Digital Communications and Signal Processing K. Vasudevan Department of Electrical Engineering Indian Institute of Technology Kanpur - 208 016 INDIA version 3.2 January 2, 2020 ii K.VasudevanFacultyofEEIITKanpurIndiaemail:[email protected] Digital Communications and Signal Processing c 2001-present K. Vasudevan. All rights reserved. No part of this book may be reproduced without prior permission from the author. This book is set in LATEX To my family Acknowledgements During the course of my master’s degree and PhD at IIT Madras, I had the opportunity to learn the fundamental concepts of digital communications from my instructors, Dr. V. G. K. Murthy, Dr. V. V. Rao, Dr. K. Radhakrishna Rao, Dr. Bhaskar Ramamurthi and Dr. Ashok Jhunjhunwalla. It is a pleasure to acknowledge their teaching. I also gratefully acknowledge the guidance of Dr. K. Giridhar and Dr. Bhaskar Ramamurthi who were jointly my Doctoral supervisors. I also wish to thank Dr. Devendra Jalihal for introducing me to the LATEX document processing system without which this book would not have been complete. I would like to express my gratitude to my instructors at IIT Kharagpur (where I had completed my undergraduate) – Dr. S. L. Maskara, Dr. T. S. Lamba, Dr. R. V. Rajkumar and Dr. S. Shanmugavel (now with Anna University, Chennai). I thank Dr. Surendra Prasad of IIT Delhi and Dr. P. Y. Kam of the National University of Singapore for encouraging me to write this book. During the early stages of my career (1991-1992), I was associated with the CAD-VLSI Group, Indian Telephone Industries Ltd., at Bangalore. I would like to express my gratitude to Mr. K. S. Raghunathan (formerly a Deputy Chief Engineer at the CAD-VLSI Group), for his supervision of the implementation of a statistical fault analyzer for digital circuits. It was from him that I learnt the concepts of good programming, which I cherish and use to this day. Special mention is also due to Dr. Bixio Rimoldi of the Mobile Communi- cations Lab, EPFL Switzerland and Dr. Raymond Knopp, now with Institute Eurecom, Sophia Antipolis France, for providing me the opportunity to imple- ment the signal processing algorithms discussed in Chapter 4, in real-time, for their software radio platform. I would like to thank many of my students for their valuable feedback on the first edition of this book. Finally, I thank my colleagues at IIT Kanpur, in particular Dr. S. C. Srivastava, Dr. V. Sinha (now retired from IITK), Dr. Govind Sharma, Dr. Pradip Sircar, Dr. R. K. Bansal, Dr K. S. Venkatesh and Dr. A. K. Chaturvedi, for their support and encouragement. Thanks are also due to the Open Source community for providing operating systems like Linux and software like Scilab, LATEX, Xfig and Gnuplot, without which this book would not have been complete. I also wish to thank the publisher, Mr. Madhu Reddy and the editor Ms. Sreelatha Menon for their skill and dedication in bringing out the third edition of this book. In spite of my best efforts, some errors might have gone unnoticed. Sugges- tions for improving the book are welcome. K. Vasudevan August 2012 Notation A B Elements of set A minus set B. a{ }\{A } a is an element of set A. a∈{ / A} a is not an element of set A. a ∈{b } Logical AND of a and b. a ∧ b Logical OR of a and b. ∨ a =? b a may or may not be equal to b. There exists. ∃! There exists uniquely. ∄∃ Does not exist. For all. ∀x Largest integer less than or equal to x. ⌊x⌋ Smallest integer greater than or equal to x. ⌈ ⌉ j √ 1 − =∆ Equal to by definition. ⋆ Convolution. δ ( ) Dirac delta function. D · δK( ) Kronecker delta function. x˜ · A complex quantity. xˆ Estimate of x. x A vector or matrix. IM An M M identity matrix. S Complex× symbol (note the absence of tilde). Real part. ℜ{·} Imaginary part. ℑ{·} xI Real or in-phase part ofx ˜. xQ Imaginary or quadrature part ofx ˜. E[ ] Expectation. erfc(· ) Complementary error function. · [x1, x2] Closed interval, inclusive of x1 and x2. [x1, x2) Open interval, inclusive of x1 and exclusive of x2. (x1, x2) Open interval, exclusive of x1 and x2. P ( ) Probability. p( )· Probability density function. Hz· Frequency in Hertz. wrt With respect to. Calligraphic Letters A A B B C C D D E E F F G G H H I I J J K K L L M M N N O O P P Q Q R R S S T T U U V V W W X X Y Y Z Z Contents Preface to the Second Edition xi Preface to the Third Edition xii List of Programs xiii 1 Introduction 1 1.1 Overview of the Book ........................ 5 1.2 Bibliography ............................. 7 2 Communicating with Points 9 2.1 Coherent Detectors for 2D Constellations ............. 9 2.1.1 Performance Analysis .................... 11 2.1.2 Optimizing the Constellation ................ 13 2.1.3 Union Bound on the Probability of Error ......... 13 2.1.4 M-ary Signalling Schemes .................. 