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  随机信号分析（英...
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本书讨论随机信号的基本概念和分析方法。其特色可归纳为：注重随机信号处理领域整体知识体系的构建、强化重点知识的分析推导、理论分析与实际应用相结合。全书共分为六章，主要内容包括概率论知识回顾、随机过程及其特征描述、随机过程的线性变换、平稳窄带随机过程、平稳随机过程的非线性变换、非平稳随机过程的特征描述等。

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• Contents
  0 Introduction 1
  0.1 Probability space 1
  0.1.1 Randomized trials 1
  0.1.2 Sample space 2
  0.1.3 Probability space 2
  0.2 Conditional probability space 3
  0.2.1 Conditional probability 3
  0.2.2 Multiplication formula 4
  0.2.3 Total probability formula 4
  0.2.4 The Bayesian formula 5
  0.3 Random variables 6
  0.3.1 The concept of random variables 6
  0.3.2 Discrete random variables 7
  0.3.3 Continuous random variables 7
  0.3.4 Multidimensional random variables 9
  0.4 Distribution of random variable functions 13
  0.4.1 Distribution of discrete random variable functions 14
  0.4.2 Distribution of continuous random variable functions 14
  0.5 Numerical characteristics of random variables 15
  0.5.1 Mathematical expectations 16
  0.5.2 Variance and standard deviation 17
  0.5.3 Covariance and correlation coefficients 18
  0.5.4 The moment of random variables 18
  0.6 Characteristic functions of random variables 22
  0.6.1 Complex random variables 22
  0.6.2 Characteristic functions of random variables 23
  0.6.3 Properties of characteristic functions 24
  0.6.4 Relationship between characteristic functions and moments 25
  0.6.5 Characteristic functions of multidimensional random variables 26
  0.7 Chebyshev inequality and the limit theorem 28
  0.7.1 Chebyshev inequality 28
  0.7.2 Central limit theorem 28
  1 Random processes 33
  1.1 Basic concepts of random processes 33
  1.1.1 Definition of random processes 33
  1.1.2 Probabilitydistribution of random processes 36
  1.1.3 The moment function of random processes 40
  1.1.4 Characteristic functions of random processes 43
  1.2 Stationary random processes 45
  1.2.1 Characteristics and classification 45
  1.2.2 Ergodic processes 49
  1.2.3 Properties of correlation functions 56
  1.2.4 Correlation coefficient and correlation time 59
  1.3 Joint stationary random processes 61
  1.3.1 Joint probability distribution and moment functions of two random processes 61
  1.3.2 Moment function of joint stationary random processes 63
  1.4 Discrete time random process 66
  1.4.1 Definition of discrete time random processes 66
  1.4.2 Probability distribution of discrete time random processes 67
  1.4.3 Digital characteristics of discrete time random processes 69
  1.4.4 Properties of correlation functions of stationary discrete time random processes 74
  1.5 Normal random processes 75
  1.5.1 General normal random processes 76
  1.5.2 Stationary normal random processes 78
  1.5.3 Vector matrix representation of normal stochastic processes 80
  1.6 Spectral analysis of stationary random processes 82
  1.6.1 Concept of spectral density 82
  1.6.2 Definition of power spectral density 84
  1.6.3 Relation between the power spectral density and correlation functions 86
  1.6.4 Properties of power spectral density 88
  1.6.5 Mutual spectral density of joint stationary random processes 91
  1.6.6 Power spectral density of discrete time random processes 93
  1.7 White noise 95
  2 Linear transformation of random processes 101
  2.1 Linear transformation and linear system overview 101
  2.1.1 Basic concepts of linear system 101
  2.1.2 Research topic of linear transformation of random processes 106
  2.2 Differentiation and integration in stochastic processes 107
  2.2.1 Limit of the random process 107
  2.2.2 Continuity of stochastic processes 108
  2.2.3 Differential of stochastic processes (derivatives) 110
  2.2.4 Differential transformation of stochastic processes 113
  2.2.5 Integrals of random processes 117
  2.2.6 Integral transformation of random processes 118
  2.3 Analysis of random processes through continuous time systems 121
  2.3.1 Impulse response method 121
  2.3.2 Spectrum method 124
  2.4 White noise through linear systems 129
  2.4.1 General relations 129
  2.4.2 Noise equivalent passband 130
  2.4.3 White noise through RC integral circuits 131
  2.4.4 White noise through ideal lowpass linear systems 133
  2.4.5 White noise through ideal bandpass linear systems 134
  2.4.6 White noise through a linear system with a Gaussian band 136
  2.5 Probability distribution of the linear transformation of random processes 137
  2.5.1 Input is normal and output is still normal 138
  2.5.2 Input is a non normal process of a broadband (relative to system’s passband), and output is an approximate normal process 140
