数字信号处理在matlab中的应用(英文版)

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Contents
Abbreviations................................................................................................................................. xiii
Author ..............................................................................................................................................xv
Chapter 1 Continuous and Discrete Signals .................................................................................1
1.1 Continuous Deterministic Signals.....................................................................1
Periodic Signals.................................................................................................1
Non-Periodic Continuous Signals .....................................................................1
Unit Step Functions..............................................................................2
Ramp Function .....................................................................................3
Rectangular Function ...........................................................................3
Triangular Pulse Function ....................................................................3
Signum Function ..................................................................................3
Sinc Function........................................................................................3
Gaussian Function ................................................................................3
Error Function ......................................................................................3
Exponential and Double Exponential Functions..................................4
Type of Signals—Even, Odd, Energy and Power.................................4
1.2 Sampling of Continuous Signals-Discrete Signals............................................6
Table 1.1: Some Useful Functions in Analog and Discrete Forms....................7
Approximation of the Derivative and Integral...................................................8
Impulse (delta) Function....................................................................................9
Table 1.2: Basic Delta Function Properties..................................................... 10
The Comb Function......................................................................................... 11
1.3 Signal Conditioning and Manipulation ........................................................... 11
Modulation ......................................................................................... 11
Shifting and Flipping.......................................................................... 12
Time Scaling....................................................................................... 12
Windowing of Signals........................................................................ 12
Table 1.3: Windows for Continuous Signal Processing................................... 12
1.4 Convolution of Analog and Discrete Signals .................................................. 13
Analog Signals ................................................................................... 13
Discrete Signals.................................................................................. 13
Table 1.4: Basic Convolution Properties ......................................................... 16
1.5 MATLAB Use for Vectors and Arrays (Matrices).......................................... 17
Examples of Array Operations........................................................... 17
Hints–Suggestions–Solutions of the Exercises .......................................................... 18
Chapter 2 Fourier Analysis of Continuous and Discrete Signals ............................................... 21
2.1 Introduction ..................................................................................................... 21
2.2 Fourier Transform (FT) of Deterministic Signals........................................... 21
2.3 Sampling of Signals.........................................................................................24
2.4 Discrete-Time Fourier Transform (DTFT)......................................................27
2.5 DTFT of Finite-Time Sequences.....................................................................30
Windowing ......................................................................................... 32
2.6 The Discrete Fourier Transform (DFT) .......................................................... 33
The Inverse DFT (IDFT) .................................................................................34
viii Contents
2.7 Properties of DFT............................................................................................34
Linearity..............................................................................................34
Symmetry ...........................................................................................34
Time Shifting...................................................................................... 35
Frequency Shifting ............................................................................. 35
Time Convolution............................................................................... 35
Frequency Convolution ......................................................................37
Parseval’s Theorem.............................................................................37
2.8 Effect of Sampling Time T..............................................................................37
2.9 Effect of Truncation.........................................................................................39
Windowing .........................................................................................40
2.10 Resolution........................................................................................................40
2.11 Discrete Systems ............................................................................................. 41
2.12 Digital Simulation of Analog Systems............................................................46
2.12.1 Second-Order Differential Equations ................................................ 52
Hints–Suggestions–Solutions of the Exercises ..........................................................54
Appendix 2.1: Fourier Transform Properties ............................................................. 61
Appendix 2.2: Fourier Transform Pairs...................................................................... 62
Appendix 2.3: DTFT Properties.................................................................................63
Appendix 2.4: DFT Properties...................................................................................64
Chapter 3 The z-Transform, Difference Equations, and Discrete Systems .................................65
3.1 The z-Transform...............................................................................................65
3.2 Properties of the z-Transform.......................................................................... 67
