Algorithms for Communications Systems and their Applications
Description
The definitive guide to problem-solving in the design of communications systems
In Algorithms for Communications Systems and their Applications, 2nd Edition, authors Benvenuto, Cherubini, and Tomasin have delivered the ultimate and practical guide to applying algorithms in communications systems. Written for researchers and professionals in the areas of digital communications, signal processing, and computer engineering, Algorithms for Communications Systems presents algorithmic and computational procedures within communications systems that overcome a wide range of problems facing system designers.
New material in this fully updated edition includes:
- MIMO systems (Space-time block coding/Spatial multiplexing /Beamforming and interference management/Channel Estimation)
- OFDM and SC-FDMA (Synchronization/Resource allocation (bit and power loading)/Filtered OFDM)
- Improved radio channel model (Doppler and shadowing/mmWave)
- Polar codes (including practical decoding methods)
- 5G systems (New Radio architecture/initial access for mmWave/physical channels)
The book retains the essential coding and signal processing theoretical and operative elements expected from a classic text, further adopting the new radio of 5G systems as a case study to create the definitive guide to modern communications systems.
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Persons
Nevio Benvenuto, Professor, DEI-Telecommunications Group, University of Padua, Italy. Nevio received his Ph.D. in engineering from the University of Massachusetts, Amherst, in 1983.
Giovanni Cherubini, IBM Research Zurich, Switzerland. Giovanni Cherubini received M.S. and Ph.D. degrees from the University of California, San Diego, in 1984 and 1986, respectively, all in Electrical Engineering.
Stefano Tomasin, Associate Professor, Department of Information Engineering, University of Padova, Italy. Stefano received the Ph.D. degree in Telecommunications Engineering from the University of Padova, Italy, in 2003.
Content
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 Cramér-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
Successive cancellation list decoding 763
Other decoding algorithms 765
11.9.4 Frozen set design 765
Genie-aided SC decoding 766
Design based on density evolution 767
Channel polarisation 770
11.9.5 Puncturing and shortening 770
Puncturing 771
Shortening 772
Frozen set design 774
11.9.6 Performance 774
11.10Milestones in channel coding 775
Bibliography 775
Appendixes 781
11.A Nonbinary parity check codes 782
Linear codes 783
Parity check matrix 784
Code generator matrix 785
Decoding of nonbinary parity check codes 786
Coset 786
Two conceptually simple decoding methods 787
Syndrome decoding 787
12 Trellis coded modulation 789
12.1 Linear TCM for one and two-dimensional signal sets 790
12.1.1 Fundamental elements 790
Basic TCM scheme 792
Example 792
12.1.2 Set partitioning 795
12.1.3 Lattices 797
12.1.4 Assignment of symbols to the transitions in the trellis 802
12.1.5 General structure of the encoder/bit-mapper 807
Computation of dfree 809
12.2 Multidimensional TCM 811
Encoding 812
Decoding 815
12.3 Rotationally invariant TCM schemes 817
Bibliography 817
13 Techniques to achieve capacity 819
13.1 Capacity achieving solutions for multicarrier systems 819
13.1.1 Achievable bit rate of OFDM 819
13.1.2 Waterfilling solution 820
Iterative solution 821
13.1.3 Achievable rate under practical constraints 821
Effective SNR and system margin in MC systems 822
Uniform power allocation and minimum rate per subchannel 823
13.1.4 The bit and power loading problem revisited 824
Transmission modes 824
Problem formulation 825
Some simplifying assumptions 826
On loading algorithms 826
The Hughes-Hartogs algorithm 827
The Krongold-Ramchandran Jones algorithm 827
The Chow-Cioffi Bingham algorithm 830
Comparison 832
13.2 Capacity achieving solutions for single carrier systems 833
Achieving capacity 837
Bibliography 838
14 Synchronization 839
14.1 The problem of synchronization for QAM systems 839
14.2 The phase-locked loop 841
14.2.1 PLL baseband model 843
Linear approximation 844
14.2.2 Analysis of the PLL in the presence of additive noise 846
Noise analysis using the linearity assumption 847
14.2.3 Analysis of a second order PLL 848
14.3 Costas loop 852
14.3.1 PAM signals 852
14.3.2 QAM signals 854
14.4 The optimum receiver 856
Timing recovery 858
Carrier phase recovery 862
14.5 Algorithms for timing and carrier phase recovery 863
14.5.1 ML criterion 863
Assumption of slow time varying channel 863
14.5.2 Taxonomy of algorithms using the ML criterion 863
Feedback estimators 865
Early-late estimators 866
14.5.3 Timing estimators 867
Non data aided 867
NDA synchronization via spectral estimation 869
Data aided and data directed 871
Data and phase directed with feedback: differentiator scheme 874
Data and phase directed with feedback: Mueller & Muller scheme 874
Non data aided with feedback 877
14.5.4 Phasor estimators 878
Data and timing directed 878
Non data aided forM-PSK signals 878
Data and timing directed with feedback 879
14.6 Algorithms for carrier frequency recovery 880
14.6.1 Frequency offset estimators 881
Non data aided 881
Non data aided and timing independent with feedback 882