14 2.1.5 Constellations with Minimum Average Power ....... 25 2.1.6 Analysis for Non-Equiprobable Symbols .......... 26 2.2 Coherent Detectors for Multi-D Orthogonal Constellations .... 28 2.2.1 Performance Analysis .................... 30 2.2.2 Union Bound on the Probability of Error ......... 32 2.2.3 Minimum SNR Required for Error-free Transmission ... 33 2.2.4 Binary Antipodal and Orthogonal Constellations ..... 41 2.3 Bi-Orthogonal Constellations .................... 42 2.4 Simplex Constellations ........................ 43 2.5 Noncoherent Detectors for Multi-D Orthogonal Constellations .. 43 2.5.1 Performance Analysis .................... 45 2.5.2 Exact Expression for Probability of Error ......... 47 2.6 Noncoherent Detectors for M-ary PSK ............... 50 2.6.1 Approximate Performance Analysis ............. 51 2.7 Coherent Detectors in Coloured Noise ............... 54 2.7.1 Performance Analysis .................... 57 2.7.2 Predictive VA for Channel Coded Symbols ........ 61 2.8 Coherent Detectors for Flat Fading Channels ........... 63 viii K.VasudevanFacultyofEEIITKanpurIndiaemail: [email protected] 2.8.1 Performance Analysis .................... 66 2.8.2 Performance Analysis for BPSK .............. 71 2.8.3 Approximate Performance Analysis ............. 75 2.8.4 Multiple Input Multiple Output (MIMO) Systems .... 76 2.9 Differential Detectors for Flat Fading Channels .......... 86 2.10 Summary ............................... 93 3 Channel Coding 96 3.1 The Convolutional Encoder ..................... 98 3.2 Are the Encoded Symbols Correlated? ...............106 3.3 Hard Decision Decoding of CC ...................107 3.3.1 The Viterbi Algorithm (VA) ................110 3.3.2 Performance Analysis of Hard Decision Decoding .. .. 114 3.4 Soft Decision Decoding of CC ....................124 3.4.1 Performance Analysis of Soft Decision Decoding .. .. 125 3.5 Trellis Coded Modulation (TCM) ..................130 3.5.1 Mapping by Set Partitioning ................131 3.5.2 Performance of TCM Schemes ...............134 3.5.3 Analysis of a QPSK TCM Scheme .............138 3.5.4 Analysis of a 16-QAM TCM Scheme ............140 3.6 Maximization of the Shape Gain ..................143 3.7 Constellation Shaping by Shell Mapping ..............147 3.7.1 The Shell Mapping Algorithm ................152 3.8 Turbo Codes .............................159 3.8.1 The Turbo Decoder .....................161 3.8.2 The BCJR Algorithm ....................164 3.8.3 Performance of ML Decoding of Turbo Codes ....... 171 3.9 Summary ...............................173 4 Transmission of Signals through Distortionless Channels 174 4.1 Linear Modulation ..........................177 4.1.1 Transmitter ..........................177 4.1.2 Power Spectral Density of the Transmitted Signal . 180 4.1.3 Proof of Proposition 3.0.1 ..................184 4.1.4 Receiver ............................186 4.1.5 Pulse Shapes with Zero ISI .................192 4.1.6 Application of Matched Filtering in CDMA ........ 194 4.1.7 Discrete-Time Receiver Implementation .......... 200 4.2 Carrier and Timing Synchronization ................209 4.2.1 Data-Aided Carrier Phase Estimation ...........210 4.2.2 Non-Data-Aided Carrier Phase Estimation ........ 213 4.2.3 Error-Rate Analysis .....................217 4.2.4 Data-Aided Timing Synchronization ............218 4.2.5 Results for Synchronization .................226 4.3 Non-Linear Modulation .......................230 4.3.1 CPFM with Full Response Rectangular Filters ...... 231 K. Vasudevan Faculty of EE IIT Kanpur India email: [email protected] ix 4.4 Summary ...............................246 5 Transmission of Signals through Distorting Channels 248 5.1 Receivers Based on Equalization ..................249 5.1.1 Linear Equalization – Symbol-Spaced Equalizers . .. 249 5.1.2 Finite Length Equalizer ...................257 5.1.3 The Steepest Descent Algorithm ..............262 5.1.4 The Least Mean Square (LMS) Algorithm ......... 265 5.1.5 Linear Equalization – Fractionally-Spaced Equalizers . 265 5.1.6 Non-Linear Equalization – The Predictive DFE ...... 269 5.1.7 Implementation of the Predictive DFE ...........271 5.1.8 The Conventional DFE ...................273 5.2 Receivers Based on MLSE ......................276 5.2.1 Symbol-Spaced MLSE ....................276 5.2.2 Fractionally-Spaced MLSE .................285 5.2.3 T -spaced and T/2-spaced ML Detectors ..........288 5.3 Multicarrier Communication .....................294 5.3.1 Channel Loading .......................300 5.3.2 The Discrete Multitone (DMT) ...............303 5.4 Summary ...............................307 A Complex Differentiation 309 B The Chernoff Bound 311 C On Groups and Finite Fields 314 C.1 Groups .................................314 C.2