Table 3.1: Summary of z-Transform Properties......................................................... 67
3.3 Inverse z-Transform ......................................................................................... 73
Table 3.2: Common z-Transform Pairs....................................................................... 74
3.4 Transfer Function.............................................................................................77
Higher-Order Transfer Functions...............................................................................79
3.5 Frequency Response of Discrete Systems.......................................................80
3.6 z-Transform Solution of Difference Equations................................................82
Hints–Suggestions–Solutions of the Exercises ..........................................................84
Chapter 4 Finite Impulse Response (FIR) Digital Filter Design ................................................89
4.1 Introduction .....................................................................................................89
4.2 Finite Impulse Response (FIR) Filters............................................................89
Discrete Fourier-Series Method .........................................................89
Commonly Used Windows.................................................................94
Discrete Fourier Transform Method...................................................95
High-Pass Filter..................................................................................96
Table 4.1: Frequency Transformations.......................................................................98
Hints–Suggestions–Solutions of the Exercises ........................................................100
Appendix 4.1: Window Characteristics and Performance........................................ 103
Chapter 5 Random Variables, Sequences, and Probability Functions ...................................... 105
5.1 Random Signals and Distributions................................................................ 105
Stochastic Processes......................................................................... 110
Stationary and Ergodic Processes..................................................... 111
5.2 Averages ........................................................................................................ 112
Contents ix
Mean Value....................................................................................... 112
Correlation........................................................................................ 113
Sample Autocorrelation Function..................................................... 113
Covariance........................................................................................ 115
Independent and Uncorrelated RVs.................................................. 116
5.3 Stationary Processes...................................................................................... 116
Table 5.1: Properties of WSS Processes........................................................ 117
Autocorrelation Matrix..................................................................... 117
Purely Random Process (WN) ......................................................... 118
Random Walk (RW) ......................................................................... 119
5.4 Probability Density Functions....................................................................... 119
Uniform Distribution........................................................................ 119
Table 5.2: Properties and Definitions ............................................................120
Gaussian (Normal) Distribution ....................................................... 121
Table 5.3: Properties of a Gaussian Random Process ................................... 121
Exponential Distribution ..................................................................124
Lognormal Distribution....................................................................126
Chi-Square Distribution ...................................................................126
Student’s Distribution....................................................................... 127
F Distribution ................................................................................... 128
Rayleigh Probability Density Function ............................................128
5.5 Transformations of PDFs............................................................................... 130
Hints, Suggestions, and Solutions for the Exercises................................................ 132
Chapter 6 Linear Systems with Random Inputs, Filtering, and Power Spectral Density ......... 137
6.1 Spectral Representation................................................................................. 137
The Wiener–Khintchine (W–K) Relations.................................................... 139
6.2 Linear Systems with Random Inputs ............................................................ 142
Table 6.1: Summary of Correlation and Spectral Densities .......................... 143
6.3 Autoregressive Moving Average Processes (ARMA) ................................... 149
6.4 Autoregressive (AR) Process......................................................................... 151 *6.5 Parametric Representations of Stochastic Processes: ARMA and
ARMAX Models........................................................................................... 154
Table 6.2: Linear Systems and Random Signals...................................................... 154
Table 6.3: ARMAX Representation ......................................................................... 159
Table 6.4: MA Representation.................................................................................. 160
Table 6.5: AR Representation................................................................................... 160
Hints–Suggestions–Solutions for the Exercises....................................................... 161
Chapter 7 Least Squares-Optimum Filtering............................................................................ 167
7.1 Introduction ................................................................................................... 167
7.2 The Least-Squares Approach ........................................................................ 167
7.3 Linear Least Squares..................................................................................... 170 *7.3.1 Matrix Formulation of Linear Least Squares (LLS)........................ 171
7.4 Point Estimation ............................................................................................ 172
7.4.1 Estimator Performance..................................................................... 173
7.4.2 Biased and Unbiased Estimators...................................................... 175
7.4.3 Cramer–Rao Lower Bound (CRLB) ................................................ 175
