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Preface 3 Acknowledgments 3 1 Elements of signal theory 7 1.1 Continuous-time linear systems 7 1.2 Discrete-time linear systems 10 Discrete Fourier transform 13 The DFT operator 14 Circular and linear convolution via DFT 15 Convolution by the overlap-save method 17 IIR and FIR filters 19 1.3 Signal bandwidth 22 The sampling theorem 24 Heaviside conditions for the absence of signal distortion 26 1.4 Passband signals and systems 26 Complex representation 26 Relation between a signal and its complex representation 28 Baseband equivalent of a transformation 36 Envelope and instantaneous phase and frequency 37 1.5 Second-order analysis of random processes 38 1.5.1 Correlation 39 Properties of the autocorrelation function 40 1.5.2 Power spectral density 40 Spectral lines in the PSD 40 Cross power spectral density 42 Properties of the PSD 42 PSD through filtering 43 1.5.3 PSD of discrete-time random processes 43 Spectral lines in the PSD 44 PSD through filtering 45 Minimum-phase spectral factorization 46 1.5.4 PSD of passband processes 47 PSD of in-phase and quadrature components 47 Cyclostationary processes 50 1.6 The autocorrelation matrix 56 Properties 56 Eigenvalues 56 Other properties 57 Eigenvalue analysis for Hermitian matrices 58 1.7 Examples of random processes 60 1.8 Matched filter 66 White noise case 68 1.9 Ergodic random processes 69 1.9.1 Mean value estimators 71 Rectangular window 74 Exponential filter 74 General window 75 1.9.2 Correlation estimators 75 Unbiased estimate 76 Biased estimate 76 1.9.3 Power spectral density estimators 77 Periodogram or instantaneous spectrum 77 Welch periodogram 78 Blackman and Tukey correlogram 79 Windowing and window closing 79 1.10 Parametric models of random processes 82 ARMA 82 MA 84 AR 84 Spectral factorization of AR models 87 Whitening filter 87 Relation between ARMA, MA, and AR models 87 1.10.1 Autocorrelation of AR processes 89 1.10.2 Spectral estimation of an AR process 91 Some useful relations 92 AR model of sinusoidal processes 94 1.11 Guide to the bibliography 95 Bibliography 95 Appendixes 97 1.A Multirate systems 98 1.A.1 Fundamentals 98 1.A.2 Decimation 100 1.A.3 Interpolation 102 1.A.4 Decimator filter 104 1.A.5 Interpolator filter 105 1.A.6 Rate conversion 108 1.A.7 Time interpolation 109 Linear interpolation 110 Quadratic interpolation 112 1.A.8 The noble identities 112 1.A.9 The polyphase representation 113 Efficient implementations 114 1.B Generation of a complex Gaussian noise 121 1.C Pseudo-noise sequences 122 Maximal-length 122 CAZAC 124 Gold 125 2 The Wiener filter 129 2.1 The Wiener filter 129 Matrix formulation 130 Optimum filter design 132 The principle of orthogonality 134 Expression of the minimum mean-square error 135 Characterization of the cost function surface 136 The Wiener filter in the z-domain 137 2.2 Linear prediction 140 Forward linear predictor 141 Optimum predictor coefficients 141 Forward prediction error filter 142 Relation between linear prediction and AR models 143 First and second order solutions 144 2.3 The least squares method 145 Data windowing 146 Matrix formulation 146 Correlation matrix 147 Determination of the optimum filter coefficients 147 2.3.1 The principle of orthogonality 148 Minimum cost function 149 The normal equation using the data matrix 149 Geometric interpretation: the projection operator 150 2.3.2 Solutions to the LS problem 151 Singular value decomposition 152 Minimum norm solution 154 2.4 The estimation problem 155 Estimation of a random variable 155 MMSE estimation 155 Extension to multiple observations 157 Linear MMSE estimation of a random variable 158 Linear MMSE estimation of a random vector 158 2.4.1 The Cramer-Rao lower bound 160 Extension to vector parameter 