使用Matlab实例的数字信号处理：第一卷——信号与数据处理

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List of Figures

Figure 1.1 A square signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Figure 1.2 The sampled square signal . . . . . . . . . . . . . . . . . . . . . . . . 5

Figure 1.3 Sinusoidal signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Figure 1.4 Sine (solid) and cosine (dashed) signals . . . . . . . . . . . . . . . 6

Figure 1.5 Square signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Figure 1.6 Sawtooth signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Figure 1.7 Different sawtooth signals . . . . . . . . . . . . . . . . . . . . . . . . 9

Figure 1.8 The sinusoidal audio signal . . . . . . . . . . . . . . . . . . . . . . . 10

Figure 1.9 Sum of three sinusoidal signals. . . . . . . . . . . . . . . . . . . . . 12

Figure 1.10 Multiplication of two sinusoidal signals . . . . . . . . . . . . . . . 13

Figure 1.11 Example of odd signal (sawtooth signal, 3 periods) . . . . . . . 14

Figure 1.12 Amplitude of the first ten harmonics of the sawtooth

signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Figure 1.13 Example of even signal (rectified sine signal, 3 periods) . . . 16

Figure 1.14 Amplitude of the first ten harmonics of the rectified

sine signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Figure 1.15 Example of signal with half-wave symmetry

(triangular signal, 3 periods) . . . . . . . . . . . . . . . . . . . . . . . 18

Figure 1.16 Amplitude of the first ten harmonics of the triangular

signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Figure 1.17 A pulse train signal (3 periods) . . . . . . . . . . . . . . . . . . . . . 20

Figure 1.18 Amplitude of the first 50 harmonics of the pulse train

signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Figure 1.19 The sinc(t) function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Figure 1.20 Effect of a low sampling frequency

(sampled sine signal) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Figure 1.21 Aliasing example (3 Hz sine signal) . . . . . . . . . . . . . . . . . 24

Figure 2.1 A probability density function . . . . . . . . . . . . . . . . . . . . . 30

Figure 2.2 A random signal with uniform PDF. . . . . . . . . . . . . . . . . . 31

Figure 2.3 Uniform PDF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Figure 2.4 Histogram of a random signal with uniform PDF . . . . . . . . 33

xvii

Figure 2.5 A random signal with normal PDF . . . . . . . . . . . . . . . . . . 34

Figure 2.6 Normal PDF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Figure 2.7 Histogram of a random signal with normal PDF . . . . . . . . . 35

Figure 2.8 A random signal with log-normal PDF . . . . . . . . . . . . . . . 37

Figure 2.9 Log-normal PDF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Figure 2.10 Histogram of a random signal with log-normal PDF . . . . . . 38

Figure 2.11 Mean, median and mode marked on a PDF . . . . . . . . . . . . 40

Figure 2.12 PSD of a random signal with log-normal PDF . . . . . . . . . . 44

Figure 2.13 The buried sinusoidal signal . . . . . . . . . . . . . . . . . . . . . . . 44

Figure 2.14 The sine+noise signal . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Figure 2.15 PSD of the sine+noise signal . . . . . . . . . . . . . . . . . . . . . . 46

Figure 2.16 The random signal with normal PDF to be heared. . . . . . . . 47

Figure 2.17 The gamma function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Figure 2.18 Gamma-type PDFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Figure 2.19 Example of chi-square PDF . . . . . . . . . . . . . . . . . . . . . . . 50

Figure 2.20 Example of beta PDF . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Figure 2.21 Example of Student’stPDF....................... 52

Figure 2.22 Example of Weibull PDF . . . . . . . . . . . . . . . . . . . . . . . . . 53

Figure 2.23 Example of Rayleigh PDF . . . . . . . . . . . . . . . . . . . . . . . . 54

Figure 2.24 Example of bivariate Gaussian PDF . . . . . . . . . . . . . . . . . 56

Figure 2.25 Example of bivariate Gaussian PDF . . . . . . . . . . . . . . . . . 56

