reduces to serving each user with the largest requested rate 0 50 100 150 200 250 300 02 03 04 05 06 07 08 09 1 Time slots Requested rates in bits s Hz Figure 615 In general with asymmetric user channel statistics the scheduling algorithm serves each user when it is near its peak within the latency timescale tc 0 50 100 150 200 250 300 02 03 04 05 06 07 08 09 1 11 12 Time slots Requested rates in bits s Hz requested rate Thus each user is scheduled when its channel is good and at the same time the scheduling algorithm is perfectly fair in the longterm In Figure 615 due perhaps to different distances from the basestation one users channel is much stronger than that of the other user on average even though both channels fluctuate due to multipath fading Always picking the user with the highest requested rate means giving all the system resources to the statistically stronger user and would be highly unfair In contrast under the scheduling algorithm described above users compete for resources not directly based on their requested rates but based on the rates normalized by their respective average throughputs The user with the statistically stronger channel will have a higher average throughput Thus the algorithm schedules a user when its instantaneous channel quality is high relative to its own average channel condition over the timescale tc 269 67 Multiuser diversity system aspects Figure 623 Comparison of the distribution of the overall channel gain with and without opportunistic beamforming using two transmit antennas Rician fading The Rayleigh distribution is also shown 0 05 1 15 2 25 3 0 02 04 06 08 10 12 14 16 18 20 Rayleigh 2 antenna Rician 1 antenna Rician Channel amplitude Density The three techniques have different system requirements Coherent space time codes like the Alamouti scheme require the users to track all the indi vidual channel gains amplitude and phase from the transmit antennas This requires separate pilot symbols on each of the transmit antennas Transmit beamforming has an even stronger requirement that the channel should be known at the transmitter In an FDD system this means feedback of the individual channel gains amplitude and phase In contrast to these two tech niques the opportunistic beamforming scheme requires no knowledge of the individual channel gains neither at the users nor at the transmitter In fact the users are completely ignorant of the fact that there are multiple transmit antennas and the receiver is identical to that in the single transmit antenna case Thus they can be termed dumb antennas Opportunistic beamforming does rely on multiuser diversity scheduling which requires the feedback of the overall SNR of each user However this only needs a single pilot to measure the overall channel What is the performance of these techniques when used in the downlink In a slow fading environment we have already remarked that opportunistic beamforming approaches the performance of transmit beamforming when there are many users in the system On the other hand spacetime codes do not perform as well as transmit beamforming since they do not capture the array power gain This means for example using the Alamouti scheme on dual transmit antennas in the downlink is 3 dB worse than using opportunistic beamforming combined with multiuser diversity scheduling when there are many users in the system Thus dumb antennas together with smart scheduling can surpass the performance of smart spacetime codes and approach that of the even smarter transmit beamforming 270 Multiuser capacity and opportunistic communication Table 61 A comparison between three methods of using transmit antennas Dumb antennas Opp beamform Smart antennas Spacetime codes Smarter antennas Transmit beamform Channel knowledge Overall SNR Entire CSI at Rx Entire CSI at Rx Tx Slow fading performance gain Diversity and power gains Diversity gain only Diversity and power gains Fast fading performance gain No impact Multiuser diversity Multiuser diversity power How about in a fast Rayleigh fading environment In this case we have observed that dumb antennas have no effect on the overall channel as the full multiuser diversity gain has already been realized Spacetime codes on the other hand increase the diversity of the pointtopoint links and consequently decrease the channel fluctuations and hence the multiuser diversity gain Exercise 631 makes this more precise Thus the use of spacetime codes as a pointtopoint technology in a multiuser downlink with rate control and scheduling can actually be harmful in the sense that even the naturally present multiuser diversity is removed The performance impact of using transmit beamforming is not so clear on the one hand it reduces the channel fluctuation and hence the multiuser diversity gain but on the other hand it provides an array power gain However in an FDD system the fast fading channel may make it very difficult to feed back so much information to enable coherent beamforming The comparison between the three schemes is summarized in Table 61 All three techniques use the multiple antennas to transmit to only one user at a time With full channel knowledge at the transmitter an even smarter scheme can transmit to multiple users simultaneously exploiting the multiple degrees of freedom existing inherently in the multiple antenna channel We will discuss this in Chapter 10 674 Multiuser diversity in multicell systems So far we have considered a singlecell scenario where the noise is assumed to be white Gaussian For wideband cellular systems with full frequency reuse such as the CDMA and OFDM based systems in Chapter 4 it is important to consider the effect of intercell interference on the performance of the system particularly in interferencelimited scenarios In a cellular system this effect is captured by measuring the channel quality of a user by the SINR signaltointerferenceplusnoise ratio In a fading environment the energies in both the received signal and the received interference fluctuate over time Since the multiuser diversity scheduling algorithm allocates resources based 273 67 Multiuser diversity system aspects of outage low for some fixed data rate The second part uses opportunistic beamforming to induce large and fast channel fluctuations and a scheduler to harness the multiuser diversity gains The performance metric on this part is to maximize the multiuser diversity gain The gains of the opportunistic beamforming and nulling depend on the probability that the received signal is near beamformed and all the interfer ence is near null In the interferencelimited regime and when PN0 1 the performance depends mainly on the probability of the latter event see Exercise 630 In the downlink this probability is large since there are only one or two basestations contributing most of the interference The uplink poses a contrasting picture there is interference from many mobiles allowing interference averaging Now the probability that the total interference is near null is much smaller Interference averaging which is one of the principle design features of the wideband full reuse systems such as the ones we saw in Chapter 4 based on CDMA and OFDM is actually unfavorable for the opportunistic scheme described here since it reduces the likelihood of the nulling of the interference and hence the likelihood of the peaks of the SINR In a typical cell there will be a distribution of users some closer to the basestation and some closer to the cell boundaries Users close to the basestation are at high SINR and are noiselimited the contribution of the intercell interference is relatively small These users benefit mainly from opportunistic beamforming Users close to the cell boundaries on the other hand are at low SINR and are interferencelimited the average interference power can be much larger than the background noise These users benefit both from opportunistic beamforming and from opportunistic nulling of intercell interference Thus the cell edge users benefit more in this system than users in the interior This is rather desirable from a system fairness pointofview as the cell edge users tend to have poorer service This feature is particularly important for a system without soft handoff which is difficult to implement in a packet data scheduling system To maximize the opportunistic nulling benefits the transmit power at the basestation should be set as large as possible subject to regulatory and hardware constraints See Exercise 6305 where this is explored in more detail We have seen the multiuser diversity as primarily a form of power gain The opportunistic beamforming technique of using an array of multiple transmit antennas has approximately an ntfold improvement in received SNR to a user in a slow fading environment as compared to the singleantenna case With an array of nr receive antennas at each mobile and say a single transmit antenna at the basestation the received SNR of any user gets an nrfold improvement as compared to a single receive antenna this gain is realized by receiver beamforming This operation is easy to accomplish since the mobile has full