第四章随机信号通过非线性系统

x(t若x(t) 0若x(t)y(t)=|x(t)

yn

n y(t)=nXt的一维概率密度为fXx;tfxx1x2;t1t2yf(xX

f(x)fx

fn(x)

x;txRyt1,x

f(x1)f(x2)fx

yyy

x;t

bn

x2n(t)

Rt,

b2Ex(t)2x(t)2

2bx2fx,x;t,tdx

由于方差为x的正态随

Exn

xxxx令n＝1n＝2

E[y(t)]b

3E[y(t)]x Ey(t)2E[y(t)]2x

y y

b2{E{x(t)2}E{x(t)2}2E2{x(t)x(t

Rx Rx

2 xx

RxRx

12 1

S/

设平方律检波器输入端的随机信号是xtstnt t,tb2E[s(t)+n(t)2s(t)+n(t)2 b2E[s2(t)s2(t)+4s(t)s(t)n(t)n(t)+s2(t)n2(t)+s2(t)n2(t)+n2(t)n2(t

=t2y y

SSSSN

R

4 例1.假设输入是等幅正弦信号s(t)=acos0t，式中a0是常数，为在02之间均匀分布的随量．输入噪声

a=2cos0t2t1aa2a2

Es2(t)s2(t)a4E

tcos2t

0 0 cos2t1t2 t2

nnnnRn2

2

222 n22

E[y(t)]ba22 n 0 2

24E[y(t)]3b8an n y2E[y(t)2]E[y(t)]y

a4

4nn nnyy x xx x

fxx

x e2xEy(t)

bxf(x)dx

22dx

x

Ey2(t)

b2x2f(x)dx

22x x bb x2

b2

11yEy

x x x Ry

2x(t))y(t)

x(t x(ty(t)的均值和相关函数。E{y(t)y(t)}1P{x(t)x(t)0}1P{x(t)x(t)x(t)x(trR(设sinR()

E{y(t)y(t)}

1

(1) R()2arcsinR( 例3

x(t)

mX、方差为

XX

解：由y

bfYy;tfXx;t 2 exp 2X2X

lnylnbmX

X2yX2

XX

x22mxm222x

exp

dx 222m24 exp XXXXX

2 x2

b2e2

XX

2mxm242x

dx

X 2224m244X exp XXX

Xxb2exp2m22Xx xx

b2e2mX2

xx例4x(t)是均值为零、方差为1的正态平稳随机过程，我们对x(t构成如下非线性运算YtbXt解：由ybxxy(x>=0)xy 所以fy;t x;ty y2

y2 b

2 2

exp

exp 22

2b2

y

00

d

2b2

2b2

y2 exp dy

d

2b2

0

2

2bb212Yt

yf(x)

12

12

F()

f(x)exejxdx

f(x)ejx sj

F(s) f(x)esx0f(x) jF(s)esxds2

若f(x)在,上不绝对可积，且当x0时f(x) 定义f(x)f(x)f(x) f(x)f

x

x00f

f

x

f(x)

f xF(s)

f(x)esx

和 的单边拉氏变换存在，且

Res

0f

F(s)

(s)

(s)

f

F(s)

(s)

(s)

fF(s)

fy=f(x) jF(s)esxds2

Y

12jD

RyEy(t)y(t E

11

F()EejX(t)jX(t)42

2

XtRyR F()d

42

R F(s F(s)M,;ds 2j2

1

s,s;e 12s,s;e 12M

1s2s22ss

R

F(s F(s)e2

12

2j2

1 将es1s2Rx

XssRkk Rk Rk

1 1 2R

F(s)ske2 F(s)ske2yk0k!2j2y

fx

x

x

Fs f(x)esxdx bxesxdx 1s2s22ss M2s1,s2

e2

1 r

2Rx E t 0Rx E t

Rx0

1 1s2s22ssr

e2

12

R

1 2ske2 ske2ss 2 ss1k0k!2j1

1 22k 2k

D kD

k

D

e2ds1

e2

rk

k

rk sk2e2ds kk

k

k

Ry

22n2!!2nx(tA(tcos0t(t)，其中A(t)是随机慢变化的时间函数，非线性系统的特性为yfy(t)fA(t)cos0tt将随机过程A(t)和(t在某一时刻tA,ttyt(t)fAtcos0tt0ttt

ytf0Atf1Atcostf2Atcos2t fA

fAcos

fA fAcos

cosn

n1,2,L,

则YtI0I1I2LInI0f0AtInfnAtcosn0tt

E[yt]EI0I1I2InLEI0EI1EI2LEIn n E[yt]EI0Ef0AtE

f0

fAAtA(t)的一维概率分布。IHI0EI00000020

EI2E

EI2 f2A

A0

yf(x)

xx

输入端的窄带正态随机过程在t时刻的状态为

AtcostbAcos yf(x)

022IHI0Eytf0AtEf0At

fAcos

I0

bAtcostdt

2bAtcostdt t tEI0E

f0

f0

f

Atttt At2tt

At EI0

bAt□At

2t 2t

22dAtt020

EI2 I2AEI2 I2At t 220

0

t tb2b22

tt

22dA

tt

22 1

2 2

b

yf(xbx2

Atcos 2ytbAtcost2E[y]E[bA2cos2]bE[A2]E[cos2 A 2 AdA A

2

22dA2