第3章隨機(jī)過(guò)程通過(guò)線性系統(tǒng)

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如果當(dāng)n<0時(shí)， h(n)=0，那么該系統(tǒng)稱為因果系統(tǒng)。￥1t*f2￥

12t-tdtf1-*=

￥￥￥

f1f2tttfft-￥ ftt=ftftt-

=f1dt-t1￥tdt=dfttdt￥

￥ ￥=ftftdt-t1=ft-t1)

出為 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)為平穩(wěn)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)是相關(guān)函數(shù)為2

N0

系統(tǒng)的沖激響應(yīng)為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由自相關(guān)函數(shù)的偶函數(shù)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沖激響應(yīng)為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也是寬遍歷性的，且Y(tX(t)聯(lián)合遍歷)mXtmX ￥mY=mX0￥

= =0h(u)RX(t-

￥￥￥

RY(t1,t2)=0

Xt￥因?yàn)閅(t)=

Y(t+T)=0

輸出Y(t+T) 和Y(t)分別是輸入X(t+T) 和X(t)與h(t)的卷積，即可以表示成級(jí)數(shù)和的形式。由于隨機(jī)信號(hào)X(t)是嚴(yán)平穩(wěn)的，所以X(t)與X(t+T)具有相同的n維概率密度函數(shù)，這樣Y(tT)與Y(t)也應(yīng)該具有相同的n也是寬遍歷性的，且Y(tX(t)聯(lián)合遍歷由X(t)X(t)=則輸出

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)是相關(guān)函數(shù)為2

N0

系統(tǒng)的沖激響應(yīng)為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，它的極點(diǎn)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的白化濾波器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

（實(shí)際系統(tǒng)P=N0

（理想系統(tǒng) 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計(jì)算低通濾波器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)=

設(shè)設(shè)GX(w)=

H(w)= 0R(t)=

￥G(w)

1 1

Dw/2N02p2

coswdw2

Dwt

sinDwt

=N0ADw 設(shè)GX(w)=設(shè)

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設(shè)設(shè)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與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=難 nY(t)= nfi隨)隨)

w

2 0d 22解:2()()(

SX

wH

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

N0b4

Yt=f

2y20 =pNbexp-Nb0

Xi為 量，它是 的

Dti?

nn

ty>>ty>>

即ty 設(shè)有一個(gè)實(shí)值函數(shù)x(t)，它 ?(t（H[x(tx(t)=?(t)

p-￥t- =p-

ttdt h(t)=1? (jw)=-jsgn(w)=

f(t)?Fjw)F(jt)?2pf(-w)因?yàn)閟gn(t)

jw

2?2psgn(-w)=-2p

hH(t)=p

?HH(jw)=-j[x(t)=H-1 ?(t[ =-p-

?

?(t (t)=-

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

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

HH(

=

(t)=-

?(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 定義：給定任一實(shí)隨機(jī)過(guò)程X定義一復(fù)隨機(jī)過(guò)程%(t)%(t)j??(t)=H[X(t)]= ￥Xp-￥t-

是X(t) （1）X(t?

?(t)=X(t)* (w)= (w)H(w) =SX(w)22?XH( X(t? （3）R

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

R?(t)=?X p￥l=1p1p￥Xt)1l=￥l=R?X X(t? （3）R

R?(t)=?XX (-t)=E[?(t)X(t-t)]=Ep￥ ￥p1Xll=p￥l=1p￥l=X=-R 3.5.2R?(-t)=-R? （5）RX?X(-t)=-RX?X

%jRj?XX[%%(tXXj?j?[RX=2[RX(t)+jRX? R R()=2[R()+

(w)=-jSX 證明：由性質(zhì)3，證明：由性質(zhì)3， (t)=R(t)*hXXH兩邊取付氏變換得SX?(w)jsgn(w)SX(w)S(w)=-jSX X

352352 證明：由性質(zhì) %X?S%jSX?=2[SX(w)+sgn(w)SX4SX(w)0 (w)=4SX 例3.14設(shè)平穩(wěn)隨機(jī)過(guò)程X(t)的功率譜密度為SX?t是X(t) 變換，求Vt=Xtt+?tst,t+t=VtVtt=Xtt+?tsw0t·Xt+tsinw0t+t+?ttcosttXtXt+t0t+?t?ttstcosw0t?tXt+tsw0tsinw0t+Xt?ttnw0tcosw0tXtXt+t0t+?t?ttstcosw0t?tXt+tsw0tsinw0t+Xt?ttnw0tcosw0t

in0tsinw0t+t+

sw0tR

sw0tsinw0t+t+R

nw0t=RXt-RXX?pw+dwsinw0t?pw+w0-dw-w0SV=SXpw+w+dw- -1 *pw+w-dw- 2pX? =1Sw+w+ w j w+w-

w-w

X? = *

?=-jSX?

ptXX S=1

w+w+ w

-sgnw

w

+sgnw

w-w0=1 w-sgnw +1 w-w+w- 例3.15設(shè)平穩(wěn)隨機(jī)過(guò)程X(t)的功率譜密度為SX?t是X(t) 變換，求Vt=Xtt-?tst,tt=VtVtt=Xtt-?tsw0t·Xt+tsinw0t+t-?ttcosttXtXt+t0t+?t?ttstcosw0t?tXt+tsw0tsinw0tXt?ttnw0tcosw0tXtXt+t0t+?t?ttstcosw0t?tXt+tsw0tsinw0tXt?ttnw0tcosw0t

0tsinw0t+t+

sw0t-R?X

st+t-

nw0t=RXt+RX?pw+dwsinw0t?pw+w0-dw-w0SV=SXpw+w+dw- +1 *pw+w-dw- 2pX? =1Sw+w+ w +j w+w-

w-w

X? = *

?=-jSX?