channel information at each of the antenna elements Hence the gains of opportunistic beamforming are about the same order as that of installing a receive antenna array at each of the mobiles 274 Multiuser capacity and opportunistic communication Thus for a system designer the opportunistic beamforming technique provides a compelling case for implementation particularly in view of the constraints of space and cost of installing multiple antennas on each mobile device Further this technique needs neither any extra processing on the part of any user nor any updates to an existing airlink interface standard In other words the mobile receiver can be completely ignorant of the use or nonuse of this technique This means that it does not have to be designed in by appropriate inclusions in the air interface standard and the receiver design and can be addedremoved at any time This is one of the important benefits of this technique from an overall system design point of view In the cellular wireless systems studied in Chapter 4 the cell is sectorized to allow better focusing of the power transmitted from the antennas and also to reduce the interference seen by mobile users from transmissions of the same basestation but intended for users in different sectors This technique is particularly gainful in scenarios when the basestation is located at a fairly large height and thus there is limited scattering around the basestation In contrast in systems with far denser deployment of basestations a strategy that can be expected to be a good one for wireless systems aiming to pro vide mobile broadband data services it is unreasonable to stipulate that the basestations be located high above the ground so that the local scattering around the basestation is minimal In an urban environment there is sub stantial local scattering around a basestation and the gains of sectorization are minimal users in a sector also see interference from the same basestation due to the local scattering intended for another sector The opportunistic beamforming scheme can be thought of as sweeping a random beam and scheduling transmissions to users when they are beamformed Thus the gains Table 62 Contrast between conventional multiple access and opportunistic communication Conventional multiple access Opportunistic communication Guiding principle Averaging out fast channel fluctuations Exploiting channel fluctuations Knowledge at Tx Track slow fluctuations No need to track fast ones Track as many fluctuations as possible Control Power control the slow fluctuations Rate control to all fluctuations Delay requirement Can support tight delay Needs some laxity Role of Tx antennas Pointtopoint diversity Increase fluctuations Power gain in downlink Multiple Rx antennas Opportunistic beamform via multiple Tx antennas Interference management Averaged Opportunistically avoided 277 68 Bibliographical notes Channel fluctuations can be sped up and their dynamic range increased by the use of multiple transmit antennas to perform opportunistic beam forming The scheme sweeps a random beam and schedules transmis sions to users when they are beamformed In a cellular system multiuser diversity scheduling performs interference avoidance as well a user is scheduled transmission when its channel is strong and the outofcell interference is weak Multiple transmit antennas can perform opportunistic beamforming as well as nulling 68 Bibliographical notes Classical treatment of the general multiple access channel was initiated by Ahlswede 2 and Liao 73 who characterized the capacity region The capacity region of the Gaussian multiple access channel is derived as a special case A good survey of the literature on MACs was done by Gallager 45 Hui 59 first observed that the sum capacity of the uplink channel with singleuser decoding is bounded by 1442 bitssHz The general broadcast channel was introduced by Cover 25 and a complete characterization of its capacity is one of the famous open problems in information theory Degraded broadcast channels where the users can be ordered based on their channel quality are fully understood with superposition coding being the optimal strategy a textbook reference is Chapter 146 in Cover and Thomas 26 The best inner and outer bounds are by Marton 81 and a good survey of the literature appears in 24 The capacity region of the uplink fading channel with receiver CSI was derived by Gallager 44 where he also showed that orthogonal multiple access schemes are strictly suboptimal in fading channels Knopp and Humblet 65 studied the sum capacity of the uplink fading channel with full CSI They noted that transmitting to only one user is the optimal strategy An analogous result was obtained earlier by Cheng and Verdú 20 in the context of the timeinvariant uplink frequencyselective channels Both these channels are instances of the parallel Gaussian multiple access channel so the two results are mathematically equivalent The latter authors also derived the capacity region in the twouser case The solution for arbitrary number of users was obtained by Tse and Hanly 122 exploiting a basic polymatroid property of the region The study of downlink fading channels with full CSI was carried out by Tse 124 and Li and Goldsmith 74 The key aspect of the study was to observe that the fading downlink is really a parallel degraded broadcast channel the capacity of which has been fully understood El Gamal 33 There is an intriguing similarity between the downlink resource allocation solution and the uplink one This connection is studied further in Chapter 10 Multiuser diversity is a key distinguishing feature of the uplink and the downlink fading channel study as compared to our understanding of the pointtopoint fading C H A P T E R 7 MIMO I spatial multiplexing and channel modeling In this book we have seen several different uses of multiple antennas in wireless communication In Chapter 3 multiple antennas were used to provide diversity gain and increase the reliability of wireless links Both receive and transmit diversity were considered Moreover receive antennas can also provide a power gain In Chapter 5 we saw that with channel knowledge at the transmitter multiple transmit antennas can also provide a power gain via transmit beamforming In Chapter 6 multiple transmit antennas were used to induce channel variations which can then be exploited by opportunistic communication techniques The scheme can be interpreted as opportunistic beamforming and provides a power gain as well In this and the next few chapters we will study a new way to use multiple antennas We will see that under suitable channel fading conditions having both multiple transmit and multiple receive antennas ie a MIMO channel provides an additional spatial dimension for communication and yields a degreeof freedom gain These additional degrees of freedom can be exploited by spatially multiplexing several data streams onto the MIMO channel and lead to an increase in the capacity the capacity of such a MIMO channel with n transmit and receive antennas is proportional to n Historically it has been known for a while that a multiple access system with multiple antennas at the basestation allows several users to simultane ously communicate with the basestation The multiple antennas allow spatial separation of the signals from the different users It was observed in the mid 1990s that a similar effect can occur for a pointtopoint channel with multiple transmit and receive antennas ie even when the transmit antennas are not geographically far apart This holds provided that the scattering environment is rich enough to allow the receive antennas to separate out the signals from the different transmit antennas We have already seen how channel fading can be exploited by opportunistic communication techniques Here we see yet another example where channel fading is beneficial to communication It is insightful to compare and contrast the nature of the performance gains offered by opportunistic communication and by MIMO techniques 290 309 73 Modeling of MIMO fading channels will have to be many wavelengths to be able to exploit this spatial multiplexing effect Summary 71 Multiplexing capability of MIMO channels SIMO and MISO channels provide a power gain but no degreeoffreedom gain Lineofsight MIMO channels with colocated transmit antennas and colocated receive antennas also provide no degreeoffreedom gain MIMO channels with farapart transmit antennas having angular separation greater than 1Lr at the receive antenna array provide an effective degree offreedom gain So do MIMO channels with farapart receive antennas having angular separation greater than 1Lt at the transmit antenna array Multipath MIMO channels with colocated transmit antennas and colocated receive antennas but with scatterersreflectors far away also provide a degreeoffreedom gain 73 Modeling of MIMO fading channels The examples in the previous section are deterministic channels Building on the insights obtained we migrate towards statistical MIMO models which capture the key properties that enable spatial multiplexing 731 Basic