统计信号处理算法（英文影印版）

.1.5 the cepstrum 53
1.6 summary and references 56
problems 56

2 algorithms for convolution and dft
2.1 modulo polynomials 61
2.2 circular convolution as polynomial multiplication mod un-i 63
2.3 a continued fraction of polynomials 64
2.4 chinese remainder theorem for polynomials 66
2.5 algorithms for short circular convolutions 67
2.6 how we count multiplications 74
2.7 cyclotomic polynomials 76
2.8 elementary number theory 77
2.8.1 greatest common divisors and euler's totient function, 78
2.8.2 the equation ax + by= l. 78
2.8.3 modulo arithmetic, 81
2.8.4 the sino representation of integers modulo m, 83
2.8.5 exponentials modulo m, 85
2.9 convolution length and dimension 88
2.10 the dft as a circular convolution 92
2.11 winograd's dft algorithm 95
2.12 number-theoretic analogy of dft 98
2.13 number-theoretic transform 100
2.13.1 mersenne number transform, 104
2.13.2 fermat number transform , 106
2.13.3 considerations for use of ntts to perform circular convolution, 107
2.13.4 use of surrogate fields for complex arithmetic, 108
2. 14 split-radix fft 110
2. 15 autogen technique 116
2. 16 summary 122
problems 123

3 linear prediction and optimum linear filters
3.1 innovations representation of a stationary random process 125
3.1.1 rational power spectra, 128
3.1.2 relationships between the filter parameters and the autocorrelation sequence. 129
3.2 forward and backward linear prediction 131
3.2. 1 forward linear prediction, 131
3.2.2 backward linear prediction. 135
3.2.3 optimum reflection coefficients for the lattice forward and backward predictors, 139
3.2.4 relationship of an ar process to linear prediction, 139
3.3 solution of the normal equations 140
3.3.1 levinson-durbin algorithm. 140
3.3.2 the schur algorithm, 144
3.4 properties of the linear prediction-error filters 148
3.5 ar lattice and arma lattice-ladder filters 152
3.5.1 ar lattice structure. 152
3.5.2 arma processes and lattice-ladder filters, 154
3.6 wiener filters for filtering and prediction 157
3.6.1 fir wiener filter 157
3.6.2 orthogonality principle in linear mean-square estimation, 160
3.6.3 iir wiener filter, 161
3.6.4 noncausal wiener filter 165
3.7 summary and references 167
problems 168

4 least-souares methods for system modeling and fiter design
4.1 system modeling and identification 178
4.1.1 system identification based on fir (ma) system model, 178
4.1.2 system identification based on aii-pole (ar) system model, 181
4.1.3 system identification based on pole-zero (ar) system model, 183
4.2 least-squares filter design for prediction and deconvolution 189
4.2.1 least-squares linear prediction filter 190
4.2.2 fir least-squares inverse filters, 191
4.2.3 predictive deconvolution, 195
4.3 solution of least-squares estimation problems 197
4.3. 1 definition and basic concepts, 198
4.3. 2 matrix formulation of last-squares estimation. 199
4.3.3 cholesky decomposition. 203
4.3.4 ldu decomposition. 205
4.3.5 qr decomposition, 207
4.3.6 gram-schmidt orthogonalization, 209
4.3.7 givens rotation, 211
4.3.8 householder reflection, 214
4.3.9 singular-value decomposition. 217
4.4 summary and references 225
problems 226

5 adaptive filters
5.1 applications of adaptive filters 231
5.1.1 system identification or system modeling, 233
5.1.2 adaptive channel equalization, 235
5.1.3 echo cancellation in data transmission over telephone channels, 238
5.1.4 suppression of narrowband interference in a wideband signal, 242
5.1.5 adaptive line enhancer 246
5.1.6 adaptive noise cancelling, 247
5.1.7 linear predictive coding of speech signals, 248
5.1.8 adaptive arrays, 251
5.2 adaptive direct-form fir filters 253
5.2.1 minimum mean-square-error criterion, 254
5.2.2 the lms algorithm, 256
5.2.3 properties of the lms algorithm, 259
5.2.4 recursive least-squares algorithms for direct-form fir filters, 265
5.2.5 properties the direct-form ris algorithms, 273
5.3 adaptive lattice-ladder filters 276
5.3.1 recursive least-squares lattice-ladder algorithms, 276
5.3.2 gradient lattice-ladder algorithm, 300
5.3.3 properties of lattice-ladder, algorithms, 304
5.4 summary and references 309
problems 309

