Signal Processing in Digital Communications

George J. Miao

A R T E C H H O U S E BOSTON|LONDON ar techhouse.com

Contents

Preface xv

1 Introduction 1 1.1 A History of Communications Using Electricity 1 1.2 Digital Communication Systems 3 1.3 Digital RF Systems 6

1.3.1 Digital Transceivers 7 1.3.2 A/D Converter Challenge 9 1.3.3 Digital Downconversion and Channelization 12

1.4 Link Budget 17 1.4.1 NoiseFigure 19 1.4.2 Receiver Sensitivity 19 1.4.3 Maximum Path Loss 20

1.5 Summary 21 References 22

2 Probability, Random Variables, and Stochastic Signal Processing 25 2.1 Introduction 25 2.2 Probability 26

2.2.1 Intuitive Probability 27 2.2.2 Axiomatic Probability 27 2.2.3 Conditional Probability 29 2.2.4 Independence 31

2.3 Random Variables 31 2.4 Probability Distributions and Densities 32

2.4.1 Probability Distributions 32 2.4.2 Probability Densities 33 2.4.3 Joint Probability Distributions and Densities 34

vii

viii Signal Processing in Digital Communications

2.4.4 Statistical Averages and Joint Moments 35 2.4.5 Moment Generation Function 37

2.5 Functions of Random Variables 39 2.5.1 Sums of Random Variables 39 2.5.2 Transformations of Random Variables 43

2.6 Discrete Distributions and Densities 45 2.6.1 Discrete Uniform Distribution 45 2.6.2 Binomial Distribution 46 2.6.3 Poisson Distribution 48

2.7 Continuous Distributions and Densities 51 2.7.1 Gaussian Density Function 51 2.7.2 Error Function 54 2.7.3 Q-Function 56 2.7.4 Multivariate Gaussian Distribution 58 2.7.5 Uniform Density Function 58 2.7.6 Chi-Square Distribution 60 2.7.7 JF Distribution 61 2.7.8 Rayleigh Distribution 63 2.7.9 Rice Distribution 64

2.8 Upper Bounds on the Probability 66 2.8.1 Chebyshev Inequality 67 2.8.2 Law of Large Numbers 68 2.8.3 Central Limit Theorem 69

2.9 Stochastic Signal Processes 70 2.9.1 Definition of Discrete-Time Random Process 71 2.9.2 Stationary Processes 73 2.9.3 Estimated Functions 73 2.9.4 Power Spectrum 74 2.9.5 Stochastic Processes for Linear Systems 76 2.9.6 Mean Square Estimation 81

2.10 Detection Theory and Optimum Receivers 83 2.10.1 Optimality Criterion 83 2.10.2 Maximum Likelihood Detector 85 2.10.3 Probability of Error 87

2.11 Summary 87 References 89

3 Sampling Theory 3.1 Introduction

91 91

Contents ix

3.2 Discrete-Time Sampled Signals 92 3.2.1 Instantaneous Sampling 92 3.2.2 Ideal Sampled Signal 94

3.3 Nyquist Sampling Theorem 94 3.3.1 Time-Domain Interpolation Formula 96 3.3.2 Frequency-Domain Interpolation Formula 98 3.3.3 Aliasing 99

3.4 Undersampling 100 3.4.1 Minimum Sampling Rate 102 3.4.2 Antialiasing Bandpass Filter 103

3.5 Stochastic Sampling Theorem 105 3.6 Summary 106 References 108

4 Channel Capacity 109 4.1 Introduction 109 4.2 Gaussian Channel Capacity 110 4.3 Bandlimited Channel Capacity 112 4.4 MIMO Channel Capacity 117 4.5 SIMO Channel Capacity 121 4.6 MISO Channel Capacity 121 4.7 Summary 122 References 123

5 Smart Antenna Systems 125 5.1 Introduction 125 5.2 Smart Antennas and Beamforming Structures 126

5.2.1 Switched Beamforming 126 5.2.2 Delay-and-Sum Beamforming 128 5.2.3 Space-Time Beamforming 129 5.2.4 Interpolation Beamforming 130