approach In the previous section we assessed the capacity of physical MIMO channels by first looking at the rank of the physical channel matrix H and then its condition number In the example in Section 724 for instance the rank of H is 2 but the condition number depends on how the angle between the two spatial signatures compares to the spatial resolution of the antenna array The twostep analysis process is conceptually somewhat awkward It suggests that physical models of the MIMO channel in terms of individual multipaths may not be at the right level of abstraction from the point of view of the design and analysis of communication systems Rather one may want to abstract the physical model into a higherlevel model in terms of spatially resolvable paths We have in fact followed a similar strategy in the statistical modeling of frequencyselective fading channels in Chapter 2 There the modeling is directly on the gains of the taps of the discretetime sampled channel rather than on the gains of the individual physical paths Each tap can be thought 325 73 Modeling of MIMO fading channels b Bins 0 0 1 0 1 1 0 1 0 1 1 0 k 1 0 L r 3 n r 2 a Bins 0 0 2 3 1 4 2 3 2 1 4 3 k 1 0 2 3 4 L r 3 n r 5 reduce the number of degrees of freedom and the diversity of the channel Figure 722 a Antennas are sparsely spaced Some of the bins contain paths from multiple angular windows b The antennas are very sparsely spaced All bins contain several angular windows of paths Placing the antennas more densely adds spurious basis vectors which do not correspond to any physical directions and does not add resolvability In terms of the angular channel matrix Ha this has the effect of adding zero rows and columns in terms of the spatial channel matrix H this has the effect of making the entries more correlated In fact the angular domain representation makes it apparent that one can reduce the densely spaced system to an equivalent 2Lt 2Lr critically spaced system by just focusing on the basis vectors that do correspond to physical directions Figure 724 Increasing the antenna separation within a given array length Lr does not increase the number of degrees of freedom in the channel What about increas ing the antenna separation while keeping the number of antenna elements nr the same This question makes sense if the system is hardwarelimited rather than limited by the amount of space to put the antenna array in Increasing the antenna separation this way reduces the beam width of the nr angular basis beamforming patterns but also increases the number of main lobes in each Figure 725 If the scattering environment is rich enough such that the received signal arrives from all directions the number of nonzero rows of the channel matrix Ha is already nr the largest possible and increasing the spacing does not increase the number of degrees of freedom in the channel On the other hand if the scattering is clustered to within certain directions increasing the separation makes it possible for the scattered signal to be C H A P T E R 8 MIMO II capacity and multiplexing architectures In this chapter we will look at the capacity of MIMO fading channels and discuss transceiver architectures that extract the promised multiplexing gains from the channel We particularly focus on the scenario when the transmitter does not know the channel realization In the fast fading MIMO channel we show the following At high SNR the capacity of the iid Rayleigh fast fading channel scales like nmin logSNR bitssHz where nmin is the minimum of the number of transmit antennas nt and the number of receive antennas nr This is a degreeoffreedom gain At low SNR the capacity is approximately nrSNRlog2 e bitssHz This is a receive beamforming power gain At all SNR the capacity scales linearly with nmin This is due to a combi nation of a power gain and a degreeoffreedom gain Furthermore there is a transmit beamforming gain together with an oppor tunistic communication gain if the transmitter can track the channel as well Over a deterministic timeinvariant MIMO channel the capacityachieving transceiver architecture is simple cf Section 711 independent data streams are multiplexed in an appropriate coordinate system cf Figure 72 The receiver transforms the received vector into another appropriate coordinate system to separately decode the different data streams Without knowledge of the channel at the transmitter the choice of the coordinate system in which the independent data streams are multiplexed has to be fixed a priori In conjunction with joint decoding we will see that this transmitter architecture achieves the capacity of the fast fading channel This architecture is also called VBLAST1 in the literature 1 Vertical Bell Labs SpaceTime Architecture There are several versions of VBLAST with different receiver structures but they all share the same transmitting architecture of multiplexing independent streams and we take this as its defining feature 332 357 83 Receiver architectures Figure 812 Performance of the decorrelator bank with and without successive cancellation at low SNR Here nt nr 8 SNR dB 20 30 Without successive cancellation With successive cancellation 01 08 07 06 05 04 03 02 Rdecorr C88 30 20 10 0 10 The main observation is that while the decorrelator bank performs well at high SNR it is really far away from the capacity at low SNR What is going on here To get more insight let us plot the performance of a bank of matched filters the kth filter being matched to the spatial signature hk of transmit antenna k From Figure 813 we see that the performance of the bank of matched filters is far superior to the decorrelator bank at low SNR although far inferior at high SNR Derivation of the MMSE receiver The decorrelator was motivated by the fact that it completely nulls out inter stream interference in fact it maximizes the output SNR among all linear Figure 813 Performance ratio of the rate to the capacity of the matched filter bank as compared to that of the decorrelator bank At low SNR the matched filter is superior The opposite is true for the decorrelator The channel is iid Rayleigh with nt nr 8 Decorrelator Matched fillter SNR dB 20 30 01 08 09 07 06 05 04 03 02 30 20 10 0 10 1 0 361 83 Receiver architectures Figure 815 Performance the ratio of rate to the capacity of a basic bank of MMSE receivers as compared to the matched filter bank and to the decorrelator bank MMSE performs better than both over the entire range of SNR The channel is iid Rayleigh with nt nr 8 Decorrelator 20 10 0 10 20 30 30 SNR dB MMSE Matched filter 0 1 09 08 07 06 05 04 03 02 01 R C88 MMSESIC Analogous to what we did in Section 832 for the decorrelator we can now upgrade the basic bank of linear MMSE receivers by allowing successive cancellation of streams as well as depicted in Figure 816 What is the performance improvement in using the MMSESIC receiver Figure 817 plots the performance as compared to the capacity of the channel with nt nr 8 for iid Rayleigh fading We observe a startling fact the bank of linear MMSE receivers with successive cancellation and equal power allocation achieves the capacity of the iid Rayleigh fading channel Figure 816 MMSESIC a bank of linear MMSE receivers each estimating one of the parallel data streams with streams successively cancelled from the received vector at each stage Subtract stream 1 2 nt 1 Stream 2 Decode stream nt Stream nt Subtract stream 1 Stream 1 Decode stream 1 Decode stream 2 Decode stream 3 Subtract stream 1 2 MMSE receiver 1 MMSE receiver nt MMSE receiver 3 MMSE receiver 2 ym Stream 3 374 MIMO II capacity and multiplexing architectures Successive cancellation Decode the data streams sequentially using the results of the decoding operation to cancel the effect of the decoded data streams on the received signal Bank of linear MMSE receivers with successive cancellation achieves the capacity of the fast fading MIMO channel at all SNR Outage performance of slow fading MIMO channels The iid Rayleigh slow fading MIMO channel provides a diversity gain equal to the product of nt and nr Since the VBLAST architecture does not code across the transmit antennas it can achieve a diversity gain of at most nr Staggered interleaving of the streams of VBLAST among the transmit antennas achieves the full outage performance of the MIMO channel This is the DBLAST architecture 86 Bibliographical notes The interest in MIMO communications was sparked by the capacity analysis of Foschini 40 Foschini and Gans 41 and Telatar 119 Foschini and Gans focused on analyzing the outage capacity of the slow fading MIMO channel while Telatar studied the capacity of fixed MIMO channels under optimal waterfilling ergodic capacity of fast fading channels under receiver CSI as well as outage capacity of slow fading channels The DBLAST architecture was introduced by Foschini 40 while the VBLAST architecture was considered by Wolniansky etal 147 in the context of pointtopoint MIMO communication The study of the linear receivers decorrelator and MMSE was initiated in the context of multiuser detection of CDMA signals The research in multiuser detection is very