0 Introduction 0.1 Probability space 0.1.1 Randomized trials 0.1.2 Sample space 0.1.3 Probability space 0.2 Conditional probability space 0.2.1 Conditional probability 0.2.2 Multiplication formula 0.2.3 Total probability formula 0.2.4 The Bayesian formula 0.3 Random variables 0.3.1 The concept of random variables 0.3.2 Discrete random variables 0.3.3 Continuous random variables 0.3.4 Multidimensional random variables 0.4 Distribution of random variable functions 0.4.1 Distribution of discrete random variable functions 0.4.2 Distribution of continuous random variable functions 0.5 Numerical characteristics of random variables 0.5.1 Mathematical expectations 0.5.2 Variance and standard deviation 0.5.3 Covariance and correlation coefficients 0.5.4 The moment of random variables 0.6 Characteristic functions of random variables 0.6.1 Complex random variables 0.6.2 Characteristic functions of random variables 0.6.3 Properties of characteristic functions 0.6.4 Relationship between characteristic functions and moments 0.6.5 Characteristic functions of multidimensional random variables 0.7 Chebyshev inequality and the limit theorem 0.7.1 Chebyshev inequality 0.7.2 Central limit theorem 1 Random processes 1.1 Basic concepts of random processes 1.1.1 Definition of random processes 1.1.2 Probabilitydistribution of random processes 1.1.3 The moment function of random processes 1.1.4 Characteristic functions of random processes 1.2 Stationary random processes 1.2.1 Characteristics and classification 1.2.2 Ergodic processes 1.2.3 Properties of correlation functions 1.2.4 Correlation coefficient and correlation time 1.3 Joint stationary random processes 1.3.1 Joint probability distribution and moment functions of two random processes 1.3.2 Moment function of joint stationary random processes 1.4 Discrete time random process 1.4.1 Definition of discrete-time random processes 1.4.2 Probability distribution of discrete-time random processes 1.4.3 Digital characteristics of discrete-time random processes 1.4.4 Properties of correlation functions of stationary discrete-time random processes 1.5 Normal random processes 1.5.1 General normal random processes 1.5.2 Stationary normal random processes 1.5.3 Vector matrix representation of normal stochastic processes 1.6 Spectral analysis of stationary random processes 1.6.1 Concept of spectral density 1.6.2 Definition of power spectral density 1.6.3 Relation between the power spectral density and correlation functions 1.6.4 Properties of power spectral density 1.6.5 Mutual spectral density of joint stationary random processes 1.6.6 Power spectral density of discrete-time random processes 1.7 White noise 2 Linear transformation of random processes 2.1 Linear transformation and linear system overview 2.1.1 Basic concepts of linear system 2.1.2 Research topic of linear transformation of random processes 2.2 Differentiation and integration in stochastic processes 2.2.1 Limit of the random process 2.2.2 Continuity of stochastic processes 2.2.3 Differential of stochastic processes (derivatives) 2.2.4 Differential transformation of stochastic processes 2.2.5 Integrals of random processes 2.2.6 Integral transformation of random processes 2.3 Analysis of random processes through continuous-time systems 2.3.1 Impulse response method 2.3.2 Spectrum method 2.4 White noise through linear systems 2.4.1 General relations 2.4.2 Noise equivalent passband 2.4.3 White noise through RC integral circuits 2.4.4 White noise through ideal lowpass linear systems 2.4.5 White noise through ideal bandpass linear systems 2.4.6 White noise through a linear system with a Gaussian band 2.5 Probability distribution of the linear transformation of random processes 2.5.1 Input is normal and output is still normal 2.5.2 Input is a non normal process of a broadband (relative to system's passband), and output is an approximate normal process 2.5.3 Input is white noise and output of the limited bandwidth system is an approximate normal process 3 Stationary and narrowband random processes 3.1 Narrowband random processes represent