[communication principle] Chapter 3 -- random process [i]

The third chapter random process

The basic concept of stochastic process

• random process ： It is a kind of process that changes randomly with time , He can't use the exact time function to describe
• Sample function ： Each record of test results , Is a definite time function x_i(t), It is called sample function
• The total of all sample functions {x₁(t),x₂(t) … x_n(t)} It's a random process , Write it down as ξ(t), A random process is a set of all sample functions
• The random process can also be regarded as a set of random variables at different times in the time process

Distribution function of random process

• set up ξ(t) Represents a random process , At any moment of his life t₁ Value ξ(t₁) It's a random variable , Its statistical characteristics can be described by distribution function or probability density , The random variable ξ(t₁) Less than or equal to a certain value x Probability P[ξ(t₁) ≤ x₁], Write it down as the following formula , And become a random process ξ(t) Of One dimensional distribution function
  $$F_1(x_1,t_1)=P[\xi (t_1)\le x_1]$$
• if F₁(x₁, t₁) Yes x₁ The partial derivative of exists , said f₁(x₁,t₁) by ξ(t) One dimensional probability density function
  $$\frac{\partial F_1(x_1,t_1)}{\partial x_1}=f_1(x_1,t_1)$$

The numerical characteristics of stochastic processes

mean value

• random process ξ(t) The mean or mathematical expectation of , Defined as
  $$E[\xi (t)]={\int }_{-\infty }^{\infty }x{f}_1(x,t)dx$$

variance

• Variance represents the random process at time t Relative to the mean a(t) The deviation process
  $$X=>Dx=E{x}^2-[Ex{]}^2$$
  $$D[\xi (t)]={\int }_{-\infty }^{\infty }{x}^2{f}_1(x,t)dx-[a(t){]}^2$$

Correlation function

• The correlation function is much like the degree of correlation between random variables obtained by a random process at any two times
  • The larger the correlation function => The greater the degree of correlation at any two moments
• random process ξ(t) The correlation function of -> Autocorrelation
  $$R(t_1,t_2)=E[\xi (t_1)\xi (t_2)]={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{x}_1{x}_2{f}_2(x_1,x_2;t_1,t2)d{x}_{1}d{x}_2$$
• The concept of correlation function is extended to two or more random processes , You can get the cross-correlation function

Stationary random process

Definition

• If a random process ξ(t) The statistical properties of are independent of the starting point of time , That is, time translation does not affect any of its statistical properties , The random process is said to be in a strict sense Stationary random process , abbreviation Strictly stationary stochastic process
• Stationary random process ξ(t) In any finite dimension, the probability density function is independent of the time starting point

1. For any positive integer n And all real numbers Δ, One dimensional probability density function and time t irrelevant
   $$f_1(x_1,t_1)=f_1(x_1)$$
2. The two-dimensional distribution function is only related to the time interval τ = t₂ - t₁ of
   $$f_2(x_1,x_2;t_1,t_2)=f_2(x_1,x2;\tau )$$

• Stationary random process ξ(t) It has concise numerical characteristics
  1. Mean and t irrelevant , Constant a
  2. The autocorrelation function is only related to the time interval τ = t₂ - t₁ of
  • The above two conditions are often used to directly judge the stationarity of stochastic processes
  • And the process that meets the above two conditions at the same time is defined as Generalized stationary stochastic process ( Wide stationary stochastic process )
  • Strictly stationary stochastic processes must be generalized stationary , On the contrary, it is not necessarily

Ergodicity of all States

• Ergodic process , Its numerical characteristics ( Are statistical average ) It can be completely replaced by the time average of any realization in the random process
• The meaning of ergodicity of States ： Any time in the random process realizes all possible states of the random process
• Be careful ：
  • A random process with ergodicity of States must be a stationary process , On the contrary, it is not necessarily true
  • Random signal and noise encountered in communication system , Generally, it can satisfy the ergodic condition of all States

Important formula

• hypothesis x(t) It's a smooth journey ξ(t) Any implementation of ( sample ), Because it is a definite function of time , You can get his time average , Its time mean and time correlation function are defined as
  $$\{\begin{array}{r}\overline{a}=\overline{x(t)}=\underset{T\to \infty }{\lim }\frac{1}{T}{\int }_{-\frac{T}{2}}^{\frac{T}{2}}x(t)dt\\ \overline{R(\tau )}=\overline{x(t)x(t+\tau )}=\underset{T\to \infty }{\lim }\frac{1}{T}{\int }_{-\frac{T}{2}}^{\frac{T}{2}}x(t)x(t+\tau )dt\end{array}$$
  $$\{\begin{array}{r}a=\overline{a}\\ R(\tau )=\overline{R(\tau )}\end{array}$$