well exposited and summarized in a book by Verdú 131 who was the pioneer in this field In particular decorrelators were introduced by Lupas and Verdú 77 and the MMSE receiver by Madhow and Honig 79 The optimality of the MMSE receiver in conjunction with successive cancellation was shown by Varanasi and Guess 129 The literature on random matrices as applied in communication theory is summa rized by Tulino and Verdú 127 The key result on the asymptotic distribution of the singular values of large random matrices used in this chapter is by Marcenko and Pastur 78 87 Exercises Exercise 81 reciprocity Show that the capacity of a timeinvariant MIMO channel with nt transmit nr receive antennas and channel matrix H is the same as that of the channel with nr transmit nt receive antennas matrix H and same total power constraint 388 MIMO III diversitymultiplexing tradeoff and universal spacetime codes Figure 91 Tradeoff curves for the single antenna slow fading Rayleigh channel Spatial multiplexing gain r R log SNR Diversity Gain d r 12 0 Fixed reliability 1 0 Fixed rate 0 1 PAM QAM Figure 92 Increasing the SNR by 6dB decreases the error probability by 14 for both PAM and QAM due to a doubling of the minimum distance pe pe 1 4 QAM PAM SNR 4 SNR 1 4 This is consistent with our observation in Section 313 that PAM uses only half the degrees of freedom of QAM The increase in data rate is due to the packing of more constellation points for a given Dmin This is illustrated in Figure 93 The two endpoints represent two extreme ways of using the increase in the resource SNR increasing the reliability for a fixed data rate or increasing the data rate for a fixed reliability More generally we can simultaneously increase the data rate positive multiplexing gain r and increase the reliability positive diversity gain d 0 but there is a tradeoff between how much of each type of gain we can get The diversitymultiplexing curve describes this tradeoff Note that the classical diversity gain only describes the rate of decay of the error probability for a fixed data rate but does not provide any information on how well a scheme exploits the available degrees of freedom For example PAM and QAM have the same classical diversity 396 MIMO III diversitymultiplexing tradeoff and universal spacetime codes Figure 97 Diversitymultiplexing tradeoff dr for the iid Rayleigh fading channel Spatial multiplexing gain r R log SNR Diversity gain d r minnt nr 0 0 nt nr r nt rnr r 2 nt 2nr 2 1 nt 1nr 1 Figure 98 Adding one transmit and one receive antenna increases spatial multiplexing gain by 1 at each diversity level Spatial multiplexing gain r R log SNR Diversity gain d r d This is because the entire tradeoff curve is shifted by 1 to the right see Figure 98 The optimal tradeoff curve is based on the outage probability so in principle arbitrarily large block lengths are required to achieve the optimal tradeoff curve However it has been shown that in fact spacetime codes of block length l nt nr 1 achieve the curve In Section 924 we will see a scheme that achieves the tradeoff curve but requires arbitrarily large block lengths 416 MIMO III diversitymultiplexing tradeoff and universal spacetime codes For the parallel channel the universal criterion is to maximize the product of the codeword differences Somewhat surprisingly this is the same as the criterion arrived at by averaging over the Rayleigh channel statistics For the MISO channel the universal criterion is to maximize the smallest singular value of the codeword difference matrices For the nt nr MIMO channel the universal criterion is to maximize the product of the nmin smallest singular values of the codeword difference matrices With nr nt this criterion is the same as that arrived at by averaging over the iid Rayleigh statistics The MIMO channel can be transformed into a parallel channel via DBLAST This transformation is universal universal parallel channel codes for each of the interleaved streams in DBLAST serve as a uni versal code for the MIMO channel The rate loss due to initialization in DBLAST can be reduced by increasing the number of interleaved streams For the MISO channel however the DBLAST transformation with only one stream ie using the transmit antennas one at a time is approximately universal within the class of channels that have iid fading coefficients 93 Bibliographical notes The design of spacetime codes has been a fertile area of research There are books that provide a comprehensive view of the subject for example see the books by Larsson Sto ica and Ganesan 72 and Paulraj etal 89 Several works have recognized the tradeoff between diversity and multiplexing gains The formulation of the coarser scaling of error probability and data rate and the corresponding characterization of their fundamental tradeoff for the iid Rayleigh fading channel is the work of Zheng and Tse 156 The notion of universal communication ie communicating reliably over a class of channel was first formulated in the context of discrete memoryless channels by Black well etal 10 Dobrushin 31 and Wolfowitz 146 They showed the existence of universal codes The results were later extended to Gaussian channels by Root and Varaiya 103 Motivated by these information theoretic results Wesel and his coau thors have studied the problem of universal code design in a sequence of works start ing with his PhD thesis 142 The worstcase code design metric for the parallel channel and a heuristic derivation of the product distance criterion were obtained in 143 This was extended to MIMO channels in 67 The general concept of approxi mate universality in the high SNR regime was formulated by Tavildar and Viswanath 118 earlier in the special case of the 2 2 MIMO channel Yao and Wornell 152 used the determinant condition 980 to show the tradeoffoptimality of their rotation based codes The conditions derived for approximate universality cf 938 953 970 and 980 are also necessary this is derived in Tavildar and Viswanath 118 The design of tradeoffoptimal spacetime codes is an active area of research and several approaches have been presented recently They include rotationbased codes for the 22 channel by Yao and Wornell 152 and Dayal and Varanasi 29 lattice spacetime LAST codes by El Gamal etal 34 permutation codes for the parallel C H A P T E R 10 MIMO IV multiuser communication In Chapters 8 and 9 we have studied the role of multiple transmit and receive antennas in the context of pointtopoint channels In this chapter we shift the focus to multiuser channels and study the role of multiple antennas in both the uplink manytoone and the downlink onetomany In addition to allowing spatial multiplexing and providing diversity to each user multiple antennas allow the basestation to simultaneously transmit or receive data from multiple users Again this is a consequence of the increase in degrees of freedom from having multiple antennas We have considered several MIMO transceiver architectures for the point topoint channel in Chapter 8 In some of these such as linear receivers with or without successive cancellation the complexity is mainly at the receiver Independent data streams are sent at the different transmit antennas and no cooperation across transmit antennas is needed Equating the transmit antennas with users these receiver structures can be directly used in the uplink where the users have a single transmit antenna each but the basestation has multiple receive antennas this is a common configuration in cellular wireless systems It is less apparent how to come up with good strategies for the downlink where the receive antennas are at the different users thus the receiver struc ture has to be separate one for each user However as will see there is an interesting duality between the uplink and the downlink and by exploiting this duality one can map each receive architecture for the uplink to a correspond ing transmit architecture for the downlink In particular there is an interesting precoding strategy which is the transmit dual to the receiverbased succes sive cancellation strategy We will spend some time discussing this The chapter is structured as follows In Section 101 we first focus on the uplink with a single transmit antenna for each user and multiple receive antennas at the basestation We then in Section 102 extend our study to the MIMO uplink where there are multiple transmit antennas for each user In Sections 103 and 104 we turn our attention to the use of multiple antennas in the downlink We study precoding strategies that achieve the capacity of 425 473 105 Multiple antennas in cellular networks precanceling the data streams The performance of this scheme linear beam forming strategies with and without Costa precoding can be related to the corresponding performance of a dual MIMO uplink channel much as in the discussion of Section 1032 with multiple antennas at the basestation alone This scheme achieves the capacity of the MIMO downlink channel 105 Multiple antennas in cellular networks a system view We have discussed the system design implications of multiple antennas in both the uplink and