第3章随机过程通过线性系统

x1(t)x2(t)aL[x1(t)]+y(t-C)=L[x(t-y(t)=x(t)*h(t)=

y(w)=H

¥x(w)¥H(w)

Y(s)=H(s)Xs=s+t0

h(t)0 y(n)=x(n)*h(n)=x(n-k)h(k)=x(k)h(n-k

¥¥

y(z)=x(z)H(z)z=e如果当n<0时， h(n)=0，那么该系统称为因果系统。¥1t*f2¥

12t-tdtf1-*=

¥¥¥

f1f2tttfft-¥ ftt=ftftt-

=f1dt-t1¥tdt=dfttdt¥

¥ ¥=ftftdt-t1=ft-t1)

出为 y(t)=h(t)*x(t)= h(t)x(t-•：•：000 0¥0=h(t)E[X(t-0¥=h(t)m(t-

X(t)为平稳SP =mX(t)*

mY=mX0

2h(u)X(t-20¥0=h(u)E[X(t1)X(t2-0¥=0h(u)RX(t1,t2-¥

00

h(u)RX(t-

RY(t1,t2)=E[Y(t1)Y(t2)]=h(t1)*h(t2)*RX(t1,t2RY(t1,t2)=E[Y(t1)Y(t2)]=h(t1)*h(t2)*RX(t1,t2 h(v)X(t2- ¥¥=00h(u)h(v)E[X(t1-u)X(t2-¥=00h(u)h(v)RX(t1-u,t2-=¥

Rt-u,t-

=X1-u,t2- 0=0Y1-u,t20

=¥X,t2-vdv=1*RXY,t2

ht2*1RY(t1,t2)=E[Y(t1)Y(t2)]=h(t1)*h(t2)*RX(t1,t2¥¥ X(tn)]*h(t1)*h(t2)**h(tn

小mt=

¥ ¥

XRY(t1,t2)=h(t1)*h(t2)*RX(t1,t2

cos20t t=10e- mt=Yt=E

XtY

¥Xt-¥0t0t+M=0mYt=50

hudu=5¥10e-10tdt=2X(t)是相关函数为2

N0

系统的冲激响应为t=be-btUtb=t=tt

mt=Yt=E

XtY

¥Xt-¥¥=mX

¥RY=YtYtt¥=E=E

Xt-uu

hv

¥ XtXt¥=00

t+u-vdudv¥=¥

N

dt+u-vdudv¥ ¥¥¥=N¥¥

dt+u-vdvdu0=N0

¥t+udu=¥

¥be-bube-bt+udu 0N0=2

e-bt0

e-2bu

4由自相关函数的偶函数特性<0RY

N0b4

RY

N0b4

e-bQ=RY

N0b4¥RXY=RX)=¥

t-¥=N¥

dt-

,=2=

=N0t£

N02 =R=

N

dt- =N0, =2 NN0

R= 0e-0e-t冲激响应为tbe-btUt

b，Dfbb RY=YtYtt¥=0

R

t+u-vdudv¥=¥b2e-bue-¥

¥N¥N bu

t+udv

t+u

¥ ¥

e-bt-e-bt2 R

e-bt-e-bt

bNb2

N0b

RY» X(t也是宽遍历性的，且Y(tX(t)联合遍历)mXtmX ¥mY=mX0¥

= =0h(u)RX(t-

¥¥¥

RY(t1,t2)=0

Xt¥因为Y(t)=

Y(t+T)=0

输出Y(t+T) 和Y(t)分别是输入X(t+T) 和X(t)与h(t)的卷积，即可以表示成级数和的形式。由于随机信号X(t)是严平稳的，所以X(t)与X(t+T)具有相同的n维概率密度函数，这样Y(tT)与Y(t)也应该具有相同的n也是宽遍历性的，且Y(tX(t)联合遍历由X(t)X(t)=则输出