162 2.5 Examples of application 164 2.5.1 Identification of a linear discrete-time system 164 2.5.2 Identification of a continuous-time system 166 2.5.3 Cancellation of an interfering signal 169 2.5.4 Cancellation of a sinusoidal interferer with known frequency 170 2.5.5 Echo cancellation in digital subscriber loops 171 2.5.6 Cancellation of a periodic interferer 172 Bibliography 173 Appendixes 174 2.A The Levinson-Durbin algorithm 175 Lattice filters 176 The Delsarte-Genin algorithm 177 3 Adaptive transversal filters 179 3.1 The MSE design criterion 180 3.1.1 The steepest descent or gradient algorithm 181 Stability 181 Conditions for convergence 183 Adaptation gain 184 Transient behaviour of the MSE 185 3.1.2 The least mean square algorithm 186 Implementation 187 Computational complexity 188 Conditions for convergence 188 3.1.3 Convergence analysis of the LMS algorithm 190 Convergence of the mean 191 Convergence in the mean-square sense: real scalar case 192 Convergence in the mean-square sense: general case 193 Fundamental results 196 Observations 197 Final remarks 199 3.1.4 Other versions of the LMS algorithm 199 Leaky LMS 199 Sign algorithm 200 Normalized LMS 200 Variable adaptation gain 201 3.1.5 Example of application: the predictor 202 3.2 The recursive least squares algorithm 208 Normal equation 209 Derivation 210 Initialization 212 Recursive form of the minimum cost function 212 Convergence 214 Computational complexity 214 Example of application: the predictor 215 3.3 Fast recursive algorithms 215 3.3.1 Comparison of the various algorithms 216 3.4 Examples of application 216 3.4.1 Identification of a linear discrete-time system 217 Finite alphabet case 219 3.4.2 Cancellation of a sinusoidal interferer with known frequency 220 Bibliography 221 4 Transmission channels 223 4.1 Radio channel 223 4.1.1 Propagation and used frequencies in radio transmission 224 Basic propagation mechanisms 224 Frequency ranges 224 4.1.2 Analog front-end architectures 226 Radiation masks 226 Conventional superheterodyne receiver 227 Alternative architectures 227 Direct conversion receiver 228 Single conversion to low-IF 229 Double conversion and wideband IF 229 4.1.3 General channel model 230 High power amplifier 230 Transmission medium 233 Additive noise 234 Phase noise 234 4.1.4 Narrowband radio channel model 235 Equivalent circuit at the receiver 237 Multipath 238 Path loss as a function of distance 240 4.1.5 Fading effects in propagation models 243 Macroscopic fading or shadowing 243 Microscopic fading 245 4.1.6 Doppler shift 245 4.1.7 Wideband channel model 247 Multipath channel parameters 249 Statistical description of fading channels 250 4.1.8 Channel statistics 252 Power delay profile 252 Coherence bandwidth 253 Doppler spectrum 254 Coherence time 255 Doppler spectrum models 256 Power angular spectrum 256 Coherence distance 256 On fading 257 4.1.9 Discrete-time model for fading channels 258 Generation of a process with a preassigned spectrum 259 4.1.10 Discrete-space model of shadowing 261 4.1.11 Multiantenna systems 264 Discrete-time model 266 4.2 Telephone channel 268 Distortion 270 Noise sources 270 Echo 270 Appendixes 272 4.A Discrete-time NB model for mmWave channels 273 Angular domain representation 273 Bibliography 274 5 Vector quantization 277 5.1 Basic concept 277 5.2 Characterization of VQ 278 Parameters determining VQ performance 278 Comparison between VQ and scalar quantization 280 5.3 Optimum quantization 281 Generalized Lloyd algorithm 282 5.4 The Linde, Buzo, and Gray algorithm 284 Choice of the initial codebook 285 Splitting procedure 286 Selection of the training sequence 287 5.4.1 k-means clustering 288 5.5 Variants of VQ 288 Tree search VQ 288 Multistage VQ 289 Product code VQ 291 5.6 VQ of channel state