7.4.4 Mean Square Error Criterion ........................................................... 178
7.4.5 Maximum Likelihood Estimator...................................................... 178
x Contents
7.5 Mean Square Error (MSE) ............................................................................ 184
7.6 Finite Impulse Response (FIR) Wiener Filter............................................... 186
7.7 Wiener Solution—Orthogonal Principle....................................................... 190
7.7.1 Orthogonality Condition .................................................................. 193
7.8 Wiener Filtering Examples............................................................................ 193
7.8.1 Linear Prediction..............................................................................204
Hints, Suggestions, and Solutions of the Exercises..................................................205
Chapter 8 Nonparametric (Classical) Spectra Estimation ........................................................ 211
8.1 Periodogram and Correlogram Spectra Estimation ...................................... 211
8.1.1 Deterministic Signals (see also Chapter 2) ...................................... 211
8.1.2 The Periodogram-Random Signals.................................................. 212
8.1.3 Correlogram ..................................................................................... 214
8.1.4 Computation of Periodogram and Correlogram Using FFT............ 215
Windowed Periodogram ................................................................................ 221
8.2 Book Proposed Method for Better Resolution Using Transformation of
the Random Variables ...................................................................................222
8.3 Daniel Periodogram.......................................................................................223
8.4 Bartlett Periodogram.....................................................................................224
8.4.1 Book-Modified Method....................................................................226
8.5 Blackman–Tukey (BT) Method.....................................................................229
8.6 Welch Method................................................................................................ 233
8.6.1 Proposed Modified Methods for Welch Method.............................. 235
Modified Method Using Different Types of Overlapping ................ 235
Modified Welch Method Using RV Transformation ........................238
Hints, Suggestions, and Solutions of the Exercises.................................................. 239
Appendix A8.1: Important Windows and Their Spectra .......................................... 241
Chapter 9 Parametric and Other Methods for Spectral Estimation ..........................................245
9.1 Introduction ...................................................................................................245
9.2 AR, MA, and ARMA Models.......................................................................245
9.3 Yule–Walker (YW) Equations......................................................................247
9.4 Least-Squares (LS) Method and Linear Prediction ...................................... 251
9.5 Minimum Variance Method..........................................................................254
9.6 Model Order ..................................................................................................256
9.7 Levinson–Durbin Algorithm......................................................................... 257
9.8 Maximum Entropy Method...........................................................................262
9.9 Spectrums of Segmented Signals..................................................................263
9.9.1 Method 1: The Average Method.......................................................264
9.9.2 Method 2: Extrapolation Method .....................................................265
9.10 Eigenvalues and Eigenvectors of Matrices (See Also Appendix 2) ..............268
9.10.1 Eigendecomposition of the Autocorrelation Matrix.........................269
Table 9.1: Eigenvalue Properties................................................................... 270
9.10.2 Harmonic Model .............................................................................. 273
9.10.3 Pisarenko Harmonic Decomposition ...............................................277
9.10.4 MUSIC Algorithm............................................................................ 278
Hints, Suggestions, and Solutions of the Exercises.................................................. 279
Contents xi
Chapter 10 Newton’s and Steepest Descent Methods .................................................................285
10.1 Geometric Properties of the Error Surface ...................................................285
10.2 One-Dimensional Gradient Search Method..................................................288
10.2.1 Gradient Search Algorithm..............................................................289
10.2.2 Newton’s Method in Gradient Search ..............................................290
10.3 Steepest Descent Algorithm.......................................................................... 291
10.3.1 Steepest Descent Algorithm Applied to Wiener Filter ....................292
10.3.2 Stability (Convergence) of the Algorithm ........................................294
10.3.3 Transient Behavior of MSE..............................................................295
10.3.4 Learning Curve ................................................................................297
10.4 Newton’s Method...........................................................................................297 *10.5 Solution of the Vector Difference Equation ..................................................299
Additional Exercises.................................................................................................302
Hints, Suggestions, and Solutions of the Exercises..................................................302
Chapter 11 The Least Mean Square (LMS) Algorithm ..............................................................307
11.1 Introduction ...................................................................................................307
11.2 The LMS Algorithm......................................................................................307
Table 11.2.1: The LMS Algorithm for an Mth-Order FIR Filter..............................309
11.3 Example Using the LMS Algorithm ............................................................. 310 *11.4 Performance Analysis of the LMS Algorithm .............................................. 318
11.4.1 Learning Curve ................................................................................320
11.4.2 The Coefficient-Error or Weighted-Error Correlation Matrix ......... 322
11.4.3 Excess MSE and Misadjustment ......................................................324
11.4.4 Stability ............................................................................................ 326