Y(t)=lim1

TYTTfi¥ T¥=lim1[h(u)X(t-T¥Tfi¥ ¥=[lim1¥

X(t-T Tfi¥ T¥

X(tY(t)Y(t+t)=lim1

Y(t)Y(tTTfi¥ T¥¥=[lim1¥¥

T Tfi¥ T¥=0 X(t)Y(t+t)=lim1

X(t)Y(tTTfi¥ TT¥=lim1[h(u)X(t+t-T¥Tfi¥ ¥=[lim1¥

T Tfi¥ T¥=0RX(t-

mYt=mXtt)mYw=mXwHw¥mY=mX0h(t)dt=mXH(w)w=0=mXH¥SY(w)=SX(w)H(w)H(-w)=SX(w)H(w)SY(s)=SX(s)H(s)H(-Y=RXX=RX

SY(w)=SXY(w)H(-w)=SYX(w)H(w)SY(s)=SXY(s)H(-s)=SYX(s)HF(s)

f(t)e-stdt¥¥

¥¥s=s+f(tf(t)e-例3.

2X(t)是相关函数为2

N0

系统的冲激响应为t=be-btUtb=t=tt

(1)0mY=mX¥0

w=0=mXH(0)=(２)2SYw=

wHw = 2b2+wRY

N0b4

e-b(３)Q=

N0b4(４) w= wHw= (

,

RYX=RXYt=N0t£ y(t)=0h(t)x(t- Y(n)=h(k)X(n-k ¥mY(n)=E[Y(n)]=h(k)E[X(n-k¥¥X(n)SP¥¥¥¥¥RXY(n,n+m)=E[X(n)Y(n+¥¥

XX

h(k)E[X(n)X(n+m-k¥

k

(nk=0+k RY(n,n+m)=E[Y(n)Y(n+ =E[(h(k)X(n-k))(h(j)X(n+m-¥ ¥=h(k)h(j)EX(n-k)X(n+m-k=0j¥=h(k)h(j)RX(n-k,n+m-¥k=0j¥RY=h(k)h(j)RX(m+k-¥k=0j¥

=h(k)h(j)RX(k-k=0j =m[h(k - k

z=1=H

SX(z)=H(z)S SX(SX(z)=H(z)S SX(z)=H(z-1z=eSYX(w)=H- X S(z)=H(z)H(z-1 (z)

S(w)=H(ejw)H(e-jw (w)=H(ejw

S2 2 =H(z-1 (z)=H RY(m)

S(z)zm-1dzYlY2R(m)=1 pH(ejw)S(w)ejmw 2p- EY2(n)= H(z)H(z-1 (z)z- 2pj 2EY2(n)=1 pH(ejw)S 2p- lz=HH-H- SY=SXHH =HH

SYw=Hw SY =H SYS，它的极点在Y-YSY)=S-+ Y-Y SYz)=S-zS+z =H- SY=S-+s YH=S-Yz=H-

SY=S-S+ YH=S-YSw

S

-25s2+

7-7+ s4

+ =sss-s+YS-Y

7+ss+YH=S-Y

7+ss+w4-

SYw=w

s4-s4-10s2+9

s-8+8- 8 ssssYS-Y-

s+8+ sss+8+ H=

ss=HH-=1

S= HHz H-=

H1

SX )S-+

S

XS-X

1XS-X

w=1.04+0.4cosw的白化滤波器SXw

1.04+e=e=

+e-jw+e-jw

z

z2 5z

z5+XS-X

X)XSXw

wSXw=w2+ww2+w2+

-s2+

3

+9

33+XS-X

33

XS-X

ss+S(w)=H2Yw)N02GY(w)

H(w)

wR(t)=N0

¥H¥

2jwt¥ 4p-¥

)

2

0h(u)h(u¥¥2p

H(w) )KDw)KDwe

H

DweY2t=N0

¥w2dw¥

2

NKYt=0 Kdw= Dwe

2¥0¥

H(w)max=H

ww

Dwe

H

H2¥0¥

Dwe

2

H-2jH)-

2t=N0¥2p¥

H2

（实际系统P=N0

（理想系统 和H(w) w

由Hw=

知H

w)=

+w

Hw

=Dwe

Hw

w2¥0¥

w¥=0

+w2

=barctanb =2 =Dwe= 2p

Hwb2b2+wb2+w2=2

Df=b例3.11计算低通滤波器Hw=2-w,w0 由Hw=2-w

Hw2=2-w，Hw)

=Dwe

Hw

Hw2¥0¥1=41

2-w2w=¥3¥ =Dwe= 2p

Hw2=(2-w(2Dw)22

Hw 22

Df=222H(w)= 0 A

GX(w)=

2

wH(w)= 0 A N0

GY(w)

