北大随机过程课件：第5章第4讲高斯随机过程通过非线性系统

1、高斯随机过程通过非线性系统(续1)高斯随机过程通过半波整流器的研究半波整流非线性函数关系：Ibx,y=10,x0 x01输入是窄带平稳实高斯随机过程输入随机过程的概率密度E;t(x)=2no2EexpI2Of丿输出随机过程的概率密度/(y)=15(y)+1n;t2t2nb2O2exp2b2O2vE丿各阶矩、方差偶数阶矩，考虑到输入窄带平稳实高斯随机过程的概率密度函数是偶函数En2m=b2mEE2m1=b2mO2m(2m1)3122奇数阶矩En2m+1=m!2n22mb2m+1C2m+1(2m1)531均值、-,.方差en(t)=1bo2nEDn(t)=En2(t)-en(t)2=b2o22Ef

2、1boEJI2n1b迂&-1，n)相关函数R(t)=R(t,t)nnnn12b22兀器（lP2（T）oo88JJxxexp一12x2+x22p(T)xx122(1-P2(T)丿g12|dxdx12Exx其中，P(T)=亠12，C2g利用典型的积分变换1，得：Rnn(t)nn11b2沁b2a2+b2R(T)+2ng4gg4na2gR2(t)ggR(t)=a2p(t)功率谱v=二倍频分量2f-Aff2f+AfPnn(f)=nn1R(T)ej2nfTdTnn8b臂(f)+1b2传(f)+b24na2g1冒门性（f一f，）df82输入信号是矩形带通窄带实平稳随机过程输入的功率谱密度：N/2,oo,-A

3、fff+-o2o2otherwise非线性器件输出信号的功率谱密度：直流分量：b2C28(f)=丄b2(AfNb(f)2ng2n(N)2(fb2(N)匚f)o2Afo12JAfJ4n12丿IAf丿b24na2g带通信号分量f-Af/2ff+Af/2cccc(N)2(fb2(N)匚f)0Af1-012JIAfJ8n12丿IAf丿b24na2g低通滤波器输出信号的功率谱密度：直流分量：1b2Q28(f)=1b2(AfNb(f)2ng2n低频分量ofAf(N)2匚f=b2(N)(f)02Af1-012JIAfJ4n12丿AfJb24na2g典型的坐标变换1原积分：R(t)=R(t,t)nnnn12b

4、2JJxx1200其中2ns(1-p2(t)gp(T)=Elx(t)x(t)/212gexpx2+x2-2p(t)xx12122o2(1p2(t)gdxdx12丿变换u=x122(1-p2(t)gx222(1p2(T)g意X2)=叩-P2(T)v=12积分R(t)=R(t,t)nnnn12b2JJxx2n52(1p2(T)/212g00exp-/c/、x2+x22p(T)Xx1212(52(1P2(T)丿g2dxdx122b2Q2(lp2(T)/2彳彳()gJJuvexp-w2+v22p(T)uv烹udv00典型的坐标变换2积分之间的关系：I=jjexpl(u2+v2-2wuv)dudv00刃

5、=2Hwvexp-C2+v2-2wuvdw00JJuvexp(2+v2一2wuv1dI2dw00典型的坐标变换3原积分：I=jjexjLC2+v2-2wuv)dudv00积分变换：从uv平面至到,8平面，参数oa=w1,aw（0,n）rcoZ+Ju=sinev=12rco(a-e1sine注意到下列关系：v=12v=12aU=0，2+=2。=八a八V=0,2-=-2丿adudv12丿12丿drdr=sinasinardudv7sinad0d0rsinrsin12JL2丿cosrcossinasina7u2+v2一2p(t)uv化+0cos2JL2丿L2丿cos2+r2sin2a一2r2cosa

6、coscosJL2JL2丿sinasinasin2a原积分：2sin2a+cos(a+20)+1+cos(a-20)-2cosacosa-2cosacos20)2sin2a+cos(a+20)+1+cos(a-20)-cos2a-1-cos+20)-cos-20)00n-a2+v2一2wuvJfrexp-r2drd0sina0n-a-2n-an/2+sin-1w2sina21-w2cosa=wa=n/2-siniw原积分：dIdn/2+sin-idwdw21-w2i-w2i+(/2+siniw)1-w22(1-w2)1-w221-w221-w23/2(/2+sin-iw)原积分：00-exp(2+v2dIdwf)+41-w241-w23/2原积分：R(T)=R(t,t)nnnn12b222(1-P2(T)VX1X2eXp-g00 x2+x2-2P(T)xx1212(J2(1-P2(t)g2dxdx122n2n2nb2Q2(-p(t)/2+p(t)(/2+sin-1P(T)gb2C2gfl-1P2(T)-1P4(T)-亠P6(T)-(224246丿+P(T)兀+P(T)+二P3(T)+，二P5(T)2324