quasi-sinusoidal oscillation 3.1.1 Formation and characteristics of narrowband stochastic processes 3.1.2 Expression of narrowband stochastic processes 3.2 Analytic signals and Hilbert transforms 3.2.1 Complex signals of sinusoidal signals 3.2.2 Complex signals of high-frequency narrowband signals 3.2.3 Analytic signals and the Hilbert transform 3.3 Analytic complex stochastic process 3.3.1 Complex random variables 3.3.2 Complex random processes 3.3.3 Correlation function and power spectral density of complex stochastic processes 3.3.4 Complex envelope and statistical properties of narrowband random processes 3.4 Probability distribution of narrowband normal process envelopes and phase 3.4.1 Probability distribution of the narrowband normal noise envelope and phase 3.4.2 Probability distribution of the envelope and phase of a narrowband normal noise plus sine (type) signal 3.5 Probability distribution of narrowband random process enveloping squares 3.5.1 Probability distribution of narrowband normal noise enveloping squares 3.5.2 Probability distribution of synthesis process enveloping squares in narrowband normal noise plus sine (type) signals 3.5.3 X2 Distribution and noncentered X2 distribution 4 The nonlinear transformation of stationary random processes 4.1 Nonlinear transformation overview 4.2 Direct method of nonlinear transformation of random processes 4.2.1 Stationary normal noise through full-wave square-law devices 4.2.2 Common noise and signals through full-wave square-law device 4.2.3 Determination of the output power spectrum with the difference beat method and the sum beat method 4.2.4 Hermite polynomial method 4.2.5 Stationary normal noise through half wave linear devices 4.3 Transformation method of random process nonlinear transformation 4.3.1 Transfer function 4.3.2 Moment functions of nonlinear device output processes 4.3.3 The price method 4.4 Slowly changing envelopment method for random process nonlinear transformation 4.4.1 Slowly changing envelope method without load reaction 4.4.2 Slowly changing envelope method with load reaction 4.5 Analysis of random processes through a limiter 4.5.1 Effect of limiting on probability distribution 4.5.2 Effect of limiting on the power spectrum 4.5.3 Noise and sinusoidal signals together through limiting IF amplifier 4.6 Calculation of SNR at the output of a radio system 5 Nonstationary random processes 5.1 Statistical description of nonstationary random signals 5.1.1 Probability density of nonstationary random signals 5.1.2 Digital characteristics of nonstationary random signals 5.1.3 The time-varying power spectrum and the short-time power spectrum 5.1.4 The Wiener process 5.2 Linear time-varying systems and nonstationary random signals 5.2.1 Description of linear time-varying discrete system characteristics 5.2.2 Characterization of linear time-varying continuous systems 5.2.3 Time-varying parameters of AR, MA and ARMA models nonstationary random signals 5.3 Wigner-Ville spectrum of nonstationary random signals 5.3.1 Overview of time frequency analysis 5.3.2 Wigner distribution and the Wigner-Ville spectrum 5.3.3 Examples of WD applications 5.3.4 WV spectra of linear time-varying system outputs 5.4 Wavelet analysis of nonstationary random signals 5.4.1 Continuous wavelet transform 5.4.2 Two-dimensional phase space 5.4.3 Time-frequency characteristics of the double window function of ψa,b(t) 5.4.4 Physical meaning of the continuous wavelet transform 5.4.5 Application of the wavelet transform in random signal analysis 5.5 Non-Gaussian processing and higher-order statistics 5.5.1 Non-Gaussian signal processing 5.5.2 Higher-order moments and high-order cumulants 5.5.3 The higher-order spectrum 5.5.4 Non-Gaussian processes and linear systems A Appendix A.1 Power spectral density of nonstationary random processes A.2 Proof of a double integral formula A.3 Derivation of the detector voltage transfer coefficient A.3.1 Half-wave linear detector A.3.2 Full-wave square-law detector A.4 Derivation of the statistical mean of random processes in Rice distribution Bibliography Index