GX(w)=

设设GX(w)=

H(w)= 0R(t)=

¥G(w)

1 1

Dw/2N02p2

coswdw2

Dwt

sinDwt

=N0ADw 设GX(w)=设

2

H(w)= 0GX(w)=

A0N0=

wH(w)= 0 A

KYKY

2

sinDwt= Dwt2

w GX(w)=H(w)=AsinDwt

¥t=r(t)dt

¥ dt=p=

Dwt2

Dww设设GX(w)=

¥x¥

dx=2

a>

H(w)=w–w0£Dw/2A

w–w0£Dw/2GX(w)=

0 0 N

w-

G(w)=H G(w)= R(t)=1 ¥G(w) = w0+Dw/2A2 cosww2p0-Dw/2 A2NDwsin(Dwt/= Dwt/2A2NDwsin(Dwt/ Dwt/2与a(tcosw0t相比，a(tcosw0t是

E[Y2(t)]

N00sinDwt

r(t)

KY

= coswt K R

Dwt 2sinDwtt=¥ dt=p=

Dwt2

DwE[Y2(t)]

N00sinDwt

r(t)

KY

= coswt K R

Dwt 2sinDwtt=¥ dt=p=

Dwt2

Dw

+exp

(w)= N 2

SX(w) Aexp

b2

b2

+exp

(w)= ¥

0+exp(w+w)20+exp

R(t)0A2exp0R(t)0A2exp0

b2

b2 e

+exp-

4p-¥

b2

b2 0N0=

¥¥

-2 w2e-2b b0

4

+exp

(w)=

+exp

(w)=

R(0)=A2N0

+exp

(w)= H(w)2

-2Dwe= dw=pb-22b2

H(w0 ¥-b t0=0

dt=难 nY(t)= nfi随)随)

w

2 0d 22解:2()()(

SX

wH

2b2+wR=1 wejwtdw YY2t=R)Y

N0b4

Yt=f

2y20 =pNbexp-Nb0

Xi为 量，它是 的

Dti‡

nn

ty>>ty>>

即ty 设有一个实值函数x(t)，它 ˆ(t（H[x(tx(t)=ˆ(t)

p-¥t- =p-

ttdt h(t)=1« (jw)=-jsgn(w)=

f(t)«Fjw)F(jt)«2pf(-w)因为sgn(t)

jw

2«2psgn(-w)=-2p

hH(t)=p

«HH(jw)=-j[x(t)=H-1 ˆ(t[ =-p-

ˆ

ˆ(t (t)=-

ˆ(t)hH1(t)x(t)=ˆ(t)*hH1(t)=x(t)*hH(t)*

ˆ(thH1(t)x(t)=ˆ(t)*hH1(t)=x(t)*hH(t)* HH(jw)HH1(jw) HH1(jw)

HH(

=

(t)=-

ˆ(t) h(t)=1/

H(w) |H(w)| H(w) H(w)=

H(w)=

= w£w0>

tsw0tHt=-a(t)cosw0t 令t=a(t)cos=pw++dw- =1w+w+w-w =-w=-jw-w-w+w t=

t0

-t-jw0t=-

t2jsinw

=tnw 令1t=a(tsins=1*pw+w-dw-w s=jw+w-w-w 11=1-jsgnw1=-2w-w0+w+w0

1tejw

t-=tcos1=- 12 定义：给定任一实随机过程X定义一复随机过程%(t)%(t)jˆˆ(t)=H[X(t)]= ¥Xp-¥t-

是X(t) （1）X(tˆ

ˆ(t)=X(t)* (w)= (w)H(w) =SX(w)22ˆXH( X(tˆ （3）R

Rˆ(t)=ˆX ˆ代入ˆ(tp¥X(h)dht-h令t-h p¥ p¥l （3）R

Rˆ(t)=ˆX p¥l=1p1p¥Xt)1l=¥l=RˆX X(tˆ （3）R

Rˆ(t)=ˆXX (-t)=E[ˆ(t)X(t-t)]=Ep¥ ¥p1Xll=p¥l=1p¥l=X=-R 3.5.2Rˆ(-t)=-Rˆ （5）RXˆX(-t)=-RXˆX

