1. 1 basic terms of probability theory
1. 1. 1 random test, sample space
1. 1.2 random events, probability and independence of events
1.2 random variables and their distribution
1.2. 1 distribution function and probability density of one-dimensional random variables
1.2.2 Distribution function and probability density of multidimensional random variables
The numerical characteristics of 1.3 random variables
1.3. 1 mathematical expectation (expectation, mean, statistical mean, collective mean)
1.3.2 variance
1.3.3 Torque function
Function and distribution of 1.4 random variables
1.4. 1 function distribution of one-dimensional random variables
1.4.2 function distribution of two-dimensional random variables
1.5 Characteristic function of random variable
1.5. 1 Definition and properties of characteristic function
1.5.2 Relationship between Characteristic Function and Probability Density
1.5.3 Relationship between Characteristic Function and Moment
1.5.4 joint characteristic function and joint cumulant
General distribution law of 1.6 random signals
Several simple distribution laws of 1.6. 1
1.6.2 Gaussian distribution (normal distribution)
Square distribution of 1.6.3 x
Chapter II Stochastic Processes
2. 1 Definition and classification of stochastic processes
2. 1. 1 definition of stochastic process
2. 1.2 Classification of stochastic processes
2.2 Statistical characteristics of stochastic processes
2.2. 1 Probability distribution of stochastic processes
2.2.2 Digital characteristics of stochastic processes
2.2.3 Characteristic function of stochastic process
2.3 Complex stochastic process and its statistical description
2.3. 1 complex random variable
2.3.2 Complex stochastic process
2.4 Differential and Integral of Stochastic Processes
2.4. 1 continuity of stochastic processes
2.4.2 Differential of stochastic processes
2.4.3 stochastic process integration
2.5 ergodicity of stationary random processes and states
2.5. 1 stationary stochastic process
2.5.2 autocorrelation function of stationary stochastic process
2.5.3 Correlation coefficient and correlation time of stationary stochastic process
2.5.4 Ergodicity of Stationary Stochastic Processes
2.6 Joint distribution and cross-correlation function of stochastic processes
2.6. 1 joint probability distribution and joint probability density
2.6.2 Cross-correlation function and its properties
2.7 Typical Stationary Stochastic Processes
2.7. 1 normal stochastic process
Poisson process
Markov process
2.8 Simulation experiment of stochastic process
Chapter 3 Frequency domain analysis of stochastic processes.
3. 1 power spectral density of stochastic process
3. 1. 1 Fourier transform and power spectrum
3. 1.2 Relationship between power spectral density and autocorrelation function
3. 1.3 Properties of power spectral density
3. 1.4 white noise and white sequence
3.2 Cross Power Spectrum of Multidimensional (Joint) Stationary Stochastic Processes
3.2. 1 cross power spectral density
3.2.2 Relationship between cross-power spectral density and cross-correlation function
3.2.3 Characteristics of Cross Power Spectrum Density
3.2.4 Power spectral density of complex stochastic processes
3.3 Simulation experiment of frequency domain characteristics of stochastic process
Chapter 4 Narrow-band Stochastic Processes
4. Complex signal representation of1random signal
4. Complex signal representation of1.1narrowband random signal
4. 1.2 Hilbert transform and its characteristics
4. 1.3 parsing process
4.2 Narrow-band stochastic process
Mathematical representation of narrow-band stochastic process
4.2.2 Characteristics of narrowband stochastic processes
4.3 Envelope and Phase Characteristics of Narrow-band Gaussian Stochastic Processes
4.3. One-dimensional probability distribution of envelope and phase of1narrowband Gaussian random process
4.3.2 Two-dimensional probability distribution of envelope and phase of narrow-band Gaussian random process
4.4 envelope and phase distribution of narrowband Gaussian process plus sine signal
4.5 Probability distribution of envelope square of narrowband Gaussian process
4.6 narrowband stochastic process simulation experiment
4.6. 1 simulation principle
Simulink simulation results
Chapter 5 analyzes random signals through linear systems.
5. Basic concepts and theories of1linear system
5. 1. 1 Time-invariant linear system
5. 1.2 continuous time-invariant linear system
5. 1.3 discrete time-invariant linear system
5.2 Analysis method of random signal passing through continuous-time system
5.2. 1 differential equation method
Impulse response method
Spectral method
5.3 Analysis method of random signal passing through discrete-time system
5.3. 1 impulse response method
Spectral method
5.4 White noise through linear system analysis
5.4. 1 equivalent noise bandwidth
5.4.2 White noise is analyzed by ideal linear system.
5.4.3 White noise is analyzed by the actual linear system.
5.5 Probability Distribution of Linear System Output
5.5. 1 Gaussian stochastic process through linear system
5.5.2 Standardization of stochastic processes
5.6 stationary random sequence through discrete-time linear system analysis
5.6. Wiener-Sinchin Theorem of1Random Sequence
5.6.2 The stationary random sequence passes through the first-order FIR filter.
5.6.3 The stationary random sequence passes through the first-order IIR filter.
5.7 Simulation experiment of random signal passing through linear system
5.7. 1 Analysis of Typical Time Series Model
5.7.2 Analysis of stochastic process with linear system.
Chapter 6 Analysis of Random Signals Passing through Nonlinear Systems
6. 1 Common nonlinear systems
6.2 Direct method of output signal analysis of nonlinear system
6.2. 1 stationary Gaussian noise acts on the square law detector.
6.2.2 The stationary Gaussian process acts on the linear semi-detector.
6.3 Characteristic Function Method for Output Signal Analysis of Nonlinear Systems
6.3. 1 laplace transform
6.3.2 General form of output autocorrelation function of nonlinear system
6.3.3 Autocorrelation function of Gaussian noise output through nonlinear system
6.3.4 autocorrelation function of cosine signal plus Gaussian noise output by nonlinear system
6.4 Series expansion method for output signal analysis of nonlinear system
6.5 Envelope Method for Output Signal Analysis of Nonlinear Systems
6.5. 1 Statistical characteristics of output signal
6.5.2 Narrow-band Gaussian process passes through a linear semi-detector.
6.5.3 Narrow-band Gaussian process passes through the square law detector.
6.6 Through the stochastic process simulation experiment of nonlinear system
Chapter VII Analysis Methods of Nonstationary Stochastic Processes
7. 1 Higher-order statistics of stochastic processes
7. 1. 1 Moment and Cumulant
7. Properties of1.2 Cumulant
7.2 Higher-order Spectrum of Stochastic Process
7.3 cyclostationary stochastic processes and cyclic spectra
7.3. 1 Second-order cyclostationary stochastic processes and cyclic spectra
7.3.2 Cyclic Cumulant and Cyclic Spectrum of Higher Order Cyclic Stationary Stochastic Processes
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