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?yàn)閟gn(t)

jw

2?2psgn(-w)=-2p

hH(t)=p

?HH(jw)=-j定義：給定任一實(shí)隨機(jī)過(guò)程X定義一復(fù)隨機(jī)過(guò)程t%(t)j??(t)=H[X(t)]= ￥Xp-￥t-

是X(t) 3.5.2X(t? （3）R

R?(t)=?XX （5）RX?X(-t)=-RX?X

%jRj?X

(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+?(t)sinb(t)=-X(t)sinw0t+?(t)

?(t)=a(t)sinw0t+b(t)

3.6將X(t)表示成解析形式tX(t? 同時(shí)又 jw0tXt+?wt- =Xtcosw0t+?tnw0t+-Xtnw0t+?tcosw0t] t=tcosw0t?tnt=tn?tcosw0t~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)為平穩(wěn)過(guò)程，且假設(shè)其均值為0 ?XSa(w)=Sb(w)= SS[SX(w+w0)+SX(w-w0Sab(w)=- SS[SX(w+w0)-SX(w-w03.63.6.3 b(t)證明：因?yàn)閄(t)和?(t)都是實(shí)過(guò)程。 a(t)=X(t)cosw0t+?(t)sinb(t)=-X(t)sinw0t+?(t)所以 b(t)都是實(shí)隨機(jī)過(guò)3.63.6.3

b(t)E[a(t)]=E[b(t)]= ?XSa(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[?(t)]=E[a(t)]=E[X(t)]cosw0t+E[?(t)]sinw0t=3.63.6.3

b(t)E[a(t)]=E[b(t)]= ?XSa(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)]??=RX(t)cosw0tcosw0(t+t)+RXX?(t)cosw0tsinw0(t+RX?X(t)sinw0tcosw0(t+t)+RX?(t)sinw0tsinw0(t因?yàn)椋?/p>

E[a2(t)]=Ra(0)=RX(0)<￥3.63.6.3 b(t)E[a(t)]=E[b(t)]= ?XSa(w)=Sb(w)= SS[SX(w+w0)+SX(w-w0Sab(w)=- SS[SX(w+w0)-SX(w-w03.63.6.3證明：由性質(zhì)3t3.63.6.3 b(t)E[a(t)]=E[b(t)]= ?X

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

=E[{X(t)cosw0t+?(t)sin?R?X

(t)sinw0tsinw0(t+t)+RXX?(t)cosw0tcosw0(tR(t)=R ?X

3.63.6.3 b(t)E[a(t)]=E[b(t)]= ?XSa(w)=Sb(w)= SS[SX(w+w0)+SX(w-w0Sab(w)=- SS[SX(w+w0)-SX(w-w03.63.6.3證明：由性質(zhì)3.63.6.3

b(t)E[a(t)]=E[b(t)]= ?XSa(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)]= ?XSa(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

SX?(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)]= ?XSa(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-w0?X =-1Rew-e-iwt

ew+e-iwt2

S?(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- 2pX? =1Sw+w+ w +j w+w-

w-w

X? = *

?=-j

X?

ptXXS=1

w+w+ w

-sgnw

w

+sgnw

S

w =1 w+w+w+ +1

w

-w ?X =-1 *pw+w-dw- 2p +1 pw+w+dw- 2pX? =-jSw+w- w +1

w+w+

w-w

X

X? 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設(shè)平穩(wěn)隨機(jī)過(guò)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工程上應(yīng)用最多的窄帶隨機(jī)過(guò)程是窄帶過(guò)程，因?yàn)椴粌H熱噪聲是過(guò)程，很多寬帶噪聲通過(guò)窄帶系統(tǒng)后也成為窄帶過(guò)程。因此，重點(diǎn)討論窄帶過(guò)程是很有必要的，當(dāng)中放定理：當(dāng)X(t為窄帶隨機(jī)過(guò)程，即X(tDww0，A(t)和f(t) b(t)是低頻限帶隨機(jī)過(guò)程即它們的功率譜只在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此式說(shuō)明：若t1，在tt+ta(t)wca(t)因?yàn)閣

，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)-

足夠小時(shí)，對(duì)于給定的e>右式趨近于0x(t為窄帶隨機(jī)過(guò)程時(shí)，在一個(gè)高頻周期T0內(nèi)，a(t)的變化大于e的概率趨于0。也就是說(shuō)，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節(jié)性質(zhì)

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