%jRjˆXX[%%(tXXjˆjˆ[RX=2[RX(t)+jRXˆ R R()=2[R()+

(w)=-jSX 证明：由性质3，证明：由性质3， (t)=R(t)*hXXH两边取付氏变换得SXˆ(w)jsgn(w)SX(w)S(w)=-jSX X

352352 证明：由性质 %XˆS%jSXˆ=2[SX(w)+sgn(w)SX4SX(w)0 (w)=4SX 例3.14设平稳随机过程X(t)的功率谱密度为SXˆt是X(t) 变换，求Vt=Xtt+ˆtst,t+t=VtVtt=Xtt+ˆtsw0t·Xt+tsinw0t+t+ˆttcosttXtXt+t0t+ˆtˆttstcosw0tˆtXt+tsw0tsinw0t+Xtˆttnw0tcosw0tXtXt+t0t+ˆtˆttstcosw0tˆtXt+tsw0tsinw0t+Xtˆttnw0tcosw0t

in0tsinw0t+t+

sw0tR

sw0tsinw0t+t+R

nw0t=RXt-RXXˆpw+dwsinw0t«pw+w0-dw-w0SV=SXpw+w+dw- -1 *pw+w-dw- 2pXˆ =1Sw+w+ w j w+w-

w-w

Xˆ = *

ˆ=-jSXˆ

ptXX S=1

w+w+ w

-sgnw

w

+sgnw

w-w0=1 w-sgnw +1 w-w+w- 例3.15设平稳随机过程X(t)的功率谱密度为SXˆt是X(t) 变换，求Vt=Xtt-ˆtst,tt=VtVtt=Xtt-ˆtsw0t·Xt+tsinw0t+t-ˆttcosttXtXt+t0t+ˆtˆttstcosw0tˆtXt+tsw0tsinw0tXtˆttnw0tcosw0tXtXt+t0t+ˆtˆttstcosw0tˆtXt+tsw0tsinw0tXtˆttnw0tcosw0t

0tsinw0t+t+

sw0t-RˆX

st+t-

nw0t=RXt+RXˆpw+dwsinw0t«pw+w0-dw-w0SV=SXpw+w+dw- +1 *pw+w-dw- 2pXˆ =1Sw+w+ w +j w+w-

w-w

Xˆ = *

ˆ=-jSXˆ

ptXX S=1

w+w+ w

-sgnw

w

+sgnw

w-w0=1 w+w+w+ +1 w-sgnw h(t)=1« (jw)=-jsgn(w)=

f(t)«Fjw)F(jt)«2pf(-w)因为sgn(t)

jw

2«2psgn(-w)=-2p

hH(t)=p

«HH(jw)=-j定义：给定任一实随机过程X定义一复随机过程t%(t)jˆˆ(t)=H[X(t)]= ¥Xp-¥t-

是X(t) 3.5.2X(tˆ （3）R

Rˆ(t)=ˆXX （5）RXˆX(-t)=-RXˆX

%jRjˆX

(w)=-jSX

(w)=4SX

x(t) (w) w-w£w£w+w (w)= wcw0)SXSXt3.6X(t)=a(t)cosw0t-b(t)sinw0tXX(t)=a(t)cosw0t-b(t)sinw0ta(a(t)=X(t)cosw0t+ˆ(t)sinb(t)=-X(t)sinw0t+ˆ(t)

ˆ(t)=a(t)sinw0t+b(t)

3.6将X(t)表示成解析形式tX(tˆ 同时又 jw0tXt+ˆwt- =Xtcosw0t+ˆtnw0t+-Xtnw0t+ˆtcosw0t] t=tcosw0tˆtnt=tnˆtcosw0t~Xt-jw0t=t+~t=at+~~

t=t+jb(t)cosw=atcosw0t-b(t)sinw0t+tnw0t+b(t)cosw0tXt=atcosw0t-b(t)sinw0t3.6 b(t)E[a(t)]=E[b(t)]=