Non data aided and timing directed with feedback 883
14.6.2 Estimators operating at the modulation rate 883
Data aided and data directed 884
Non data aided forM-PSK 885
14.7 Second-order digital PLL 885
14.8 Synchronization in spread-spectrum systems 885
14.8.1 The transmission system 885
Transmitter 885
Optimum receiver 886
14.8.2 Timing estimators with feedback 887
Non data aided: non coherent DLL 888
Non data aided modified code tracking loop 888
Data and phase directed: coherent DLL 891
14.9 Synchronization in OFDM 891
14.9.1 Frame synchronization 891
Effects of STO 891
Schmidl and Cox algorithm 893
14.9.2 Carrier frequency synchronization 894
Estimator performance 895
Other synchronization solutions 895
14.10Synchronization in SC-FDMA 896
Bibliography 899
15 Self-training equalization 901
15.1 Problem definition and fundamentals 901
Minimization of a special function 904
15.2 Three algorithms for PAM systems 908
The Sato algorithm 908
Benveniste-Goursat algorithm 909
Stop-and-go algorithm 909
Remar
Chapter 1
Elements of signal theory
In this chapter, we recall some concepts on signal theory and random processes. For an in-depth study, we recommend the companion book [1]. First, we introduce various forms of the Fourier transform. Next, we provide the complex representation of passband signals and their baseband equivalent. We will conclude with the study of random processes, with emphasis on the statistical estimation of first- and second-order ergodic processes, i.e. periodogram, correlogram, auto-regressive (AR), moving-average (MA), and auto-regressive moving average (ARMA) models.
1.1 Continuous-time linear systems
A time-invariant continuous-time continuous-amplitude linear system, also called analog filter, is represented in Figure 1.1, where and are the input and output signals, respectively, and denotes the filter impulse response.
Figure 1.1 Analog filter as a time-invariant linear system with continuous domain.
The output at a certain instant , where denotes the set of real numbers, is given by the convolution integral
(1.1)denoted in short
(1.2)We also introduce the Fourier transform of the signal , ,
(1.3)where . The inverse Fourier transform is given by
(1.4)In the frequency domain, (1.2) becomes
(1.5)where is the filter frequency response. The magnitude of the frequency response, , is usually called magnitude response or amplitude response.
General properties of the Fourier transform are given in Table 1.1,1 where we use two important functions
(1.6) (1.7)Moreover, we denote by the Dirac impulse or delta function,
(1.8)where the derivative is taken in the generalized sense.
Definition 1.1
We introduce two functions that will be extensively used:
(1.9) (1.10)The following relation holds
(1.11)as illustrated in Figure 1.2. ?
Further examples of signals and relative Fourier transforms are given in Table 1.2.
We reserve the notation to indicate the Laplace transform of , :
(1.12)with complex variable; is also called the transfer function of the filter. A class of functions often used in practice is characterized by the ratio of two polynomials in , each with a finite number of coefficients.
It is easy to observe that if the curve in the -plane belongs to the convergence region of the integral in (1.12), then is related to by
(1.13)1.2 Discrete-time linear systems
A discrete-time time-invariant linear system, with sampling period , is shown in Figure 1.3, where and are, respectively, the input and output signals at the time instant , , where denotes the set of integers. We denote by or the entire discrete-time signal, also called sequence. The impulse response of the system is denoted by , , or more simply by .
Table 1.1 Some general properties of the Fourier transform.
property signal Fourier transform linearity duality time inverse complex conjugate real part imaginary part time scaling , time shift frequency shift modulation differentiation integration convolution correlation product real signal , Hermitian, even, odd, even imaginary signal real and even signal , real and even real and odd signal , imaginary and odd Parseval theorem Poisson sum formulaFigure 1.2 Example of signal and Fourier transform pair.
Figure 1.3 Discrete-time linear system (filter).
The relation between the input sequence and the output sequence is given by the convolution operation:
(1.14)denoted as . In the discrete time, the delta function is simply the Kronecker impulse
(1.15)Here are some definitions holding for time-invariant linear systems.
The system is causal (anticausal) if , (if , ).
Table 1.2 Examples of Fourier transform signal pairs.
signal Fourier transform 1 1 (constant) , , , ,The transfer function of the filter is defined as the z-transform2 of the impulse response , given by
(1.16)Let the frequency response of the filter be defined as
(1.17)The inverse Fourier transform of the frequency response yields
(1.18)We note the property that, for , where is a complex constant, the output is given by . In Table 1.3, some further properties of the z-transform are summarized.
For discrete-time linear systems, in the frequency domain (1.14) becomes
(1.19)where all functions are periodic of period .
Table 1.3 Properties of the z-transform.
property sequence z transform linearity time shift complex conjugate time inverse z-domain...