11.4.5 The LMS and Steepest-Descent Method.......................................... 327 *11.5 Complex Representation of the LMS Algorithm .......................................... 327
Hints, Suggestions, and Solutions of the Exercises.................................................. 330
Chapter 12 Variants of Least Mean Square Algorithm .............................................................. 333
12.1 The Normalized Least Mean Square Algorithm........................................... 333
Table 12.1: Some Variants of the LMS Formulas..................................................... 334
Table 12.2: Normalized Real and Complex LMS Algorithms................................. 334
12.2 Power NLMS................................................................................................. 337
12.3 Self-Correcting LMS Filter........................................................................... 341
12.4 The Sign-Error LMS Algorithm.................................................................... 342
12.5 The NLMS Sign-Error Algorithm................................................................. 343
12.6 The Sign-Regressor LMS Algorithm ............................................................344
12.7 Self-Correcting Sign-Regressor LMS Algorithm .........................................345
12.8 The Normalized Sign-Regressor LMS Algorithm........................................346
12.9 The Sign–Sign LMS Algorithm.................................................................... 347
12.10 The Normalized Sign–Sign LMS Algorithm................................................349
12.11 Variable Step-Size LMS................................................................................ 350
Table 12.3: The VSLMS Algorithm ......................................................................... 351
12.12 The Leaky LMS Algorithm .......................................................................... 352
12.13 The Linearly Constrained LMS Algorithm................................................... 354
Table 12.4: Linearly Constrained LMS Algorithm .................................................. 357
xii Contents
12.14 The Least Mean Fourth Algorithm ............................................................... 358
12.15 The Least Mean Mixed Normal (LMMN) LMS Algorithm ........................ 358
12.16 Short-Length Signal of the LMS Algorithm ................................................. 359
12.17 The Transform Domain LMS Algorithm......................................................360 *12.17.1 Convergence ................................................................................. 363
12.18 The Error Normalized Step-Size LMS Algorithm........................................364
12.19 The Robust Variable Step-Size LMS Algorithm...........................................368
12.20 The Modified LMS Algorithm...................................................................... 372
12.21 Momentum LMS Algorithm ......................................................................... 373
12.22 The Block LMS Algorithm ........................................................................... 374
12.23 The Complex LMS Algorithm ...................................................................... 375
Table 12.5: Complex LMS Algorithm...................................................................... 375
12.24 The Affine LMS Algorithm .......................................................................... 377
Table 12.6: The Affine Projection Algorithm........................................................... 378
12.25 The Complex Affine LMS Algorithm........................................................... 379
Table 12.7: Complex Affine Algorithm.................................................................... 379
Hints, Solutions, and Suggestions of the Exercises..................................................380
Chapter 13 Nonlinear Filtering ................................................................................................... 385
13.1 Introduction ................................................................................................... 385
13.2 Statistical Preliminaries................................................................................. 385
13.2.1 Signal and Noise Model-Robustness............................................... 385
13.2.2 Point Estimation ..............................................................................386
13.2.3 Estimator Performance ....................................................................386
13.2.4 Biased and Unbiased Estimator.......................................................388
13.2.5 Cramer–Rao Lower Bound..............................................................388
13.2.6 Mean Square Error Criterion...........................................................390
13.2.7 Maximum Likelihood Estimator .....................................................390
13.3 Mean Filter....................................................................................................396
13.4 Median Filter................................................................................................. 398
13.5 Trimmed-Type Mean Filter ...........................................................................400
13.5.1 (r−s)-Fold Trimmed Mean Filters...................................................400
13.5.2 (r,s)-Fold Winsorized Mean Filter...................................................403
13.5.3 Alpha-Trimmed Mean Filter and Alpha-Winsorized Mean Filter...403
13.5.4 Alpha-Trimmed Winsorized Mean Filter.........................................404
13.6 L-Filters.........................................................................................................405
13.7 Rank-Order Statistic Filter ............................................................................406
13.8 Edge-Enhancement Filters ............................................................................408
13.9 R-Filters.........................................................................................................409
Additional Exercises................................................................................................. 411
Problems, Solutions, Suggestions, and Hints........................................................... 411
Appendix 1: Suggestions and Explanations for MATLAB Use............................................... 415
Appendix 2: Matrix Analysis ......................................................................................................427
Appendix 3: Mathematical Formulas ........................................................................................437
Appendix 4: MATLAB Functions ..............................................................................................443
Bibliography .................................................................................................................................447
Index .......................... 449