X(t)为平稳过程，且假设其均值为0 ˆXSa(w)=Sb(w)= SS[SX(w+w0)+SX(w-w0Sab(w)=- SS[SX(w+w0)-SX(w-w03.63.6.3 b(t)证明：因为X(t)和ˆ(t)都是实过程。 a(t)=X(t)cosw0t+ˆ(t)sinb(t)=-X(t)sinw0t+ˆ(t)所以 b(t)都是实随机过3.63.6.3

b(t)E[a(t)]=E[b(t)]= ˆXSa(w)=Sb(w)= SS[SX(w+w0)+SX(w-w0Sab(w)=- SS[SX(w+w0)-SX(w-w03.63.6.3E[a(t)]=E[b(t)]=E[X(t)]= E[ˆ(t)]=E[a(t)]=E[X(t)]cosw0t+E[ˆ(t)]sinw0t=3.63.6.3

b(t)E[a(t)]=E[b(t)]= ˆXSa(w)=Sb(w)= SS[SX(w+w0)+SX(w-w0Sab(w)=- SS[SX(w+w0)-SX(w-w03.63.6.3 Ra(t)E[a(t)a(t+t)]ˆˆ=RX(t)cosw0tcosw0(t+t)+RXXˆ(t)cosw0tsinw0(t+RXˆX(t)sinw0tcosw0(t+t)+RXˆ(t)sinw0tsinw0(t因为：

E[a2(t)]=Ra(0)=RX(0)<¥3.63.6.3 b(t)E[a(t)]=E[b(t)]= ˆXSa(w)=Sb(w)= SS[SX(w+w0)+SX(w-w0Sab(w)=- SS[SX(w+w0)-SX(w-w03.63.6.3证明：由性质3t3.63.6.3 b(t)E[a(t)]=E[b(t)]= ˆX

Sa(w)=Sb(w)= SS[SX(w+w0)+SX(w-w0Sab(w)=- SS[SX(w+w0)-SX(w-w03.63.6.3ˆX

=E[{X(t)cosw0t+ˆ(t)sinˆRˆX

(t)sinw0tsinw0(t+t)+RXXˆ(t)cosw0tcosw0(tR(t)=R ˆX

3.63.6.3 b(t)E[a(t)]=E[b(t)]= ˆXSa(w)=Sb(w)= SS[SX(w+w0)+SX(w-w0Sab(w)=- SS[SX(w+w0)-SX(w-w03.63.6.3证明：由性质3.63.6.3

b(t)E[a(t)]=E[b(t)]= ˆXSa(w)=Sb(w)= SS[SX(w+w0)+SX(w-w0Sab(w)=- SS[SX(w+w0)-SX(w-w03.63.6.3X

=E[{a(t)cosw0t-b(t)sinw0t}•{a(t+t)cosw0(t+t)-b(t+t)sinw0(t=Ra(t)cosw0tcosw0(t+t)-Rba(t)sinw0tcosw0(t-Rab(t)cosw0tsinw0(t+t)+Rb(t)sinw0tsinw0(t 3.63.6.3

b(t)E[a(t)]=E[b(t)]= ˆXSa(w)=Sb(w)= SS[SX(w+w0)+SX(w-w0Sab(w)=- SS[SX(w+w0)-SX(w-w03.63.6.3Sa(w)=Sb(w)= SS[SX(w+w0)+SX(w-w0 1Rew+e-iw

ew-e-iwt 2

SXˆ(w)=-jsgn(w)SX

3.63.6.3Sa(w)=Sb(w)= SS[SX(w+w0)+SX(w-w0S(w)=1S(w-w)+ +1-jsgn(w-w 2 =1 +1-sgn(w-w SS[SX(w+w0)+SX(w-w03.63.6.3Sa(w)=Sb(w)= SS[SX(w+w0)+SX(w-w0 0 2 0 2 23.63.6.3 b(t)E[a(t)]=E[b(t)]= ˆXSa(w)=Sb(w)= SS[SX(w+w0)+SX(w-w0Sab(w)=- SS[SX(w+w0)-SX(w-w03.63.6.3Sab(w)=- SS[SX(w+w0)-SX(w-w0ˆX =-1Rew-e-iwt

ew+e-iwt2

Sˆ(w)=-jsgn(w)SX(w)

3.6Sab(w)=- SS[SX(w+w0)-SX(w-w0 (w)=-1 (w-w)- (w+w 2 +1-jsgn(w-w

)SX(w+w0=-j{-1

(w+w0+1sgn(w-w

)SX(w+w0=- SS[SX(w+w0)-SX(w-w03.6Sab(w)=- SS[SX(w+w0)-SX(w-w0ttcosw0ttnff和Sabf)： f0= f0=

