随机过程英文论文

姓名：李范佩专业：031041202 学号：031041202Random Signal Analysischief contents1. Introduction of the random process2. Definition of the random process3. The digital characteristic of the random process4. Stationary random process and ergodic property5. The normal random process6. Markov chain7. Spectrum analysis of the stationary random process8. Analysis of the random signal through the linear system9. Analysis of the random signal through the nonlinear systemIntroductiona. 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 disorderThe definition of the random processWe 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 timeProbability Distribution of Random ProcessThe definition of the probability distributionIf 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 functionfx(x1, x2 , …,xn , t1,t2, …,tn)= is called the n-dimensional distribution function of the random process {X(t),t 121212(,,,;,,,)X n n nF 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 ProcessIn practical application we cannot fully determine the finte dimensional distribution gens functions to analyse it.Thus we just exploit the digitalcharacteristics 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 signalc. Variance The variance shows the rate of deviation of sample function value related to mx(t) Correlation FunctionIf 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∞∞-∞-∞==⎰⎰Correlation function shows the degree of correlation of the sample value in the different time .For t1 = t2,Thus we can conclude that mx(t) and correlation function are the basic digital characteristics.Stationary Random Process and Ergodic PropertyIf the probability property of the random process is independent of the time shifting ,then we remember this random process as stationa ry random process. a. Stationnary random process is also classified into two types,the one is called sensu stricto random process, the other is called generalized random process. b. For the random process X(t),if it ’s n-dimensional probability density function is independent of the time start, just meet the following equation:this random process is called sensu stricto random process.c. For the random process X(t), if it meets these properties:⑴ (constant)2212(,)(,)[(()())][()]()x x x x K t t K t t E X t m t D X t t σ==-==12121212(,,,;,,,)(,,,;,,,)x n n x n n f x x x t t t f x x x t t t τττ=+++ (())()x E X t m t =。