  2.5.3 Input is white noise and output of the limited bandwidth system is an approximate normal process 142
  3 Stationary and narrowband random processes 149
  3.1 Narrowband random processes represent quasi sinusoidal oscillation 149
  3.1.1 Formation and characteristics of narrowband stochastic processes 149
  3.1.2 Expression of narrowband stochastic processes 150
  3.2 Analytic signals and Hilbert transforms 151
  3.2.1 Complex signals of sinusoidal signals 151
  3.2.2 Complex signals of high frequency narrowband signals 152
  3.2.3 Analytic signals and the Hilbert transform 155
  3.3 Analytic complex stochastic process 158
  3.3.1 Complex random variables 158
  3.3.2 Complex random processes 159
  3.3.3 Correlation function and power spectral density of complex stochastic processes 161
  3.3.4 Complex envelope and statistical properties of narrowband random processes 163
  3.4 Probability distribution of narrowband normal process envelopes and phase 168
  3.4.1 Probability distribution of the narrowband normal noise envelope and phase 169
  3.4.2 Probability distribution of the envelope and phase of a narrowband normal noise plus sine (type) signal 174
  3.5 Probability distribution of narrowband random process enveloping squares 179
  3.5.1 Probability distribution of narrowband normal noise enveloping squares 179
  3.5.2 Probability distribution of synthesis process enveloping squares in narrowband normal noise plus sine (type) signals 180
  3.5.3 x2 Distribution and noncentered/2 distribution 180
  4 The nonlinear transformation of stationary random processes 189
  4.1 Nonlinear transformation overview 189
  4.2 Direct method of nonlinear transformation of random processes 192
  4.2.1 Stationary normal noise through full wave square law devices 192
  4.2.2 Common noise and signals through full wave square law device 196
  4.2.3 Determination of the output power spectrum with the difference beat method and the sum beat method 200
  4.2.4 Hermite polynomial method 202
  4.2.5 Stationary normal noise through half wave linear devices 207
  4.3 Transformation method of random process nonlinear transformation 210
  4.3.1 Transfer function 211
  4.3.2 Moment functions of nonlinear device output processes 215
  4.3.3 The price method 220
  4.4 Slowly changing envelopment method for random process nonlinear transformation 224
  4.4.1 Slowly changing envelope method without load reaction 224
  4.4.2 Slowly changing envelope method with load reaction 230
  4.5 Analysis of random processes through a limiter 237
  4.5.1 Effect of limiting on probability distribution 237
  4.5.2 Effect of limiting on the power spectrum 241
  4.5.3 Noise and sinusoidal signals together through limiting IF amplifier 245
  4.6 Calculation of SNR at the output of a radio system 247
  5 Nonstationary random processes 255
  5.1 Statistical description of nonstationary random signals 255
  5.1.1 Probability density of nonstationary random signals 255
  5.1.2 Digital characteristics of nonstationary random signals 255
  5.1.3 The time varying power spectrum and the short time power spectrum 258
  5.1.4 The Wiener process 259
  5.2 Linear time varying systems and nonstationary random signals 260
  5.2.1 Description of linear time varying discrete system characteristics 260
  5.2.2 Characterization of linear time varying continuous systems 263
  5.2.3 Time varying parameters of AR, MA and ARMA models nonstationary random signals 264
  5.3 Wigner Ville spectrum of nonstationary random signals 265
  5.3.1 Overview of time frequency analysis 265
  5.3.2 Wigner distribution and the Wigner Ville spectrum 266
  5.3.3 Examples ofWD applications 268
  5.3.4 WV spectra of linear time varying system outputs 270
  5.4 Wavelet analysis of nonstationary random signals 273
  5.4.1 Continuous wavelet transform 273
  5.4.2 Two dimensional phase space 275
  5.4.3 Time frequency characteristics of the double window function of 277
  5.4.4 Physical meaning of the continuous wavelet transform 280
  5.4.5 Application of the wavelet transform in random signal analysis 281
  5.5 Non Gaussian processing and higher order statistics 282
  5.5.1 Non Gaussian signal processing 282
  5.5.2 Higher order moments and high order cumulants 284
  5.5.3 The higher order spectrum 287
  5.5.4 Non Gaussian processes and linear systems 291
  A Appendix 295
  A.l Power spectral density of nonstationary random processes 295
  A.2 Proof of a double integral formula 297
  A.3 Derivation of the detector voltage transfer coefficient 299
  A.3.1 Half wave linear detector 300
  A.3.2 Full wave square law detector 301
  A.4 Derivation of the statistical mean of random processes in Rice distribution 302
  Bibliography 305
  Index 307