=1f+f解：

S=1 pw+w+dw- 2p +1 *pw+w-dw- 2pXˆ =1Sw+w+ w +j w+w-

w-w

Xˆ = *

ˆ=-j

Xˆ

ptXXS=1

w+w+ w

-sgnw

w

+sgnw

S

w =1 w+w+w+ +1

w

-w ˆX =-1 *pw+w-dw- 2p +1 pw+w+dw- 2pXˆ =-jSw+w- w +1

w+w+

w-w

X

Xˆ 由于S

=-j

=-jSw+w- w -sgnw

w

-sgnw

S

w =1Sw+w- w +1w+w

w

+w

S

w =1 w+w+w+ -1 w-sgnw 例3.17设平稳随机过程Xt=twt+q-twt+q) p解 RX=ttt=twt+q-twt+qt+tw0t+t+q-t+tw0tt+qwt+qw0t+t+q+wt+qnw0tt+qwt+qnw0t+t+q+wt+qw0tt+q=Rnw-Rnwt+w0t+q=RnwRX=nw w=1Sw+w+Sw-w 3.6X(t)=A(t)cos[w0t+f(t)]

~t-

t+

t+Atejfttt+~

t=t)a(t))Xt-jw0t=tejft)~t=tejfteAt[cosw0t+jt+w0t+jtt=twt+3.6X(t)=A(t)cos[w0t+f(t)] X(t)=A(t)cos[w0t+f(t)]=A(t)cosw0tcosf(t)-A(t)sinw0tsina(t)=A(t) b(t)=A(t)sinX(t)=a(t)cosw0t-b(t)sinw0t

A(t)=a2(t)+b2f(t)=arctg工程上应用最多的窄带随机过程是窄带过程，因为不仅热噪声是过程，很多宽带噪声通过窄带系统后也成为窄带过程。因此，重点讨论窄带过程是很有必要的，当中放定理：当X(t为窄带随机过程，即X(tDww0，A(t)和f(t) b(t)是低频限带随机过程即它们的功率谱只在0£w£wc wcw0 E{[a(t+t)-a(t)]2}=E[a2(t+t)+a2(t)-2a(t)a(t=2Ra(0)-2Ra

=1wc

wc wc

jwt

p-wc Sa(w)dw-

Sa dw]

1

cc

Sac= S(w)(1-e cp-

aaSa(w)(1-

2 2 p p-

S(w)2sin

wt2

E{[a(t+t)-a(t)]}£wctRa(0)=wctE[aa a

S(w)2(wt )p- w此式说明：若t1，在tt+ta(t)wca(t)因为w

，即T2pT

，令t t=T<< wt=w

由 不等式：P{x-E(x)

令xa(t+T0a(t)，注意E(x)E[a(t+T0E[a(t

P{[a(t+T0)-a(t)]-

E{[a(t+T0)-

w2T2E[a2 P{a(t+T0)-

足够小时，对于给定的e>右式趋近于0x(t为窄带随机过程时，在一个高频周期T0内，a(t)的变化大于e的概率趋于0。也就是说，a(t t X(t)=a(t)cosw0t-b(t)sinw0t

a(t)=A(t)b(t)=令t固定，

tb= t

3.6节性质

fab(at,bt)=fa(at)fb(bt

+b2 t A2 exp- J=

= =

A2

A exp-

A‡0,0£f£22

0

A2 A2= exp- 2p=texp-

¥s¥

ff (A,f)dA=

0£

(A,f)=f(f)f(f Xt=twt-twt=twt+ A

fAt=texp- ¥

A2t=

AtfAtt

texp- 0s

A2

=-Ad

=2ps

AtA

=

=ps 2

A2fA

A2¥ =¥

t

exp- 02 2

A2

=-A2d

AA

t=¥2At

=¥ AAt AAt

2s ¥

2s2 0 0s =2-E2A=2s

2 X(t)=A(t)cos[w0t

A2

u=

= texp(- ),At‡ At=h(ut)=+

At‡f

)=

h(u

exp(-

exp(-ut t t

fA,At,At

f,f