the downlink These discussions have been in the context of multiple access within a single cell and are spread throughout the chapter Sections 1013 1016 1022 1035 and 104 In this section we take stock of these implications and consider the role of multiple antennas in cellular networks with multiple cells Particular emphasis is on two points the use of multiple antennas in suppressing intercell interference how the use of multiple antennas within cells impacts the optimal amount of frequency reuse in the network Summary 103 System implications of multiple antennas on multiple access Three ways of using multiple receive antennas in the uplink Orthogonal multiple access Each user gets a power gain but no change in degrees of freedom Opportunistic communication one user at a time Power gain but the multiuser diversity gain is reduced Space division multiple access is capacity achieving users simultane ously transmit and are jointly decoded at the basestation Comparison between orthogonal multiple access and SDMA Low SNR performance of orthogonal multiple access comparable to that of SDMA High SNR SDMA allows up to nr users to simultaneously transmit with a single degree of freedom each Performance is significantly better than that with orthogonal multiple access An intermediate access scheme with moderate complexity performs com parably to SDMA at all SNR levels blocks of approximately nr users in SDMA mode and orthogonal access for different blocks MIMO uplink Orthogonal multiple access each user has multiple degrees of freedom SDMA the overall degrees of freedom are still restricted by the number of receive antennas 479 105 Multiple antennas in cellular networks freedom when it is scheduled The discussion of the role of frequency reuse earlier now carries over to this case The nature of the tradeoff is similar there is a loss in spectral degrees of freedom due to less reuse but an increase in the spatial degrees of freedom due to the availability of multiple transmit antennas at the users 1054 Downlink with multiple receive antennas In the downlink the interference comes from a few specific locations at fixed transmit powers the neighboring basestations that reuse the same frequency Thus the interference pattern can be empirically measured at each user and the array of receive antennas used to do linear MMSE as discussed in Section 1051 and boost the received SINR For orthogonal systems the impact on frequency reuse analysis is similar to that in the uplink with the SINR from the MMSE receiver replacing the earlier simpler expression as in 520 for the uplink example If the basestation has multiple transmit antennas as well the interference could be harder to suppress in the presence of substantial scattering each of the basestation transmit antennas could have a distinct receive spatial signa ture at the mobile and in this case an appropriate model for the interference is white noise On the other hand if the scattering is only local at the base station and at the mobile then all the basestation antennas have the same receive spatial signature cf Section 723 and interference suppression via the MMSE receiver is still possible 1055 Downlink with multiple transmit antennas With full CSI ie both at the basestation and at the users the uplink downlink duality principle see Section 1032 allows a comparison to the reciprocal uplink with the multiple receive antennas and receiver CSI In particular there is a onetoone relationship between linear schemes with and without successive cancellation for the uplink and that for the downlink Thus many of our inferences in the uplink with multiple receive antennas hold in the downlink as well However full CSI may not be so practical in an FDD system having CSI at the basestation in the downlink requires substantial CSI feedback via the uplink Example 101 SDMA in ArrayComm systems ArrayComm Inc is one of the early companies implementing SDMA technology Their products include an SDMA overlay on Japans PHS cellular system a fixed wireless local loop system and a mobile cellular system iBurst 480 MIMO IV multiuser communication An ArrayComm SDMA system exemplifies many of the design features that multiple antennas at the basestation allow It is TDMA based and is much like the narrowband system we studied in Chapter 4 The main difference is that within each narrowband channel in each time slot a small number of users are in SDMA mode as opposed to just a single user in the basic narrowband system of Section 42 The array of antennas at the basestation is also used to suppress outofcell interference thus allowing denser frequency reuse than a basic narrowband system To enable successful SDMA operation and interference suppression in both the uplink and the downlink the ArrayComm system has several key design features The time slots for TDMA are synchronized across different cells Fur ther the time slots are long enough to allow accurate estimation of the interference using the training sequence The estimate of the color of the interference is then in the same time slot to suppress outofcell interference Channel state information is not kept across slots The small number of SDMA users within each narrowband channel are demodulated using appropriate linear filters for each user this operation suppresses both the outofcell interference and the incell interference from the other users in SDMA mode sharing the same narrowband channel The uplink and the downlink operate in TDD mode with the down link transmission immediately following the uplink transmission and to the same set of users The uplink transmission provides the base station CSI that is used in the immediately following downlink trans mission to perform SDMA and to suppress outofcell interference via transmit beamforming and nulling TDD operation avoids the expen sive channel state feedback required for downlink SDMA in FDD systems To get a feel for the performance improvement with SDMA over the basic narrowband system we can consider a specific implementation of the ArrayComm system There are up to twelve antennas per sector at the basestation with up to four users in SDMA mode over each narrowband channel This is an improvement of roughly a factor of four over the basic narrowband system which schedules only a single user over each narrowband channel Since there are about three antennas per user sub stantial outofcell interference suppression is possible This allows us to increase the frequency reuse ratio this is a further benefit over the basic narrowband system For example the SDMA overlay on the PHS system increases the frequency reuse ratio of 18 to 1 In the Flash OFDM example in Chapter 4 we have mentioned that one advantage of orthogonal multiple access systems over CDMA systems is that users can get access to the system without the need to slowly ramp up 481 105 Multiple antennas in cellular networks the power The interference suppression capability of adaptive antennas provides another way to allow users who are not power controlled to get access to the system quickly without swamping the existing active users Even in a nearfar situation of 4050 dB SDMA still works successfully this means that potentially many users can be kept in the hold state when there are no active transmissions These improvements come at an increased cost to certain system design features For example while downlink transmissions meant for specific users enjoy a power gain via transmit beamforming the pilot signal is intended for all users and has to be isotropic thus requiring a propor tionally larger amount of power This reduces the traditional amortization benefit of the downlink pilot Another aspect is the forced symmetry between the uplink and the downlink transmissions To successfully use the uplink measurements of the channels of the users in SDMA mode and the color of the outofcell interference in the following downlink transmission the transmission power levels in the uplink and the down link have to be comparable see Exercise 1024 This puts a strong constraint on the system designer since the mobiles operate on batter ies and are typically much more power constrained than the basestation which is powered by an AC supply Further the pairing of the uplink or downlink transmissions is ideal when the flow of traffic is symmetric in both directions this is usually true in the case of voice traffic On the other hand data traffic can be asymmetric and leads to wasted uplink downlink transmissions if only downlink uplink transmissions are desired Chapter 10 The main plot Uplink with multiple receive antennas Space division multiple access SDMA is capacityachieving all users simultaneously transmit and are jointly decoded by the basestation Total spatial degrees of freedom limited by number of users and number of receive antennas Rule of thumb is to have a group of nr users in SDMA mode and different groups in orthogonal access mode Each of the nr user transmissions in a group obtains the full receive diversity gain equal to nr Uplink with multiple transmit and receive antennas The overall spatial degrees of freedom