6 recusive least-souares algorithms for array signal processing
6.1 qr decomposition for least-squares estimation 315
6.2 gram-schmidt orthogonalization for least-squares estimation 318
6.2.1 least-squares estimation using the mgs algorithm, 319
6.2.2 physical meaning of the quantities in the mgs algorithm, 320
6.2.3 time-recursive form of the modified gram-schmidt algorithm, 321
6.2.4 variations of the rmigs algorithm, 328
6.2.5 implementation of the rmgs algorithm using vlsi arrays, and its relationship to the least-squares lattice algorithm, 332
6.3 givens algorithm for time-recursive least-squares estimation 337
6.3.1 time-recursive givens algorithm, 337
6.3.2 givens algorithm without square roots, 340
6.3.3 the cordic approach to givens transformations, 344
6.4 recursive least-squares estimation based on the householder transformation 358
6.4.1 block time-recursive least-squares estimation using the householder transformation, 358
6.5 order-recursive least-squares estimation a1gorithms 363
6.5.1 fundamental relations of orls estimation, 364
6.5.2 canonical structures for orls estimation algorithms, 370
6.5.3 variations in the basic processing cells of orls algorithms, 376
6.5.4 systematic investigation and derivation of orls algorithms, 381
6.6 summary and references 382
problems 384

7 qrd-based fast adaptive filter algorithms
7.l background 388
7.1.1 signal flow graphs, 388
7.1.2 qrd-based rls, revisited, 390
7.1.3 residual extraction, 392
7.2 qrd lattice 394
7.3 multichannel lattice 402
7.4 fast qr algorithm 4ll
7.5 multichannel fast qr algorithm 4l6
7.6 summary and references 427
problems 429

8 power spectrum estimation
8.l estimation of spectra from finite-duration observations of signals 433
8.1.1 computation of the energy density spectrum, 433
8.1.2 estimation of the autocorrelation and power spectrum of random signals: the periodogram, 438
8.1.3 use of the dft in power spectrum estimation, 443
8.2 nonparametric methods for power spectrum estimation 445
8.2.1 bartlett method: averaging periodograms, 446
8.2.2 welch method: averaging modified periodograms. 447
8.2.3 blackman and tukey method: smoothing the periodogram, 449
8.2.4 performance characteristics of nonparametric power spectrum estimators. 452
8.2.5 computational requirements of nonparametric power spectrum estimates, 456
8.3 parametric methods for power spectrum estimation 457
8.3.i relationships between the autocorrelation and the model parameters, 459
8.3.2 yule-walker method for the ar model parameters, 461
8.3.3 burg method for the ar model parameters. 462
8.3.4 unconstrained least-squares method for the ar model parameters, 465
8.3.5 sequential estimation methods for the ar model parameters, 467
8.3.6 selection of ar model order, 468
8.3.7 ma model for power spectrum estimation, 469
8.3.8 arma model for power spectrum estimation. 470
8.3.9 experimental results, 473
8.4 minimum-variance spectral estimation 481
8.5 eigenanalysis algorithms for spectrum estimation 483
8.5.1 pisarenko harmonic decomposition method, 484
8.5.2 eigendecomposition of the autocorrelation matrix for sinusoids in white noise. 486
8.5.3 music algorithm. 488
8.5.4 esprit algorithm. 489
8.5.5 order selection criteria, 492
8.5.6 experimental results, 492
8.6 summary and references 495
problems 496

9 signal analysis with higher-order spectra
9.1 use of higher-order spectra in signal processing 504
9.2 definition and properties of higher-order spectra 506
9.2.1 moments and cumulants of random signals. 506
9.2.2 higher-order spectra (cumulant spectra), 508
9.2.3 linear non-gaussian processes, 510
9.2.4 nonlinear processes. 512
9.3 conventional estimators for higher-order spectra 514
9.3.1 indirect method, 514
9.3.2 direct method, 516
9.3.3 statistical properties of conventional estimators, 517
9.3.4 test for aliasing with the bispectrum, 518
9.4 parametric methods for higher-order spectrum estimation 520
9.4.1 ma methods, 522
9.4.2 noncausal ar methods, 525
9.4.3 arma methods, 526
9.4.4 ar methods for the detection of quadratic phase coupling, 528
9.5 cepstra of higher-order spectra 531
9.5.1 preliminaries. 531
9.5.2 complex and differential cepstra, 532
9.5.3 bicepstrum, 533
9.5.4 cepstrum of the power spectrum. 535
9.5.5 cepstrum of the bicoherence, 536
9.5.6 summary of cepstra and key observations, 537
9.6 phase and magnitude retrieval from the bispectrum 537
9.7 summary and references 540
problems 541
references
index