information 292 MISO channel quantization 292 Channel feedback with feedforward information 294 5.7 Principal component analysis 295 5.7.1 PCA and k-means clustering 297 Bibliography 299 6 Digital transmission model and channel capacity 301 6.1 Digital transmission model 301 6.2 Detection 305 6.2.1 Optimum detection 306 ML 307 MAP 307 6.2.2 Soft detection 309 LLRs associated to bits of BMAP 309 Simplified expressions 312 6.2.3 Receiver strategies 314 6.3 Relevant parameters of the digital transmission model 314 Relations among parameters 315 6.4 Error probability 317 6.5 Capacity 320 6.5.1 Discrete-time AWGN channel 321 6.5.2 SISO narrowband AWGN channel 322 6.5.3 SISO dispersive AGN channel 322 6.5.4 MIMO discrete-time NB AWGN channel 325 6.6 Achievable rates of modulations in AWGN channels 326 6.6.1 Rate as a function of the SNR per dimension 327 6.6.2 Coding strategies depending on the signal-to-noise ratio 329 Coding gain 330 6.6.3 Achievable rate of an AWGN channel using PAM 331 Bibliography 333 Appendixes 334 6.A Gray labelling 335 6.B The Gaussian distribution and Marcum functions 336 6.B.1 The Q function 336 6.B.2 Marcum function 338 7 Single-carrier modulation 341 7.1 Signals and systems 341 7.1.1 Baseband digital transmission (PAM) 341 Modulator 342 Transmission channel 343 Receiver 343 Power spectral density 344 7.1.2 Passband digital transmission (QAM) 346 Modulator 346 Power spectral density 347 Three equivalent representations of the modulator 348 Coherent receiver 349 7.1.3 Baseband equivalent model of a QAM system 349 Signal analysis 349 7.1.4 Characterization of system elements 353 Transmitter 353 Transmission channel 354 Receiver 355 7.2 Intersymbol interference 356 Discrete-time equivalent system 356 Nyquist pulses 357 Eye diagram 361 7.3 Performance analysis 365 Signal-to-noise ratio 365 Symbol error probability in the absence of ISI 366 Matched filter receiver 367 7.4 Channel equalization 367 7.4.1 Zero-forcing equalizer 367 7.4.2 Linear equalizer 368 Optimum receiver in the presence of noise and ISI 369 Alternative derivation of the IIR equalizer 370 Signal-to-noise ratio at detector 374 7.4.3 LE with a finite number of coefficients 375 Adaptive LE 376 Fractionally spaced equalizer 378 7.4.4 Decision feedback equalizer 381 Design of a DFE with a finite number of coefficients 384 Design of a fractionally spaced DFE 387 Signal-to-noise ratio at the decision point 389 Remarks 390 7.4.5 Frequency domain equalization 390 DFE with data frame using a unique word 390 7.4.6 LE-ZF 394 7.4.7 DFE-ZF with IIR filters 394 DFE-ZF as noise predictor 400 DFE as ISI and noise predictor 400 7.4.8 Benchmark performance of LE-ZF and DFE-ZF 402 Comparison 402 Performance for two channel models 403 7.4.9 Passband equalizers 404 Passband receiver structure 405 Optimization of equalizer coefficients and carrier phase offset 407 Adaptive method 408 7.5 Optimum methods for data detection 410 7.5.1 Maximum-likelihood sequence detection 412 Lower bound to error probability using MLSD 413 The Viterbi algorithm 414 Computational complexity of the VA 419 7.5.2 Maximum a posteriori probability detector 419 Statistical description of a sequential machine 420 The forward-backward algorithm 421 Scaling 425 The log likelihood function and the Max-Log-MAP criterion 426 LLRs associated to bits of BMAP 427 Relation between Max-Log-MAP and Log-MAP 428 7.5.3 Optimum receivers 428 7.5.4 The Ungerboeck’s formulation of MLSD 430 7.5.5 Error probability achieved by MLSD 433 Computation of the minimum distance 437 7.5.6 The reduced-state sequence detection 441 Trellis diagram 442 The RSSE algorithm 444 Further simplification: DFSE 446 7.6 Numerical results obtained by simulations 447 QPSK over a minimum-phase channel 447 QPSK over a non minimum