are still restricted by the number of receive antennas but the diversity gain is enhanced 482 MIMO IV multiuser communication Downlink with multiple transmit antennas Uplinkdownlink duality identifies a correspondence between the down link and the reciprocal uplink Precoding is the analogous operation to successive cancelation in the uplink A precoding scheme that perfectly cancels the intracell interference caused to a user was described Precoding operation requires full CSI hard to justify in an FDD system With only partial CSI at the basestation an opportunistic beamforming scheme with multiple orthogonal beams utilizes the full spatial degrees of freedom Downlink with multiple receive antennas Each users link is enhanced by receive beamforming both a power gain and a diversity gain equal to the number of receive antennas are obtained 106 Bibliographical notes The precoding technique for communicating on a channel where the transmitter is aware of the channel was first studied in the context of the ISI channel by Tomlinson 121 and Harashima and Miyakawa 57 More sophisticated precoders for the ISI channel designed for use in telephone modems were developed by Eyuboglu and Forney 36 and Laroia etal 71 A survey on precoding and shaping for ISI channels is contained in an article by 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153 W Yu and J Cioffi Sum capacity of Gaussian vector broadcast channels IEEE Transactions on Information Theory 509 2004 18751892 154 R Zamir S Shamai and U Erez Nested linearlattice codes for structured multiterminal binning IEEE Transactions on Information Theory 48 2002 12501276 155 L Zheng and D Tse Communicating on the Grassmann manifold a geometric approach to the noncoherent multiple antenna channel IEEE Transactions on Information Theory 482 2002 359383 156 L Zheng and D Tse Diversity and multiplexing a fundamental tradeoff in multiple antenna channels IEEE Transactions on Information Theory 482 2002 359383 Index ad hoc network 5 additive white Gaussian noise AWGN 29 30 166 241 channel capacity 167 capacityachieving AWGN channel codes 170 171 packing spheres 16872 168 169 channel resources 172 bandwidth reuse in cellular systems 1758 178 continuoustime AWGN channel 172 power and bandwidth 1735 downlink channel 2356 236 general case of superposition coding achieves capacity 23840 239 symmetric case of two capacityachieving schemes 2368 formal derivation of capacity 526 5279 infinite bandwidth 3456 uplink channel 2401 capacity via successive interference cancellation SIC 22932 229 230 compared with conventional CDMA 232 233 compared with orthogonal multiple access 2324 234 general Kuser uplink capacity 2345 advanced mobile phone service AMPS 4 aggregate interference 141 aggregate interference level 130 CDMA uplink 133 Alamouti scheme for transmit diversity 734 1914 192 analog memoryless channels 5267 angular domain representation of signals 31113 313 angular bases 31314 angular domain transformation as DFT 314 degrees of freedom 31822 318 diversity 3223 323 MIMO channels 31516 316 statistical modeling 317 317 antenna diversity 71 multiple input multiple output MIMO channels 7782 78 receive diversity 713 72 transmit and receive diversity 72 transmit diversity 72 73 Alamouti scheme 734 determinant criterion for spacetime code design 747 antennas arrays with only a lineofsight path 299300 directional 121 122 dumb antennas for opportunistic beamforming 2636 264 265 dumb smart and smarter 26870 269 270 fast fading 2668 slow fading 266 geographically separated antennas receive antennas 3056 305 resolvability in angular domain 3015 303 304 305 transmit antennas 3001 300 multiple antennas in cellular networks 4734 downlink with multiple receive antennas 479 482 downlink with multiple transmit antennas 479 482 intercell interference management 4746 554 555 Index MIMO uplink 4789 uplink with multiple receive antennas 4768 481 uniform linear antenna arrays 296 approximate universality 398 400 code properties 4045 405 array gain 72 ArrayComm systems 47981 asymmetric fading downlink channels 251 bandwidth reuse in cellular systems 1758 178 bandwidthlimited systems 174 174 baseband equivalent model for wireless channels 225 23 24 discretetime model 258 27 28 29 beam width 304 beamforming configuration 266 beamforming patterns 303 304 305 Bernoulli coinflipped sequence 133 binary antipodal signaling 50 binary entropy 51819 binary entropy function 519 519 binary erasure channels 517 517 capacity 5245 524 binary phaseshiftkeying BPSK 50 60 coherent detection 53 54 degrees of freedom 569 differential BPSK 58 60 signaltonoise ratio SNR 56 binary symmetric channels 517 517 capacity 524 524 block fading 199 200 Bluetooth 5 burstiness averaging 141 143 capacityachieving AWGN channel codes 170 capacity of wireless channels 166 214 see also multiuser capacity AWGN channel capacity 167 capacityachieving AWGN channel codes 170 171 packing spheres 16872 168 169 repetition coding 1678 AWGN channel resources 172 bandwidth reuse in cellular systems 1758 178 continuoustime AWGN channel 172 power and bandwidth 1735 fading channels 1867 21314 fast fading 199203 216 frequencyselective fading channels 213 rate adaptation in IS856 20913 210 receive diversity 18990 slow fading 1879 187 21516 time and frequency diversity 1959 transmit diversity 1915 transmitter side information 20313 fading downlink channel 250 channel side information at receiver only 2501 full channel side information 2512 fading uplink channel 243 250 fast fading 2457 full channel side information 24750 slow fading 2434 245 linear timeinvariant Gaussian channels 179 21415 frequencyselective channels 1816 181 184 185 MIMO channels 3323 345 373 CSI at receiver 3368 performance gainss 33846 multiple input single output MISO channels 17980 reliable rate of communication 171 single input multiple output SIMO channels 179 capacity regions 428 429 5379 539 corner points 53940 540 carrier frequency 34 cellular networks 3 3 1203 bandwidth reuse 1758 178 capacity of cells 19 coverage of cells 19 frequency reuse 1278 historical development first generation systems 34 second generation systems 4 third generation systems 4 interference between adjacent cells 19 interference management 121 multiple access 121 multiple antennas 4734 downlink with multiple receive antennas 479 482 downlink with multiple transmit antennas 479 intercell interference management 4746 MIMO uplink 4789 uplink with multiple receive antennas 4768 481 narrowband allocations in GSM 1245 performance 125 signal characteristics and receiver design 1256 narrowband systems 1234 124 128 network and system design 1267 US frequency bands 11 wideband systems 12831 CDMA downlink 1456 146 CDMA uplink 13145 132 556 Index cellular networks Cont OFDM 14852 sectorization 153 system issues 147 chain rule for entropies 520 chain rule for mutual information 521 channel inversion 204 compared with waterfilling 209 channel modeling 2901 3289 angular domain representation of signals 31113 313 angular bases 31314 angular domain transformation as DFT 314 degrees of freedom 31822 318 diversity 3223 323 MIMO channels 31516 316 statistical modeling 317 317 MIMO channels 2956 antenna arrays with only a lineofsight path 299300 geographically separated antennas 3006 300 305 lineofsight MISO channels 2989 lineofsight plus one reflected path 3069 307 308 lineofsight SIMO channels 2968 296 MIMO fading channels 309 basic approach 30910 310 dependency on antenna spacing 3237 324 325 326 327 iid Rayleigh fading model 3278 multipath channels 311 physical modeling free space fixed transmit and receive antennas 1213 free space moving antenna 1314 moving antenna multiple reflectors 1920 power decay with distance and shadowing 1819 reflecting ground plate 1718 18 reflecting wall fixed antenna 1415 14 15 reflecting wall moving antenna 1517 16 17 channel side information CSI 207 207 channel side information at the receiver CSIR 207 207 MIMO channels 3368 346 capacity 346 performance analysis 3478 transceiver architecture 347 multiuser communications with MIMO systems uplink with multiple receive antennas 4367 437 uplink with multiple transmit and receive antennas 4457 446 channel uncertainty 102 110 channel estimation 1057 noncoherent detection for DS spreadspectrum 1035 104 105 other diversity scenarios 1078 108 channeldependent scheduling 258 259 channelstate independent coding scheme 366 chip rate 91 chipsynchronous users 132 circulant matrices 98 circular symmetric complex Gaussian random variables 2930 circular symmetry 29 500 Clarkes model clustered response models 319 flat fading 3840 40 clustered response models Clarkes model 319 effect of carrier frequency 3212 321 total angular spread 322 general model 31921 indoor channel measurements 320 multipath environment 320 code division multiple access CDMA 4 122 12831 1478 compared with AWGN uplink channel 232 233 downlink 1456 146 interference averaging and system capacity 1415 multiuser detection and ISI equalization 3645 365 system issues 147 uplink 1312 132 generation of pseudonoise sequences 1323 interference statistics 1334 IS95 