随机过程英文论文

姓名：李范佩专业：031041202 学号：031041202

Random Signal Analysis

chief contents

1. Introduction of the random process

2. Definition of the random process

3. The digital characteristic of the random process

4. Stationary random process and ergodic property

5. The normal random process

6. Markov chain

7. Spectrum analysis of the stationary random process

8. Analysis of the random signal through the linear system

9. Analysis of the random signal through the nonlinear system

Introduction

a. Random process which is aim at the dynamical phenomenon that varies with the time, is the quantitative description for the relationship of the series of random events.

b. Application: Atmosphere field, communication engineering,computer science and so on.

c. Target: To find the inherent law from the events which is seeming external disorder

The definition of the random process

We suppose the sample space of the random expriment is S= {ξ},if there exists a corresponding function X(ξ,t),t ∈T for each ξ ,thus we can get gens function {X(ξ,t),t ∈T} about ‘t ’ for all the ξ,these function famlily about ‘t ’ are called random process,and recorded X(ξ,t).

The random process can be redfined as follow:

If X(ξ, ti) is random variable for each preset ti of the time (i = 1,2,3, …),then X(ξ,t) is called random process.

Individual comprehension:

Random process can be taken for the extension in the time –domain of the random variable. It is the combination of the random variable which is continuous and varying with time

Probability Distribution of Random Process

The definition of the probability distribution

If we suppose {X(t),t T } is random process,for arbitrary fixed t1,t2, …,tn ∈T,and real number x1, x2 , …,xn ∈ R,then we mark Fx(x1,x2 , …,xn ,t1,t2, …,tn ) = P{X(t1) ≤ x1, X(t2) ≤x2 , … X(tn) ≤ xn } as the n-dimensional distribution function of the random process {X(t),t ∈ T } .

n-dimensional probability density function

fx(x1, x2 , …,xn , t1,t2, …,tn)= is called the n-dimensional distribution function of the random process {X(t),t 121212(,,,;,,,)X n n n

F x x x t t t x x x ∂∂∂∂

∈ T } .

Finite dimensional distribution gens functions or n-dimensional probability density functions can fully determine the whole statistical property of the random process.

The Digital Characteristics of The Random Process

In practical application we cannot fully determine the finte dimensional distribution gens functions to analyse it.Thus we just exploit the digital

characteristics to describe the random process.The digital characteristics includes mathematical expectation, variance, correlation function.

a. Mathematical expectation The random process {X(t),t ∈ T }

b. mx(t) = E[X(t)] = ,mx(t) shows the average of sample function value in the time t.

b. Squared value is called squared value it shows the power of the random signal

c. Variance The variance shows the rate of deviation of sample function value related to mx(t) Correlation Function

If x ,y obey the same distribution ,then it is called the autocorrelation function,otherwise called cross-correlation.

(;)x xf x t dx ∞

-∞⎰222()[()](;)x x t E X t x f x t dx ψ∞

-∞==⎰22()[()][(()())]

x x t D X t E X t m t σ==-12{(),}{(),} are the random process.For arbitray fixed t ,X t t T and Y t t T t T ∈∈∈121212(,)[()()](,;,)xy xy R t t E X t Y t x yf x y t t dxdy

∞

∞-∞-∞==⎰⎰