Figure 2.26 Example of binomial PDF . . . . . . . . . . . . . . . . . . . . . . . . 58

Figure 2.27 Example of Poisson PDF . . . . . . . . . . . . . . . . . . . . . . . . . 59

Figure 2.28 Example of geometric PDF . . . . . . . . . . . . . . . . . . . . . . . 59

Figure 2.29 Normal probability plot . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Figure 2.30 Weibull probability plot . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Figure 2.31 The log-likelihood of the Gaussian distribution,

using 20, or 60, or 200, or 500 data values . . . . . . . . . . . . 64

Figure 2.32 Bimodal distribution and mixture of Gaussians . . . . . . . . . . 68

Figure 2.33 Bimodal distribution example . . . . . . . . . . . . . . . . . . . . . . 69

Figure 2.34 Parzen estimation of PDF. . . . . . . . . . . . . . . . . . . . . . . . . 70

Figure 2.35 Kernel-based estimation of PDF . . . . . . . . . . . . . . . . . . . . 71

Figure 2.36 Area covered by a curve . . . . . . . . . . . . . . . . . . . . . . . . . 73

Figure 2.37 Same as previous figure, but with random points . . . . . . . . 73

Figure 2.38 Example of integral of the product gðxÞpðxÞ........... 75

Figure 2.39 Example of un-importance sampling . . . . . . . . . . . . . . . . . 78

Figure 2.40 Example of f(x), and p(x) for importance sampling . . . . . . . 78

Figure 2.41 The inversion procedure. . . . . . . . . . . . . . . . . . . . . . . . . . 81

Figure 2.42 Example of desired distribution function and PDF. . . . . . . . 81

Figure 2.43 Histogram of random data generated using analytical

inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

Figure 2.44 Generation of random data with a certain PDF . . . . . . . . . . 83

Figure 2.45 Histogram of random data generated using numerical

inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

xviii List of Figures

Figure 2.46 Example of desired f(x) PDF and proposal g(x) PDF. . . . . . 85

Figure 2.47 Histogram of random data generated using the rejection

method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

Figure 2.48 Histogram of sum of signals. . . . . . . . . . . . . . . . . . . . . . . 89

Figure 2.49 The PDFs of two variables A and B . . . . . . . . . . . . . . . . . 92

Figure 2.50 The two products of interest . . . . . . . . . . . . . . . . . . . . . . . 93

Figure 2.51 The product of the two PDFs . . . . . . . . . . . . . . . . . . . . . . 93

Figure 2.52 A simple Bayes network . . . . . . . . . . . . . . . . . . . . . . . . . 94

Figure 2.53 Another example of Bayes network . . . . . . . . . . . . . . . . . . 95

Figure 2.54 Two parents in a Bayes network . . . . . . . . . . . . . . . . . . . . 95

Figure 2.55 An example of Markov chain FSM . . . . . . . . . . . . . . . . . . 97

Figure 2.56 Example of Markov Chain result. . . . . . . . . . . . . . . . . . . . 100

Figure 2.57 The case considered in the Metropolis example. . . . . . . . . . 102

Figure 2.58 Histogram of random data generated by the Metr opolis

algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Figure 2.59 An example of HMM (speech generator) . . . . . . . . . . . . . . 104

Figure 2.60 Results of HMM example . . . . . . . . . . . . . . . . . . . . . . . . 104

Figure 2.61 Another HMM example . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Figure 2.62 An experiment with the HMM example . . . . . . . . . . . . . . . 105

Figure 2.63 A generic HMM path . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Figure 2.64 Initial disttool screen . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Figure 2.65 Initial randtool screen . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

Figure 2.66 Example of box plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Figure 3.1 Example of electronic filter . . . . . . . . . . . . . . . . . . . . . . . 118

Figure 3.2 Solution of equation (3.1) . . . . . . . . . . . . . . . . . . . . . . . . 118

Figure 3.3 Another example of electronic filter. . . . . . . . . . . . . . . . . . 119