link design 1367 136 pointtopoint link design 1346 power control 134 1378 power control in IP95 1389 139 soft handoff 134 13941 139 coding 59 coding gains 49 59 66 67 10910 coherence bandwidth 15 33 34 coherence distance 15 coherence time 16 31 34 coherent combining 61 coherent detection in Rayleigh fading channels 526 54 communication bandwidth 34 complex baseband equivalent 22 conditional entropy 51920 reliable communication 5212 558 Index fast fading channels Cont downlink with multiple transmit antennas 468 full CSI 468 receiver CSI 4689 receiver CSI and partial CSI at basestation 46971 multiuser capacity AWGN uplink 2457 full channel side information 24750 multiuser communications with MIMO systems 4369 full CSI 4389 receiver CSI 4367 437 uplink with multiple transmit and receive antennas 445 receiver CSI 4457 446 flashOFDM 1534 flat fading channels 33 34 Clarkes model 3840 40 foward channel 4 121 frequency coherence 32 frequency diversity 1001 basic concept 834 directsequence DS spreadspectrum 91 92 1012 performance analysis 935 Rake receiver 913 93 error probability analysis 868 extensions 1989 geometric view 1978 198 implementing MLSD 8891 88 orthogonal frequency division multiplexing OFDM systems 959 102 108 108 block length 99100 outage performance of parallel channels 1957 singlecarrier with ISI equalization 845 101 frequencyselective channel viewed as MISO channel 85 85 frequency division duplex FDD systems 69 121 frequency hopping 71 frequency reuse 122 1278 frequencyselective channels fading channels 33 34 capacity 213 multiuser capacity 2523 linear timeinvariant Gaussian channels coding across subcarriers 1856 transformation to parallel channel 1813 181 waterfilling power allocation 1835 184 185 Gaussian noise detection in complex vector space detection 5079 scalar detection 5034 504 vector space detection 5047 505 506 Gaussian noise estimation in complex vector space estimation 51113 scalar estimation 50910 vector space estimation 51011 Gaussian random variables complex Gaussian random vectors 5003 real Gaussian random vectors 497500 498 499 scalar real Gaussian random variables 4967 497 Global System for Mobile GSM communication systems 4 narrowband allocations 1245 performance 125 signal characteristics and receiver design 1256 time diversity 6971 70 Hadamard sequences 146 handoff 121 see also soft handoff Hermitian matrices 75 hopping patterns 1502 151 iid Gaussian code 170 ideal interweaving 5334 imperfect power control averaging 141 impulse response baseband equivalent 25 fading multipath channel 21 information theory 166 167 516 capacity of fast fading channels MIMO channels 5346 scalar channels 5334 discrete memoryless channels 51618 517 518 entropy conditional entropy and mutual information 51821 formal derivation of AWGN capacity 526 5279 analog memoryless channels 5267 multiple access channels capacity region 5379 539 capacity region corner points 53940 540 fast fading uplink 5401 noisy channel coding theorem 521 achieving upper bound 5235 operational interpretation 5256 reliable communication and conditional entropy 5212 outage formulation 5367 559 Index receiver optimality fading channels 364 MMSE is information lossless 3623 timeinvariant channel 3634 spherepacking interpretation 529 achievability 5302 531 converse 52930 530 timeinvariant parallel channel 5323 inner codes 194 intercell interference 1456 interference 1 interference averaging 141 interference avoidance 271 interference diversity 141 interference nuller 81 350 interferencelimited rate 235 interferencelimited systems 129 capacity 142 interleaving 59 60 61 intersymbol interference ISI 83 equalization and CDMA multiuser detection 3645 365 IS856 downlink 20910 210 prediction uncertainty 21113 rate control 21011 rate versus power control 210 IS95 link CDMA downlink 146 146 CDMA uplink 1367 136 power control 1389 139 Jensen inequality 2023 245 295 338 KuhnTucker condition 183 largescale fading 10 11 40 Latin squares 150 orthogonal 151 linear decorrelator 434 geometric derivation 34952 350 351 performance for deterministic H matrix 352 performance in fading channels 3524 353 354 linear equalizers 90 linear timeinvariant LTI channel 13 linear timevarying system model for wireless channels 202 local area networks LANs ad hoc network 5 wireless systems 5 loglikelihood ratio 51 lowcomplexity detection 801 82 macrodiversity 59 130 matched filter 61 maximal ratio combining 61 140 maximum length shift register MLSR 1323 maximum likelihood ML rule 51 503 504 maximum likelihood sequence detection MLSD 86 Viterbi algorithm 8891 88 memoryless channels 5267 minimum Hamming distance 69 minimum mean square error MMSE equalizers 90 333 information theoretic optimality fading channels 364 MMSE is information lossless 3623 timeinvariant channel 3634 linear MMSE receiver decorrelator limitations 3567 357 derivation 35760 358 MMSESIC 3612 361 362 performance 360 361 performance enhancement by MMSE decoding 45961 460 mobile switching center MSC see mobile telephone switching office MTSO mobile telephone switching office MTSO 34 multipath fading 11 16 multiple input multiple output MIMO channels 2901 3289 see also multiuser communications with MIMO systems antenna diversity degrees of freedom 778 78 lowcomplexity detection 801 82 spacial multiplexing 7980 summary of 22 schemes 82 capacity 3323 345 373 CSI at receiver 3368 performance gainss 33846 DBLAST archicture 368 coding across transmit antennas 3712 371 suboptimality 36870 diversitymultiplexing tradeoffs 383 384 22 MIMO Rayleigh channel 3925 formulation 3846 MIMO Rayleigh channel 392 393 393 MISO Rayleigh channel 3912 nn MIMO iid Rayleigh channel 3958 396 397 parallel Rayleigh channel 3901 391 scalar Rayleigh channel 38690 388 389 full CSI 346 capacity 346 performance analysis 3478 transceiver architecture 347 560 Index multiple input multiple output MIMO channels Cont modeling fading channels 309 angular domain transformation 31516 316 basic approach 30910 310 dependency on antenna spacing 3237 324 325 326 327 iid Rayleigh fading model 3278 multipath channels 311 multiplexing architectures 3323 373 fast fading channels 3356 VBLAST 3335 multiplexing capability 291 309 capacity via singular value decomposition 2914 293 rank and condition number 2945 physical modeling 2956 antenna arrays with only a lineofsight path 299300 geographically separated antennas 3006 300 303 304 305 lineofsight MISO channels 2989 lineofsight plus one reflected path 3069 307 308 lineofsight SIMO channels 2968 296 receiver architectures 3489 information theoretic optimality 3624 linear decorrelator 34954 350 351 353 354 linear MMSE receiver 35662 357 358 361 362 successive cancellation 3556 355 slow fading channels 3668 high SNR 368 universal spacetime codes 383 398 411 41516 design criterion 41213 properties of approximately universal codes 41315 QAM is approximately universal for scalar channels 398400 universality of DBLAST 41112 multiple input single output MISO channels 73 frequencyselective channels 85 85 large transmit antenna arrays 344 345 linear timeinvariant Gaussian channels 17980 modeling 2989 Rayleigh fading 3912 universal code design 407 410 conversion to parallel channels 4089 design criterion 40910 viewed as parallel channels 4078 multiplexing DBLAST architecture 368 coding across transmit antennas 3712 371 suboptimality 36870 MIMO architectures 3323 373 fast fading channels 3356 VBLAST 3335 receiver architectures 3489 information theoretic optimality 3624 linear decorrelator 34954 350 351 353 354 linear MMSE receiver 35662 357 358 361 362 successive cancellation 3556 355 slow fading MIMO channels 3668 high SNR 368 multiuser capacity 2289 see also capacity of wireless channels AWGN downlink 2356 236 241 general case of superposition coding achieves capacity 23840 239 symmetric case of two capacityachieving schemes 2368 AWGN fading downlink 250 channel side information at receiver only 2501 full channel side information 2512 AWGN fading uplink 243 250 fast fading 2457 slow fading 2434 245 AWGN uplink 2401 capacity via successive interference cancellation SIC 22932 229 230 compared with conventional CDMA 232 233 compared with orthogonal multiple access 2324 234 general Kuser uplink capacity 2345 frequencyselective fading channels 2523 multiuser communications with MIMO systems 4256 downlink with multiple receive and transmit antennas 4713 471 472 481 downlink with multiple transmit antennas 448 448 degrees of freedom 4489 fast fading 46871 precoding for downlink 4658 precoding for interference known at transmitter 45465 455 456 457 uplinkdownlink duality and transmit beamforming 44953 multiple antennas in cellular networks uplink 4789 561 Index uplink with multiple receive antennas 426 426 fast fading 4369 multiuser diversity 43942 slow fading 4336 435 436 spacedivision multiple access SDMA 4267 spacedivision multiple access SDMA capacity region 42830 429 system implications 4312 432 uplink with multiple transmit and receive antennas 442 fast fading 4457 SDMA 4424 443 444 system implications 4445 445 multiuser diversity 228 229 2767 channel prediction and feedback 