Autocorrelation function of stationary process

• Autocorrelation function is a particularly important function to express the characteristics of stationary processes , It can not only be used to describe the numerical characteristics of stationary processes , It is also intrinsically related to the spectral characteristics of stationary processes
• set up ξ(t) Is a real stationary random process , Then its autocorrelation function
  $$R(\tau )=E[\xi (t)\xi (t+\tau )]$$
• The main properties ：
  1. R(0) = E[ξ²(t)], Express ξ(t) Of Average power
  2. R(τ) = R(-τ), Express τ Even function of
  3. R(∞) = E²[ξ(t)] = a², Express ξ(t) Of DC power
  4. R(0) - R(∞) = σ²,σ² Is variance , Represents a stationary process ξ(t) Of AC power

Power spectral density of stationary process

• The spectral characteristics of random process can be expressed by its power spectral density , Any sample in the random process is a certain power type signal , For any definite power signal x(t), Its power spectral density is defined as
  $$P_x(f)=\underset{T\to \infty }{\lim }\frac{|X_T(f){|}^2}{T}$$
• ++ Power spectral density of stationary process P_ξ(f) Its autocorrelation function R(τ) It is also a pair of Fourier transform relations ++
  $$\{\begin{array}{r}{P}_{\xi }(\omega )={\int }_{-\infty }^{\infty }R(\tau )e^{-j\omega \tau }dt\\ R(\tau )=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }{P}_{\xi }(\omega )e^{j\omega \tau }d\omega \end{array}$$
  or
  $$\{\begin{array}{r}{P}_{\xi }(f)={\int }_{-\infty }^{\infty }R(\tau )e^{-j\omega \tau }dt\\ R(\tau )=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }{P}_{\xi }(f)e^{j\omega \tau }df\end{array}$$

Gaussian random process

• Gauss process , Also known as normal random process , It is the most important and common process in the field of communication

Definition

• If the random process ξ(t) Arbitrarily n The dimensional distribution obeys the normal distribution , It is called normal process or Gaussian process

Important nature

1. Gaussian process n Dimensional distribution only depends on the mean value of each random variable 、 Variance and normalized covariance
   • therefore , Just study its digital characteristics
2. Generalized stationary Gaussian processes are also strictly stationary
3. If the value of Gaussian process at different times is irrelevant , Then they are also statistically independent
4. The process generated by Gaussian process after linear transformation is still Gaussian process
   • If the input of a linear system is a Gaussian process , Then the system output is also a Gaussian process

Gaussian random variables

• The value of Gaussian process at any time is a random variable with normal distribution , Also known as Gaussian random variable , Its one-dimensional probability density function is
  $$f(x)=\frac{1}{\sqrt{2\pi }\sigma }{e}^{-\frac{(x-a{)}^2}{2{\sigma }^2}}$$
• characteristic
  • f(x) Symmetrical to x = a This line , namely f(a+x) = f(a-x)
  • $${\int }_{-\infty }^{\infty }f(x)dx=1And{\int }_{-\infty }^{a}f(x)dx={\int }_a^{\infty }f(x)dx=\frac{1}{2}$$
  • a Represents the distribution center ,σ It is called standard deviation , Indicates the degree of concentration ,f(x) The graph will follow σ Become higher and narrower with the decrease of
• skill
  • $$P(x>A)=\frac{1}{2}erfc\sqrt{\frac{d_1^2}{2{\sigma }^2}}$$
  • $$P(x<B)=\frac{1}{2}erfc\sqrt{\frac{d_2^2}{2{\sigma }^2}}$$
  • $$P(a<x<A)=\frac{1}{2}-\frac{1}{2}erfc\sqrt{\frac{d_1^2}{2{\sigma }^2}}$$
  • $$\{\begin{array}{r}{d}_1=A-a\\ d_2=a-B\end{array}$$

Stationary stochastic processes pass through linear systems

• Linear time invariant systems can be characterized by their unit impulse response h(t) Or its frequency response H(f) characterization , If order v_i Is the input signal ,v₀ For the output signal , Then the relationship between input and output can be expressed as convolution
  $$v_0(t)=v_i(t)\ast h(t)={\int }_{-\infty }^{\infty }{v}_i(\tau )h(t-\tau )d\tau$$
  or
  $$v_0(t)=h(t)\ast v_i(t)={\int }_{-\infty }^{\infty }h(\tau )v_i(t-\tau )d\tau$$
  The corresponding Fourier transform relation is
  $$V_0(t)=H(f)V_i(f)$$

nature

1. Output process ξ_o(t) The average of E[ξ_o(t)] It's a constant
   $$E[{\xi }_o(t)]=a\cdot {\int }_{-\infty }^{\infty }h(\tau )d\tau =a\cdot H(0)$$
2. Output process ξ_o(t) The autocorrelation function of
   • If the input process of a linear system is stable , Then the output process is also stable
3. Output process ξ_o(t) Power spectral density
   • The power spectral density of the output process is the power spectral density of the input process multiplied by the square of the system efficiency response
4. Output process ξ_o(t) Probability distribution of
   • If the input process of a linear system is Gaussian , Then the output process of the system is also Gaussian
   • The process of Gaussian process after linear transformation is still Gaussian process