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 Analyt