phase channel 448 8-PSK over a minimum phase channel 449 8-PSK over a non minimum phase channel 449 7.7 Precoding for dispersive channels 451 7.7.1 Tomlinson-Harashima precoding 452 7.7.2 Flexible precoding 454 7.8 Channel estimation 456 7.8.1 The correlation method 456 7.8.2 The LS method 458 Formulation using the data matrix 459 7.8.3 Signal-to-estimation error ratio 460 7.8.4 Channel estimation for multirate systems 464 7.8.5 The LMMSE method 465 7.9 Faster-than-Nyquist Signalling 467 Bibliography 467 Appendixes 470 7.A Simulation of a QAM system 471 7.B Description of a finite-state machine 477 7.C Line codes for PAM systems 478 7.C.1 Line codes 478 Non-return-to-zero format 478 Return-to-zero format 479 Biphase format 480 Delay modulation or Miller code 481 Block line codes 481 Alternate mark inversion 481 7.C.2 Partial response systems 482 The choice of the PR polynomial 485 Symbol detection and error probability 489 Precoding 491 Error probability with precoding 492 Alternative interpretation of PR systems 493 7.D Implementation of a QAM transmitter 497 8 Multicarrier modulation 499 8.1 MC systems 499 8.2 Orthogonality conditions 500 Time domain 501 Frequency domain 501 z-transform domain 501 8.3 Efficient implementation of MC systems 502 MC implementation employing matched filters 502 Orthogonality conditions in terms of the polyphase components 505 MC implementation employing a prototype filter 505 8.4 Non-critically sampled filter banks 510 8.5 Examples of MC systems 515 OFDM or DMT 515 Filtered multitone 516 8.6 Analog signal processing requirements in MC systems 517 8.6.1 Analog filter requirements 517 Interpolator filter and virtual subchannels 517 Modulator filter 519 8.6.2 Power amplifier requirements 520 8.7 Equalization 521 8.7.1 OFDM equalization 521 8.7.2 FMT equalization 524 Per-subchannel fractionally-spaced equalization 524 Per-subchannel T -spaced equalization 524 Alternative per-subchannel T -spaced equalization 525 8.8 Orthogonal time frequency space modulation 526 OTFS equalization 527 8.9 Channel estimation in OFDM 527 Instantaneous estimate or LS method 528 LMMSE 530 The LS estimate with truncated impulse response 531 8.9.1 Channel estimate and pilot symbols 532 8.10 Multiuser access schemes 532 8.10.1 OFDMA 533 8.10.2 SC-FDMA or DFT-spread OFDM 534 8.11 Comparison between MC and SC systems 535 8.12 Other MC waveforms 536 Bibliography 537 9 Transmission over multiple input multiple output channels 539 9.1 The MIMO NB channel 539 Spatial multiplexing and spatial diversity 544 Interference in MIMO channels 544 9.2 CSI only at the receiver 545 9.2.1 SIMO combiner 545 Equalization and diversity 548 9.2.2 MIMO combiner 548 Zero-forcing 549 MMSE 550 9.2.3 MIMO nonlinear detection and decoding 550 V-BLAST system 550 Spatial modulation 552 9.2.4 Space-time coding 553 The Alamouti code 553 The Golden code 555 9.2.5 MIMO channel estimation 556 The least squares method 556 The LMMSE method 557 9.3 CSI only at the transmitter 558 9.3.1 MISO linear precoding 558 MISO antenna selection 559 9.3.2 MIMO linear precoding 560 ZF precoding 561 9.3.3 MIMO nonlinear precoding 562 Dirty paper coding 562 TH precoding 564 9.3.4 Channel estimation for CSIT 564 9.4 CSI at both the transmitter and the receiver 565 9.5 Hybrid beamforming 566 Hybrid beamforming and angular domain representation 567 9.6 Multiuser MIMO: broadcast channel 568 9.6.1 CSI at both the transmitter and the receivers 569 Block diagonalization 570 User selection 571 Joint spatial division and multiplexing 572 9.6.2 Broadcast channel estimation 573 9.7 Multiuser MIMO: multiple-access channel 573 9.7.1 CSI at both the transmitters and the receiver 574 Block diagonalization 575 9.7.2 Multiple-access channel estimation 575 9.8 Massive MIMO 575 9.8.1 