随机信号-第二章过程

2.1.1RP的基本概念一.RP的定义每个元素S，总有一个确知的时间函数X(t,),tT与它相对应，这样，对于所有的S，就X(t,)xiiii 1.连续型RP离散型RP:连续随机序列:时间离散，状态为连续型 离散随机序列:时间离散，状态为离散型 2.1)X(t)Acos(wt2)不确定的随机过程:无规律可循3.按分布函数或概率密度及数字特征的不同特性分 遍历 平稳宽平稳(数字特征严平稳(概率特性 随机过程随机过程2.2.RP因素，将随机过程X(t,)简记为X(t).为防止 1对于RP，X(t,)取任意n个时刻t1,t2 tn,则X(t1,)X(t1),X(t2,)X(t2 X(tn,)X(tnn个状态X(ti) RP一、RP1.RP }如果FX(x1t1)对x1的偏导数存在，则有f(xtFX(x1t1)称作随机过程的一维概率密度函X11如果存在二元函数f1x1,t1)使下式F(x,t) fX1x,tX1 fX(x1t1)RPX(t)的一维概率密2.RPRP一维概率分布只描述了随机过程在各个孤立时刻的统计特性而 起在不同时刻的状之间的联系。1)RPX(t,),t1,t2,X(t1,)X(t1),X(t2,)X(t2FX(x1,x2;t1,t2)P{X(t1)x1;X(t2)含义是：二随机事件{X(t1x1和{X(t2x2}RV对照Fxx(x1x2P{X1x1X2)fX(x1,x2;t1,t2) X1212F(x,x;t,tF(x,x;t,t)xX121X121 1(u,u;t,t)dudu3.RP X(t):t1,t2, tn,X(t1),X(t2), n维分布函数：FX(x1,x2 xn;t1,t2，tn) X(tn)n维概率密度：fX(x1, F(x, x;t,t，tn X1 n1 (或一维概率密度是不同的3、fX(x1;t1)和FX(x1;t1)简记为:fX(x;tFX2 x1f(x,x;t,t)dxX121 1x1 tn tn) b)fX(x1,x2f(xX1f(xX1)fX(x1;t1) fX(x1, tn tn)fX(xi;tinn tn)FX(xi;tinP{X(t1)x1,X(t2) X(tn)xn}P{X(ti)RVX:mE[X] xfXXRPX(t):mX(t)E[X(t)]xfXRVX:E[X2 xf2X例)(t)E[X(t)]x2f22XXRVX:2E[(xm)2](xm)2fX RPX(t):D[X(t2XE[(X(t)m(t))2X (xm(t))f2 D[X(t)]2(t)2(t)m2(t)E[X2(t)]m2 XX(t)2(t)D[X XX(t)和方差随机过程的概率分布族能完整地描述随过程的统计特性，但在实际应用中，要确定随机过程的概率分布族常常是 的;随机过程是随时间t而变化的一族随 量,所以将随 量的数字特征推广至随机过程中去。34.自相关函数(Auto-correlation4.自相关函数(Auto-correlationFunction,RPX(t)t1,t2RVX(t1),X(t2(1)R(t,t)E[X(t)X(t)] xxf(x,x;t,t)dxX1 12X121 1RVX,X E[XX] xx (x,x)dx12XX1 11 X1RVX(t1X(t2ACF(2)t1t2tR(t,t)R(t,t)E[X(t)X(t)]E[X2(t)]2X1 X(3)如果RX(t1t20(t1t2则RPX(t)X(t1)，X(t25自协方差函数(中心化自相关函数RVX1,X2KXXcov(X1,X2)E[(X1mX)(X2mX(二阶混合中心矩 [(xm(t))(xm(t))]f(x,x;t,t)dxKX(t1,t2)E[X(t1)X(t2)]E[(X(t1)mX(t1))(X(t2)mX(t2 X X X121 1((2)KX(t1,t2)RX(t1,t2)mX(t1)mX(t2K(t,t)D[X(t)]2(t)E[(X(t)m(t))2XXX(t)E[X(t)]m(t)R(t,t)E[X2222XXXKX(t1t2)0,RPX(t)两时刻状态不相关6自相关系r(t,t)K(t,t) K(t,t K(t,t(t)(t D(t)D(t K(t,t)K(t,tr(tt0,则RP例1.RVX:CX(u)] f XCX(u;t) ]fX jux(tf(x;t)X2X(u;t)eRPRPX(t)的n阶原点矩函E[X(t)] xfX(x;t)dx(nn CnunCXE[X(t)] xf(x;t)e Xuuxf(x;t)dxjCXXuE[X(t)] xf(x;t)dx22 C2Xu2. C(u,u;t,t)E[ej[uX(t)uX(t)] 121 121(x,x;t,t)ej(uXuX)dx1fX(x1,x2;t1,t2 2 1C(u,u;t,t 121j(uxux(2du14RX(t1,t

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t)dudu(1)X(t)X(t1)X(t2 X(tnY(t)Y(t1)Y(t2 Y(tn定义两个过程的nmF(x, x;y, y;t, t;t',t t'XY1 n1 n1 n1 P{X(t)x X(t)x;Y(t')y Y(t')y XY n m n 若有F(x x;y y;tXY n m n X n nY m F(x x;t t)F( X n nY m XY n m n 或f(x x;y y;tXY n m n X n nY m f(x x;t t)f(X n nY m 则称RPX(t)和Y(t)是相互独立

二统计平 1221.互相关函数(二阶矩函数为两个过程之间定义为：RXY(t1,t2)E[X(t1)Y(t2xyfXY(x,y;t1,t2RYX(t1,t2)E[Y(t1)X(t211则两个随机过程的二维联合分布函数为FXY(x,y;t1,t2)P{X(t1)x,Y(t2)y(2)RPX(t)tX(t)Y(t)tY(t2 (x,y;t,t)dxdy12.联合概率密度函(1) (x x;y y;tmt;t'nt'mnm(x1 x;yy;t t;t'm nt'm(2)fXY(x,y;t1,t2)2 (x,y;t,t 152.中心化互相关函数/K2.中心化互相关函数/KXY(t1,t2)E[(X(t1)mX(t1))(Y(t2)mY(t2 (xm(t))(ym(t))f(x,y;t,tX1Y 13.互相关系数rXY1(t,t)KXY(t1,t2) KXY(t1,t2 X KXY(t1,t2 KX(t1,t1)KY(t2,t2(t)(t D(t)D(tX 4.KXY(t1,t2)RXY(t1,t2)mX(t1)mY(t2三四独立,1RXY(t1,t2) 212 (u,u;t,tuu例(1)FXY(x1''FX(x1xn;t1 t)F(y y;t txn;y1 ym;t1 tn;t1 tmnY1fXYx1 m fX(x1xn;y1 ym;t1xn;t1 t)f(ytn;t1y;ttm't 1m'mRPXtYt,P{X(t1) X(tn)xn}P{X(ti)nnfX(x1,x2 xn;t1,t2 t) f(x;t iFX(x1,x2xn;t1,t2 tn)FX(xi;tin2.不相关(两者关系非线性(1)Xt)rX(t1t2)0KX(t1,t2)0RX(t1,t2)mX(t1)mX(t2E[X(t1)X(