Figure 3.4 Visualization of poles and zeros of G(s)

on the complex plane . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Figure 3.5 Comparison of sinusoidal input and output. . . . . . . . . . . . . 122

Figure 3.6 Visualization of the frequency response of example A . . . . . 123

Figure 3.7 Alternative visualization of the frequency response

of example A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

Figure 3.8 Visualization of the frequency response of example B . . . . . 124

Figure 3.9 Alternative visualization of the frequency response

of example B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Figure 3.10 Step response of filter example A . . . . . . . . . . . . . . . . . . . 127

Figure 3.11 Step response of filter example B . . . . . . . . . . . . . . . . . . . 128

Figure 3.12 Three cases of zero-pole maps . . . . . . . . . . . . . . . . . . . . . 129

Figure 3.13 Response of example A to sinusoidal input . . . . . . . . . . . . 130

Figure 3.14 Response of example B to sinusoidal input . . . . . . . . . . . . 131

Figure 3.15 Response of example B to a square signal . . . . . . . . . . . . . 132

Figure 3.16 Response of example B to a high-frequency

square signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

List of Figures xix

Figure 3.17 Step response of a discrete system. . . . . . . . . . . . . . . . . . . 134

Figure 3.18 Example of zero-pole map for a discrete system . . . . . . . . . 135

Figure 3.19 PSD of u(t), and PSD of y(t) . . . . . . . . . . . . . . . . . . . . . . 136

Figure 3.20 Real and estimated frequency responses of G(s) . . . . . . . . . 137

Figure 3.21 State space model diagram . . . . . . . . . . . . . . . . . . . . . . . . 138

Figure 3.22 A two-tank system example . . . . . . . . . . . . . . . . . . . . . . . 139

Figure 3.23 System state evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

Figure 3.24 Autonomous behaviour of the system . . . . . . . . . . . . . . . . 142

Figure 3.25 Forced response of the system, unit step input . . . . . . . . . . 142

Figure 3.26 Propagation of initial state perturbations. . . . . . . . . . . . . . . 144

Figure 3.27 Evolution of mean and variance of the state

for the previous example . . . . . . . . . . . . . . . . . . . . . . . . . 145

Figure 3.28 Influence of process perturbations . . . . . . . . . . . . . . . . . . . 146

Figure 3.29 Evolution of mean and variance of the state

for the example with process noise . . . . . . . . . . . . . . . . . . 147

Figure 3.30 Histograms of x(2), x(3) and process noise. . . . . . . . . . . . . 148

Figure 3.31 A beta PDF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

Figure 3.32 Influence of process perturbations . . . . . . . . . . . . . . . . . . . 149

Figure 3.33 Histograms of process noise and x(10). . . . . . . . . . . . . . . . 150

Figure 3.34 Influence of the process and the measurement noises. . . . . . 151

Figure 3.35 Histograms of y(10) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

Figure 3.36 Influence of the process and the measurement noises. . . . . . 152

Figure 3.37 Histograms of y(10) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Figure 3.38 Behaviour of a DARMA model example . . . . . . . . . . . . . . 156

Figure 3.39 Semi-annual Sunspot activity . . . . . . . . . . . . . . . . . . . . . . 160

Figure 3.40 Periodogram of semi-annual Sunspot activity . . . . . . . . . . . 161

Figure 3.41 Behaviour of an MA process . . . . . . . . . . . . . . . . . . . . . . 163

Figure 3.42 Roots of the lag polynomial . . . . . . . . . . . . . . . . . . . . . . . 164

Figure 3.43 Covariances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

Figure 3.44 Behaviour of an AR process. . . . . . . . . . . . . . . . . . . . . . . 165

Figure 3.45 Roots of the lag polynomial . . . . . . . . . . . . . . . . . . . . . . . 166

Figure 3.46 Covariances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

Figure 3.47 Drift and trend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

Figure 3.48 Behaviour of an ARMA model . . . . . . . . . . . . . . . . . . . . . 171