5.3 Beamforming Algorithms 133 5.3.1 MMSE Beamformer 133 5.3.2 Maximum SNR of the B eamformer 13 5 5.3.3 Minimum Variance Beamformer 137

5.4 Summary 139 References 140

X Signal Processing in Digital Communications

141 141 143 143 146 148 150 156 157 159 163 163 168 169 171 172 174 178 180 181 182 183 184 187

Channel Estimation and Blind Identification 189 7.1 Introduction 189 7.2 Discrete-Time Channel Models 191 7.3 Channel Estimators 194

7.3.1 Maximum Likelihood Estimator 194 7.3.2 Least Squares Estimator 198 7.3.3 Generalized Least Squares Estimator 201 7.3.4 MMSE Estimator 202

7.4 Adaptive Channel Estimation and Algorithms 208 7.4.1 The LMS Algorithms 210 7.4.2 The LMS Algorithm Convergence 214 7.4.3 The LMS EMSE Analysis and Misadjustment 217 7.4.4 The RLS Algorithms 219 7.4.5 The RLS Algorithm Convergence 223

Channel Characterization and Distortion 6.1 6.2

6.3

6.4

6.5

6.6 6.7

Introduction Wireless Channels 6.2.1 Free Space Propagation 6.2.2 Fiat Surface Propagation 6.2.3 Multipath Propagation 6.2.4 Parameters of Multipath Channels 6.2.5 Fading Characteristics 6.2.6 Large-Scale Fading 6.2.7 Small-Scale Fading Wired Channels 6.3.1 Transmission Loop 6.3.2 Crosstalk 6.3.3 Simulation Loop Model Channel Distortion 6.4.1 Intersymbol Interference 6.4.2 Eye Diagrams 6.4.3 Ny quist Criterion Pulse Shaping 6.5.1 Raised-Cosine Pulse 6.5.2 Gaussian Shaping Pulse Matched Filtering Summary

References

Contents xi

7.4.6 The RLS EMSE Analysis and Misadjustment 224 7.4.7 Comparison of the Adaptive Algorithms 225

7.5 Channel Models and HOS Estimations 226 7.5.1 SISO Channel Model and Estimation 226 7.5.2 SIMO Channel Model and Estimation 229 7.5.3 MIMO Channel Model and Estimation 233

7.6 Blind Channel Identification 238 7.6.1 Blind Identification for SISO Channel 239 7.6.2 Subspace Blind Identification for SIMO

Channel 242 7.6.3 Blind Identification for MIMO Channel 246

7.7 Summary 252 References 253

Adaptive Equalizers in Communication Receivers 257 8.1 Introduction 257 8.2 Linear Equalizer 260

8.2.1 Channel Equalizer 262 8.2.2 Mean-Square-Error Criterion 266

8.3 Adaptive Linear Equalizer 272 8.3.1 Adaptive Algorithms for an Equalizer 274 8.3.2 Training Methodology 277 8.3.3 Tap Length of Equalizer Coefficients 278

8.4 Fractionally Spaced Equalizer 279 8.4.1 Multirate Communication System Model 280 8.4.2 Multichannel Model-Based Equalizer 283 8.4.3 FSE-MMSE Function 287 8.4.4 FSE Constant Modulus Algorithm 293 8.4.5 FSE-CM Noisy Cost Function 297 8.4.6 FSE-CM Performances 298

8.5 Decision Feedback Equalizer 299 8.5.1 MMSEforDFE 301 8.5.2 Predictive DFE 307 8.5.3 FSE-DFE 310 8.5.4 Error Propagation 312

8.6 Space-Time Equalizer 315 8.6.1 Time-Only Equalizer 316 8.6.2 Space-Only Equalizer 318 8.6.3 Space-Time MMSE Equalizer 320

xii Signal Processing in Digital Communications

8.7 Diversity Equalizer 323 8.7.1 Fundamentals of a Rake Receiver 324 8.7.2 Adaptive Rake Receiver 330 8.7.3 Equalized Rake Receiver 334

