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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
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