随机过程-第一章在一个时刻数字特征

《随机过程》电子第六节—随机过程在一个mX(t)E[X(t)]xdFX(x;tX2(t)D[X(t)]X

[x (t)]2 (x;t 二随机过程在两个1自相关函数：stT

RX(s,t)E[X(s)X(t2协方差函数（或称为中心自相关函数s,tT CX(s,t)E{[X(s)mX(s)][X(t)mX自相关函数与协CX(s,t)RX(s,t)mX(s)mX特别）mX(t)0CX(s,t)RX(s,t)2）st2(t)C(t,t)R(t,t)

均值函数和自相关函数是基本数字特例1已知g(t)为确定信号 的 量， - X(t)Yg(t m(t 2(t R(s,t (s,t g2(t g(s)g(t g(s)g(t例2已知X(t)acos(ta0,~U(0,2mX(t RX(s,tX解 m(t)X

acos(t)

dXR(s,t)Xa2

0

2

{cos[（st）2]cos[（s0cos[(st)]，只与st例 X(t)1

0 0已知随机点在(a,b]内出现的次数N(a,服从Poisson分布，且etP{N(t0,t0t]k} mX RX(s,t).解 mX(t)1P{X(t)1}(1)P{X(t)1(1

)et

e2t不妨设stRX(s,t)E[X(s)X(tP{X(s)1,X(t)1}P{X(s)1,X(t)P{X(s)1,X(t)1}P{X(s)1,X(t)P{X(s)1,X(t)X(s)P{X(s)1,X(t)X(s)P{X(s)1,X(t)X(s)P{X(s)1,X(t)X(s)P{X(t)X(s)1}P{X(t)X(s)PN(st偶数次PN(st奇数次e2(ts)2s去掉条件st， RX(s,t)三两随机过程的数 RXY(s,t)E[X(s)Y(t2协方差函s,tT,CXY(s,t)E{[X(s)mX()]Yt)mY定 对于两个随机过程{X(t),tT}和{Y(t),tT如果对s,tT 都RXY(s,t)则称X(t)与Y(t)为正交过程两随机过程的几X(t)与Y(t)正交s,tT RXY(s,t)X(t)与Y(t)不相关s,tT CXY(s,t)X(t)与Y(t)独立,等价m,nN,ti,tjFX,Y(x1,,xm;t1,,tm;y1,,yn;t1,,tnFX(x1,,xm;t1,,tm)FY(y1,,ynX(t)与Y(t)独立s,tTE[X(s)Y(t)]mX(s)mYX(t)与Y(t)独立X(t)与Y(t)不相关信号的对于输入信号X(t),信道噪声N(t),输出信RW(s,t)RX(s,t)RXN(s,t)RNX(s,t)RN(s,t)特别地，当X(t)与N(t)正交时，RW(s,t)RX(s,t)RN(s,t)四复随机过程的数字 复随机过程Z(t)为Z(t)X(t)jY(t数学期望（mZ(t)E[Z(t)]mX(t)jmY方差（方差函数2(t)EZ(t)m(t) E{[Z(t)mZ(t)][Z(t)Z(t)}E[Z(t)Z(t)]mZ(t)mZ自相关函数RZ(s,t)E{[Z(s)][Z(t协方差函数CZ(s,t)E{[Z(s)mZ(s)][Z(t)mZ(t性质2(t)2

(t)

2(t)E|Z(t)

|m(t) RZ(s,t)RZ(t, CZ(s,t)CZ(t,CZ(s,t)RZ(s,t)mZ(s)mZ(t作业 —二阶1定义任意时刻均值和方差均存在的过程称性质二阶矩过程的自二阶矩过程的自几种常用的二阶定 满mX(t)常数 (b)RX(s,t)RX(st的过程称为宽平稳过程。如第六节例 定义若二阶矩过程X(t)满足:ttt E{[X(t2)X(t1)][X(t4)X(t3)]}则称X(t (s,t)R(s,t)

(min(s,t 记min(st其中

XX2(min(s,t))2(XX2独立增量过程.若过程X(t)对t1t2t3t4t2kt2k1T,X(t2)X(t1),X(t4)X(t3),,X(t2k1)X(t2k相互独立,则称X(t (s,t)2(min(s,t 特别地，再当mx(t)=0 (s,t)R(s,t)

(min(s,t 3稳增量过X(t2X(t1)与X(t2sX(t1s)同分则称X(t)为平稳增量过程,或称X(t)为齐次的Markov过程1定义E离散型：设随机过程{X(t),tT}的状态空间是可数集E,若对于任意正整数 n及t1t2tntT, P{X(t1)i1,X(t2)i2,,X(tn)in}P{X(t)i/X(t1)i1,X(t2)i2,,X(tn)in}P{X(t)i/X(tn)in}，t

链。式称E连续型：设{X(t),tT}为连续型随机过程 于任意正整数n及t1t2tntT, P{X(t)x/X(t1)x1,X(t2)x2,,X(tn)P{X(t)x/X(tn) Markov2若独立增量过程X(t)的X(0)=0,则X(t)为Markov证明:以Markov设X(ttT为独立增量对于任意n及t1t2tntT,因为X(0)P{X(t1)X(0)i1,X(t2)X(t1)i2i1,,X(tn)X(tn1)inin1，X(t)X(tn)iP{X(t1)X(0)i1,X(t2)X(t1)i2i1,,X(tn)X(tn1)inP{X(t)X(tn)i另一方面，PX(ti/X(tninXt)P{X(t)X(tn)i所以{X(ti/X(t1i1X(t2i2,X(tninP{X(t)i/X(tn)in即X(t)，tT}为Markov3Markov Poisson定义设X(t(t0)A在(0t次数且满足X(0)X(t)为独立、平稳0stP{X(t)

X(s)

k}

((ts))k

e(ts)，k特别地，当s0P{X(t)X(0)k}

(t

et,k则称X(t)为Poisson过程,也称为强度的Poisson设X(t(t0)为Poisson过程,EX(t)DX(t)XR(s,t)2stXXC(s,t)XPoisson过程为Markov链（2）Wiener

定义：设X(t(t0)是一个实过 X(0 X(t)为独立、平稳增量过s,t0X(tX(s~N0，2ts则称X(t)为Wiener过程，也叫Brown性质设X(t(t0)为Wiener过程,EX(t)0,DX(t)2RX(s,t)CX(s,t) 2Wiener过程为Markov过程Wiener过程为Gauss过程的证明 任取正整数n及0 则X(t1X(0X(t2X(t1),X(tnX(tn1X(t)

X(t1)X(0)

X(t)X(t

0

X(tn)

X(t)

n1X(t1)X(0