8.8 Summary 338 References 342

9 Multicarrier Modulation, DMT, and OFDM 345 9.1 Introduction 345 9.2 Fundamentals of Discrete Multitone Modulation 347

9.2.1 Multitone Transmission 347 9.2.2 Geometrie SNR 351 9.2.3 Optimum of Energy Minimum and

Bit Loading Maximum 353 9.3 FFT-Based OFDM 359

9.3.1 OFDM System 359 9.3.2 OFDM Modulation by IFFT 361 9.3.3 OFDM Demodulation by FFT 363 9.3.4 ADC Resolution for the OFDM Modulation 364 9.3.5 Equalized OFDM 369

9.4 Filter Bank-Based OFDM 373 9.4.1 Filter Bank Transmultiplexer 373 9.4.2 The DFT Filter Bank 375 9.4.3 Polyphase-Based DFT Filter Bank 377 9.4.4 Maximally Decimated DFT Transmitter

Filter Bank 378 9.4.5 Perfect Reconstruction of

the DFT Filter Bank 379 9.5 Summary 382 References 384

10 Discrete-Time Synchronization 387 10.1 Introduction 387 10.2 Discrete-Time Phase Locked Loop 389

10.2.1 Discrete-Time Loop Filter 391 10.2.2 Phase Detector 397 10.2.3 Discrete-Time VCO 400

10.3 Timing Recovery 402 10.3.1 Early-Late Gate Synchronizer 402

Contents xiii

10.3.2 Bandedge Timing Recovery 406 10.3.3 Decision-Directed Timing Recovery 407 10.3.4 Multirate Timing Recovery 410 10.3.5 Polyphase Filter Bank Timing Recovery 414 10.3.6 Multicarrier Modulation Timing Recovery 418

10.4 Carrier Recovery 424 10.4.1 Carrier Phase Error 425 10.4.2 Open-Loop Carrier Recovery 428 10.4.3 Carrier Recovery for Multiple Phase Signals 430 10.4.4 Decision-Directed Carrier Recovery 432

10.5 Summary 435 References 435

Appendix A: The z-Transform 437 A. 1 Introduction 437 A.2 The z-Transform 437

A.2.1 The z-Transform Properties 442 A.2.2 Common Pairs of the z-Transform 445 A.2.3 The z-Transform Transfer Function 447

A.3 The Inverse z-Transform 448 A.3.1 The Contour Integration 449 A.3.2 The Examination Approach 449 A.3.3 The Partial Fraction Expansion 450

A.4 The z-Transform All-Pass and Phase Systems 452 A.4.1 All-Pass Systems 452 A.4.2 Phase Systems 453 A.4.3 Decomposition of Phase Systems 454 A.4.4 Compensation Systems 454 A.4.5 FIR Systems to Minimum-Phase Systems 455

References 456

Appendix B: Matrix Theory 457 B.l Introduction 457 B.2 Vector Definitions 457 B.3 Matrix Definitions 459 B.4 Orthogonal Matrices 461 B.5 Trace 461 B.6 Matrix Differentiation 462 B.7 Determinants 462

xiv Signal Processing in Digital Communications

B.8 Matrix Inversion 464 B.9 Eigenanalysis 464 B.10 Spectral Decomposition Theorem 465 B. 11 Singular Value Decomposition 468 B.12 Quadratic Forms 470 B.13 Maximization and Minimization Analysis 471 References 473

Appendix C: The Discrete Fourier Transform 475 C.l Introduction 475 C.2 DFT Operation 475 C.3 IDFT Operation 476 C.4 DFT Matrix Operation 477 References 478

Appendix D: The Fast Fourier Transform 479 D.l Introduction 479 D.2 FFTMethods 479

D.2.1 Decimation-in-Time FFT Algorithm 480 D.2.2 Decimation-in-Frequency FFT Algorithm 482 D.2.3 Computational Complexity 484

D.3 Fixed-Point FFT Algorithm 486 D.3.1 Quantization 486 D.3.2 Fixed-Point Overflow, Scaling, and SNR 487 D.3.3 Quantization Analysis of the FFT Algorithm 489

References 493

Appendix E: Discrete Mathematical Formulas 495 E. 1 Complex Exponential Formulas 495 E.2 Discrete Closed-Form Formulas 495 E.3 Approximation Formulas 496 E.4 Logarithmic Formulas 497

About the Author 499

Index 501