2623 fair scheduling 258 multiuser diversity gain in practice 2612 261 262 proportional fair scheduling 25860 259 superposition coding 2601 261 multicell systems 2702 multiuser communications with MIMO systems 439 one user at a time policy 43940 optimal power allocation policy 4402 441 multiuser diversity gain 2536 254 multiuser versus classical diversity 256 system aspects 2568 system view 2725 mutual information 5201 chain rule 521 narrowband systems 122 1234 124 128 allocation in GSM system 1245 performance 125 signal characteristics and receiver design 1256 nearest neighbor rule 504 505 nearfar problem 129 232 nested lattice codes 463 463 noise spheres 169 529 530 noncoherent detection directsequence DS spreadspectrum 1035 104 105 Rayleigh fading channels 502 51 54 onering model 39 opportunistic beamforming 229 2756 469 469 dumb antennas 2636 264 265 dumb smart and smarter 26870 269 270 fast fading 2668 slow fading 266 opportunistic communications 166 2289 442 opportunistic nulling 271 opportunistic orthogonal coding 4645 optimality principle of dynamic programming 90 90 orthogonal codes 175 orthogonal frequency division multiplexing OFDM systems 84 959 102 108 108 122 148 allocations design principles 14850 block length 99100 flashOFDM 1534 hopping pattern 1502 signal characteristics and receiver design 152 transmission and reception schemes 99 orthogonal Latin squares 151 orthogonal multiple access compared with AWGN uplink channel capacity 2324 234 uplink with multiple receive antennas 476 481 orthogonality principle 510 orthonormal set of waveforms 29 outage 138 187 190 formulation 5367 parallel channels 199 Rayleigh fading 188 time and frequency diversity 1957 outer codes 194 outofcell interference averaging 141 pairwise error probability 75 parallel channels linear timeinvariant Gaussian channels 1813 181 outage 199 time and frequency diversity 1957 timeinvariant parallel channel 5323 universal spacetime codes 4006 402 403 405 4067 waterfilling power allocation 2045 206 2079 Parseval theorem for DFTs 182 passband spectrum 23 peak to average power ratio PAPR 126 peak transmit power 126 performance gains in MIMO fading channels 338 348 high SNR regime 33840 large antenna array regime 3413 342 343 low SNR regime 340 341 periodic hopping patterns 150 151 phasedarray antenna 298 power decay 1819 562 Index power gain 72 179 powerlimited systems 174 174 processing gain 91 135 pseudocovariance matrix 500 501 pseudonoise PN 91 Q function 496 497 quadrature amplitude modulation QAM 234 approximately universal for scalar channels 398400 quadrature phaseshiftkeying QPSK 60 degrees of freedom 569 differential QPSK 60 quarter circle law 342 342 quasistatic scenario 187 radio broadcast systems AM FM etc 5 Rake receiver 913 93 performance analysis 935 ratesplitting 231 ray tracing 14 Rayleigh fading 367 22 MIMO Rayleigh channel four schemes 3924 392 393 393 optimal tradeoff 3945 channel detection coherent detection 526 54 noncoherent detection 502 51 54 dumb antennas for opportunistic beamforming 267 268 MIMO capacity 3389 339 3924 MISO channels 3912 multiuser diversity gain 2534 253 nn MIMO iid Rayleigh channel geometric interpretation 3978 397 optimal tradeoff 3956 396 outage probability 188 parallel channels 3901 391 scalar channels optimal tradeoff 38990 PAM and QAM 3869 388 389 Rayleigh random variables 501 receive beamforming 179 273 358 449 receive diversity 18990 195 receiver architectures 3489 information theoretic optimality fading channels 364 MMSE is information lossless 3623 timeinvariant channel 3634 linear decorrelator geometric derivation 34952 350 351 performance for deterministic H matrix 352 performance in fading channels 3524 353 354 linear MMSE receiver decorrelator limitations 3567 357 derivation 35760 358 MMSESIC 3612 361 362 427 42930 performance 360 361 successive cancellation 3556 355 reliability of air interface 2 repetition coding 49 59 604 65 AWGN channel capacity 1678 packing spheres 168 169 transmit diversity 1945 reverse channel 4 121 richly scattered environment 328 Rician fading 37 dumb antennas for opportunistic beamforming 2678 268 multiuser diversity gain 2534 253 rotation coding 646 65 scattering reflections 20 scheduler 258 259 sectorization 1212 122 selection combining 140 separation of timescales 145 shadowing 19 signaltointerference plus noise ratio SINR 122 CDMA uplink 135 signaltonoise ratio SNR binary phaseshiftkeying BPSK 56 quadrature phaseshiftkeying QPSK 56 Rayleigh fading channels 109 coherent detection 53 54 55 noncoherent detection 52 sinct function 25 single input multiple output SIMO channels large receive antenna arrays 344 345 linear timeinvariant Gaussian channels 179 modeling 2968 296 singular value decomposition SVD 2914 293 slow fading channels 31 34 capacity 1879 187 21516 transmitter side information 204 dumb antennas for opportunistic beamforming 266 multiplexing architecture for MIMO 3668 high SNR 368 multiuser capacity AWGN uplink 2434 245 multiuser communications with MIMO systems 4336 435 436 smallscale fading 10 41 soft capacity limit 130 563 Index soft handoff 130 see also handoff CDMA downlink 146 CDMA uplink 13941 139 softer handoff 140 spacedivision multiple access SDMA 4267 ArrayComm systems 47981 capacity region 42830 429 orthogonal multiple access 4323 uplink with multiple receive antennas 4768 481 uplink with multiple transmit antennas 4424 443 444 spacetime codes 73 determinant criterion 747 spatial multiplexing 7980 2901 308 see also VBLAST multiplexing spatial signature 297 spectral efficiency 2 1434 144 172 173 specular path 37 sphere covering 458 sphere hardening effect 169 sphere packing 16872 168 169 458 529 upper bound 52930 530 squared product distance 66 squarelaw detectors 51 stationary ergodic fading 534 statistical multiplexing 130 144 successive cancellation 228 successive interference cancellation SIC 228 275 333 AWGN uplink channel 22932 229 230 implementation issues 2412 MMSESIC receivers 3612 361 362 427 42930 receiver architectures 3556 355 sum capacity 230 superposition coding 228 275 general case 23840 239 multiuser diversity 2601 261 symmetric case 2378 238 symbolbysymbol precoding 4547 455 456 457 461 decoding 462 performance 4589 transmitter knowledge of interference 4613 462 symmetric capacity 230 235 system capacity 141 system view 2 tap gain autocorrelation function 378 time diversity 60 61 code design criterion 68 extensions 1989 geometric view 1978 198 Global System for Mobile GSM systems 6971 70 other coding systems 647 65 outage performance of parallel channels 1957 repetition coding 604 65 time division duplex TDD 121 timedivision multiple access TDMA 4 Global System for Mobile GSM systems 69 transition probabilities 516 transmit beamforming 180 340 4523 transmit diversity 191 195 Alamouti scheme 1914 192 repetition coding 1945 transmit power control 137 transmittercentric scheme 466 trellis representation 89 89 ultrawideband UWB 5 32 uncertainty sphere 531 531 underspread channels 22 34 uniform linear antenna arrays 296 universal frequency reuse 12930 universal spacetime codes 3834 398 400 4067 bitreversal scheme 4056 design criterion 4002 402 403 high SNR 4034 MIMO channels 411 41516 design criterion 41213 downlink 415 properties of approximately universal codes 41315 universality of DBLAST 41112 MISO channels 407 410 conversion to parallel channels 4089 design criterion 40910 viewed as parallel channels 4078 408 properties of approximately universal codes 4045 405 QAM is approximately universal for scalar channels 398400 universal frequency reuse 122 upconversion 22 24 uplink 4 121 uplinkdownlink duality 4502 451 VBLAST multiplexing 332 333 see also spatial multiplexing MIMO architecture 3335 virtual channels 150 151 Viterbi algorithm 834 8891 88 voice communications 4 waterfilling power allocation 1835 184 185 2046 206 2079 compared with channel inversion 209 564 Index wellconditioned matrices 295 white Gaussian noise WGN 2930 35 wideband systems 122 12831 CDMA downlink 1456 146 CDMA uplink 13145 132 system issues 147 OFDM 14852 sectorization 153 wireless channels 10 inputoutput modeling 20 41 additive white noise 2930 baseband equivalent model 225 23 24 discretetime baseband equivalent model 258 27 28 29 linear timevarying system model 202 physical modeling 1011 channel quality variation 11 free space fixed transmit and receive antennas 1213 free space moving antenna 1314 moving antenna multiple reflectors 1920 power decay with distance and shadowing 1819 reflecting ground plate 1718 18 reflecting wall fixed antenna 1415 14 15 reflecting wall moving antenna 1517 16 17 statistical modeling 412 Clarkes model for flat fading channels 3840 40 modeling philosophy 345 Rayleigh and Rician fading 367 tap gain autocorrelation function 378 summary of defining characteristics 34 summary of physical parameters 34 time and frequency coherence delay spread and coherence bandwidth 313 Doppler spread and coherence time 301 wireless LANs local area networks 5 wireless systems historical perspective 25 zero crosscorrelation property 103 zeroforcing equalizers 90 zeroforcing receiver 81 3501