Channel hardening 576 9.8.2 Multiuser channel orthogonality 576 Bibliography 576 10 Spread-spectrum systems 581 10.1 Spread-spectrum techniques 581 10.1.1 Direct sequence systems 581 Classification of CDMA systems 589 Synchronization 590 10.1.2 Frequency hopping systems 590 Classification of FH systems 592 10.2 Applications of spread-spectrum systems 593 10.2.1 Anti-jamming 594 10.2.2 Multiple access 596 10.2.3 Interference rejection 597 10.3 Chip matched filter and rake receiver 597 Number of resolvable rays in a multipath channel 597 Chip matched filter 598 10.4 Interference 601 Detection strategies for multiple-access systems 603 10.5 Single-user detection 603 Chip equalizer 603 Symbol equalizer 605 10.6 Multiuser detection 606 10.6.1 Block equalizer 606 10.6.2 Interference cancellation detector 608 Successive interference cancellation 608 Parallel interference cancellation 610 10.6.3 ML multiuser detector 610 Correlation matrix 611 Whitening filter 611 10.7 Multicarrier CDMA systems 612 Bibliography 613 Appendixes 615 10.A Walsh codes 616 11 Channel codes 619 11.1 System model 620 11.2 Block codes 622 11.2.1 Theory of binary codes with group structure 622 Properties 622 Parity check matrix 625 Code generator matrix 628 Decoding of binary parity check codes 628 Cosets 629 Two conceptually simple decoding methods 630 Syndrome decoding 631 11.2.2 Fundamentals of algebra 633 modulo-q arithmetic 634 Polynomials with coefficients from a field 637 Modular arithmetic for polynomials 638 Devices to sum and multiply elements in a finite field 640 Remarks on finite fields 642 Roots of a polynomial 646 Minimum function 648 Methods to determine the minimum function 650 Properties of the minimum function 652 11.2.3 Cyclic codes 653 The algebra of cyclic codes 653 Properties of cyclic codes 654 Encoding by a shift register of length r 658 Encoding by a shift register of length k 661 Hard decoding of cyclic codes 662 Hamming codes 663 Burst error detection 666 11.2.4 Simplex cyclic codes 666 Relation to PN sequences 668 11.2.5 BCH codes 669 An alternative method to specify the code polynomials 669 Bose-Chaudhuri-Hocquenhemcodes 671 Binary BCH codes 674 Reed-Solomon codes 675 Decoding of BCH codes 676 Efficient decoding of BCH codes 681 11.2.6 Performance of block codes 689 11.3 Convolutional codes 690 11.3.1 General description of convolutional codes 693 Parity check matrix 695 Generator matrix 696 Transfer function 696 Catastrophic error propagation 700 11.3.2 Decoding of convolutional codes 702 Interleaving 702 Two decoding models 703 Decoding by the Viterbi algorithm 704 Decoding by the forward-backward algorithm 705 Sequential decoding 706 11.3.3 Performance of convolutional codes 710 11.4 Puncturing 711 11.5 Concatenated codes 711 The soft-output Viterbi algorithm 711 11.6 Turbo codes 713 Encoding 713 The basic principle of iterative decoding 718 FBA revisited 719 Iterative decoding 728 Performance evaluation 730 11.7 Iterative detection and decoding 730 11.8 Low-density parity check codes 734 11.8.1 Representation of LDPC codes 735 Matrix representation 735 Graphical representation 736 11.8.2 Encoding 737 Encoding procedure 737 11.8.3 Decoding 738 Hard decision decoder 738 The sum-product algorithm decoder 741 The LR-SPA decoder 744 The LLR-SPA or log-domain SPA decoder 745 The min-sum decoder 747 Other decoding algorithms 748 11.8.4 Example of application 748 Performance and coding gain 748 11.8.5 Comparison with turbo codes 749 11.9 Polar codes 751 11.9.1 Encoding 752 Internal CRC 753 LLRs associated to code bits 754 11.9.2 Tanner graph 755 11.9.3 Decoding algorithms 757 Successive cancellation decoding - the principle 758 Successive cancellation decoding - the algorithm 760

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