Figure 3.49 Weekly toothpaste sales: (left) data, (right) increments. . . . . 173

Figure 3.50 An AR(1) process simulation . . . . . . . . . . . . . . . . . . . . . . 174

Figure 3.51 Quarterly US personal consumption expenditure:

(left) data, (right) increments . . . . . . . . . . . . . . . . . . . . . . 175

Figure 3.52 Gold prices (monthly): (left) data, (center) log(data),

(right) increments of log(data) . . . . . . . . . . . . . . . . . . . . . 177

Figure 3.53 Monthly beer production in Australia . . . . . . . . . . . . . . . . 178

Figure 3.54 Classical decomposition : (top) beer production,

(below) linear drift, (next below) cosine component,

(bottom) residual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

xx List of Figures

Figure 3.55 Cosine fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

Figure 4.1 A first order low-pass filter . . . . . . . . . . . . . . . . . . . . . . . 186

Figure 4.2 Frequency response of the first order low-pass filter . . . . . . 186

Figure 4.3 A first order high-pass filter . . . . . . . . . . . . . . . . . . . . . . . 187

Figure 4.4 Frequency response of the first order high-pass filter . . . . . . 187

Figure 4.5 Response of the first order low-pass filter

to a square signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

Figure 4.6 Response of the first order high–pass filter

to a square signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

Figure 4.7 A desired low-pass frequency response . . . . . . . . . . . . . . . 191

Figure 4.8 A desired high-pass frequency response . . . . . . . . . . . . . . . 191

Figure 4.9 A desired band-pass frequency response. . . . . . . . . . . . . . . 192

Figure 4.10 A desired band-stop frequency respon se. . . . . . . . . . . . . . . 192

Figure 4.11 A desired frequency response: approximation and reality . . . 193

Figure 4.12 A filter structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

Figure 4.13 Foster forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

Figure 4.14 Cauer forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

Figure 4.15 Radio tuning with an ideal filter . . . . . . . . . . . . . . . . . . . . 198

Figure 4.16 Specification of a low-pass filter . . . . . . . . . . . . . . . . . . . . 199

Figure 4.17 Frequency response of 5

order Butterworth filter. . . . . . . . 200

Figure 4.18 Poles of 5

order Butterworth filter. . . . . . . . . . . . . . . . . . 201

Figure 4.19 Step response of 5

order Butterworth filter . . . . . . . . . . . . 202

Figure 4.20 Effect of n on the frequency response of the Butterworth

filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

Figure 4.21 Response of 5

order Butterworth filter

to a square wave. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

Figure 4.22 Frequency response of 5

order Chebyshev 1 filter . . . . . . . 204

Figure 4.23 Poles of 5

order Chebyshev 1 filter . . . . . . . . . . . . . . . . . 205

Figure 4.24 Step response of 5

order Chebyshev 1 filter . . . . . . . . . . . 205

Figure 4.25 Effect of n on the frequency response of the Chebyshev

1 filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

Figure 4.26 Response of 5

order Chebyshev 1 filter

to a square wave. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

Figure 4.27 Frequency response of 5

order Chebyshev 2 filter . . . . . . . 208

Figure 4.28 Poles and zeros of 5

order Chebyshev 2 filter. . . . . . . . . . 209

Figure 4.29 Step response of 5

order Chebyshev 2 filter . . . . . . . . . . . 209

Figure 4.30 Effect of n on the frequency response

of the Chebyshev 2 filter . . . . . . . . . . . . . . . . . . . . . . . . . 210

Figure 4.31 Response of 5

order Chebyshev 2 filter

to a square wave. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

Figure 4.32 Frequency response of 5

order elliptic filter . . . . . . . . . . . 212

Figure 4.33 Poles of 5

order elliptic filter . . . . . . . . . . . . . . . . . . . . . 212

Figure 4.34 Step response of 5

order elliptiic filter . . . . . . . . . . . . . . . 213

List of Figures xxi

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