北大隨機過程課件：第5章第4講高斯隨機過程通過非線性系統(tǒng).

1、高斯隨機過程通過非線性系統(tǒng)(續(xù)1)高斯隨機過程通過半波整流器的研究?bx,半波整流非線性函數(shù)關系：y二?？ 0,x > 0 x<01 輸入是窄帶平穩(wěn)實高斯隨機過程 輸入隨機過程的概率密度f E ;t(x)=x2? exp? -2 ? 22 n(?學 c 2 1輸出隨機過程的概率密度? 12? U(yt) f n ；t(y)= 各階矩、方差偶數(shù)階矩，考慮到輸入窄帶平穩(wěn)實高斯隨機過程的概率密度函數(shù)是偶函數(shù)，En 2m=奇數(shù)階矩12m1bE E 2m=b2mr 2m(2m-1)"3> 1 22E n 2m+1 =均值m!2 n 22mb2m+C E 2m+1(2m-1)&

2、quot;5 3? 1E n (t)=方差12b cED n (t)=E n 2(t)-E n (t)22|和r bx.x> 0: 低通濾波器43 _|0.A <0? 122? 1122=b cE (1-1/n?=cE?- 22?2相關函數(shù)Rnn ( t )=R nn (t1,t2)=2 ncE22?x12+x2 -2 p ( t )x?2 x1x2?exp-dx1dx222 ?f2-2 c (1 p ( t )(1- p?r )00 E b2其中，P ( T )=Ext1xt2cE2利用典型的積分變換1,得：b212212Rnn ( t ) q+bREE ( t )+R2 EE

3、( t )22 n 44 ncEREE ( t )= (tE 2 p ( t)功率譜ooPnn (f)=-ooj Rnn ( t )ej2 n f td tOOb1221b tE$ +b2P EE +2 n 44 ncE2-ooj PEE (f ')p EE (f-f ' )df '2 輸入信號是矩形帶通窄帶實平穩(wěn)隨機過程 輸入的功率譜密度：? N0/2,PEE (f?=? 0,f0- f Af<f<f0+22 otherwisetE2 A ? N0非線性器件輸出信號的功率譜密度： 直流分量：12212b tES (f)=b( N0) B (f) 2 n 2

4、n低頻分量0W f <A? fb2? N0?21f A-? A f4 n ?2?2? b2? N0? f?=?1-? 4 n 2? A? ?帶通信號分量fc- f/2 < f < fc+ f/212N0b42二倍頻分量 2fc- f < f < 2fc+ Af?fb2?N0? fl A? A f4 n 2 2?2 2? b2? N0? f?=?1-? 8 n? 2? A? ? ?低通濾波器輸出信號的功率譜密度：直流分量：12212b cES (f)=?(NCA$(f) 2 n 2n低頻分量0W f <A?fb2? N0? 21f A-? A f4 n 2E?

5、22?b2?N0? f?=?1-? 4n?2? A?典型的坐標變換1原積分：Rnn ( t )=R nn (t1,t2)=2ncE?222+x2-2 p ( t )x1X2? xxexpdxdx? -121222 M 22 c (1 p ( t )-(1-p?rb?00其中，p ( t )=Ex(t1)x(t2)/ 變換cE 2u=v=x12 cE 2(1- p 2( t )x22cE 2(1- p 2( t )?(x1,x2)=2 cE 2(1- p 2( t ) ?(u,v)積分Rnn ( t )=R nn (t1,t2)b2=2ncE 2(1- p 2( t )1/2=oo2x12+x2

6、-2 p ( t )x1x2? dx1dx2 expxx? -1222 j?S. cE (1- p?(0£)? 0022uv? exp-u+v-2 p ( t )uvdudv / 002b2 cE 2(1- p 2( t )3/2 n () 典型的坐標變換2積分之間的關系：0001= / exp-u2+v2-2wuvdudv00()dI=2 exp-u2+v2-2wuvdudvdw00 000 ()22/J?uexp-u+v-2wuvdudv=00()1dl2dw典型的坐標變換3原積分：XX1= / exp-u2+v2-2wuvdudv00()積分變換：從u,v平面到r,平面，參數(shù)c

7、os a =w< 1,a(0, n)? a? rcos? + (P? 2? u=sin ,a ? a? rcos? - (? 2? v=sin a注意到下列關系：u=0,v=0, a 2+( =- 9?=«n, (=?2?22? a2? n ? , ( #-?2?22?u?r?u? (? a os? + (? ?v? 2sin a ?r=? a rsi? + (?(? 2-sin ? a rcos - (? 2rsi n a ? sin ? a rsi? - (? 2sin au2+v2-2 p ( t )uv? a? a? a? a cos2+ ( cos2? - ( cos

8、? + ( cos? - (? +r2? 2? 2? 2? -2r2 cos ? 2=r2sin a sin a sin2 a sin2 ar2(1+cos( a +2( )+1+cos( a -2 ( )-2cos a cos a -2cos a cos2 ( ) =2sin2 a r2(1+cos( a +2( )+1+cos( a -2 ( )=2sin2 a-cos2 a -1-cos( a +2( )-COS( a -2 ()=r2原積分：XXl= / exp-u2+v2-2wuvdudv00X () n - a 2=/ 0- /n a -2rexp-r2drd ( sin an -

9、 a n /2+sin-1w=2sin a 2-w2COS a =W a = n /2-sin-1w原積分：did n /2+sin-1w=dwdw2-w21w-w2+n /2+sin-1w-w2-w2=21-w21w-1=+ n /2+sinw23/2221-w21-w ()()原積分：22?-+-2wuvdudv=uvexpuv ff 00()1dl2dw-1=1w+41-w241-w23/2( n /2+sinw)原積分：Rnn ( t )=R nn (t1,t2) (t?XX ?x12+x2-2p ( T)X? dx1dx2x1x2?exp-=2222 f f ncE (1- p ( t )002cE (1- p1221/2b o?&1- p ( t )+ p ( t ) n /2+sin-1 p ( t )2 n122? 1211?bl ? 1- p ( t )- p 4( t )- ? =2 t2-42? 4? 6? ? 2=()131? 3? n? p ( t )+ p?5( +p (?r+p ( t )+22? 4? 5? 2?=