北郵通信原理第3章隨機(jī)過程

3.13.1 WirelessSupportPeopleF1(x1,t1)=P{X(t1)￡x1?F1(x1,t1)=p(x,t?x1

WirelessSupportPeopleFn(x1,x2,",xn;t1,t2,",tn=P{X(t1)￡x1,X(t2)￡x2,",X(tn)￡xn?Fn(x1,x2,",xn;t1,t2,",tn)=?x1?x2 ?x

n(x1,x

,",

n;t1,t2,",tn WirelessSupportPeopleE[X(t)]=-

xp1(x,t)dx=mX(tD[X(t)]=E{X(t)-E[X(t)]}2=s2

(t

sX(t WirelessSupportPeopleCX(t1,t2)=E{[X(t1)-mX(t1)][X(t2)-mX(t2)]} RX(t1,t2)=E[X(t1)X(t2)]=-￥-￥x1x2p2(x1,x2;t1,t2)dx1dxr(t,t)

CX(t1,t2

s(t)s(t WirelessSupportPeopleFn,m(x1,x2,",xn,t1,t2,",tn;y1,y2,",ym,,",=P{X(t1)￡x1,X(t2)￡x2,",X(tn)￡xnY()￡y1,Y(t)￡y2,",Y(t)￡ym?Fn,m(x1,x2,",xn,t1,t2,",tn;y1,y2,",ym,t1,t,",t?x1?x2 ?xn?y1?y2 ?y=pn,m(x1,x2,",xn,t1,t2,",tn;y1,y2,",ym,t1,t,",t WirelessSupportPeopleCXY(t1,t2)=E{[X(t1)-mX(t1)][Y(t2)-mY(t2)]}RXY(t1,t2)=E[X(t1)Y(t2)] WirelessSupportPeople WirelessSupportPeoplepnx1,x2,…xn;t1,t2,…tn=fnx1,x2,…xn;t1+t,t2+t,…tn+t，"一維分布與時(shí)間無關(guān)，二維分布只與時(shí)間間隔t1t211+tx1x1x1+tx1 x2;t1-t2X t1,t2=Rt1=X)=

Ct

=

=C 1 WirelessSupportPeopleX

Rt1,t2=RXY(t1,t2)=E[X(t1)Y(t2)]=RXY(tt=t2- WirelessSupportPeople

R(t)￡tt 統(tǒng)計(jì)平均功率：EX2ttfit

tfitttfittt WirelessSupport WirelessSupportPeople

=R(0)-R(￥TT

xt)=lim1

xtdtTfi

PEXt=xt=

ttt=

tt+tdtTTfi TP

xt)xt+t=1，則稱X(t) WirelessSupportPeople ￥ ￥

<￥ WirelessSupportPeopleP(f)=

(f)x(t)=x(t

t?FT(f2FT(f2Tfi

tˇ (f)=E (f)=lim

EF EF

(f)2

Tfi WirelessSupportPeople RX PX(fEF(f2 TT X(t)e-j2pft 2X(tej2pftdtTTT T =E =E

- XX(t)X(te-j2pf(t-t)dttTT TT XT=T XT=T

t)e WirelessSupportPeople 令t=t-t

2

2T2

T-t ￠T

Fw

=

--t

R(t)e-j2p TtTtT1T T1

T-T-

R(t)e-j2pft

2

R(t)e-jwt(1-X X

2T

dtt

X (t)e-j2pft(1X

T WirelessSupportPeopleTT)1E )T

R(t)e-j2pft(1-

￥(f2 R(t)e-j2pftdt￥ Tfi￥

WirelessSupportPeople?

0=￥￥

PX

0￥￥

RXt單邊功率譜密度

(f)=2PX(f f>

f< WirelessSupportPeople tsinw0,0, 量.求其數(shù)字特征。tcosq+sinq1sinw0=1E{coswt-t-coswt+t+ =1cosw -t Xt是廣義平穩(wěn)隨機(jī)過程 + 4

WirelessSupportPeople

f(x)

11

x2fx)

s2

a對稱：f(a+xf(a-1

s1>s

f(x)dx

f(x)dx=21afx左右平 sfx

WirelessSupportPeople n維正態(tài) x exp-1x-aTC-1x-aXpnX

C1 其中：a=E[x]是 T-

DX WirelessSupportPeople

)與X(t)兩兩不相關(guān)，即： ,s

exp-1x-aTC-1x-a

Xpn2X1

1 s x-a2s

ipni

i

2i

x- 2 i exp- ii

n= Wireless WirelessSupportPeople

xi -x2/2 xx F(x)

e-z2/2

x-a

Fx)=

dz=F s

xp

xe-z20

(2x-xx￥

Q(x)=

e-z2/2

=1-Fx

1

x WirelessSupportPeoplex(Xt

y(t)=Ytt

tth￥￥

htdt=mXH0)t,tYt ￥ t+vhuhvdudv =RX X-￥- WirelessSupportPeopleXt和Ytt1t1Yt2 =E[X(t

X(t2-￥ E[X(t)X ￥=-￥R(t2-u-t1==RX*t=RXYPXY(fFRXYPX(fH(f WirelessSupportPeople例:若輸入Xt為白噪聲, (f)=N0,(-￥<f<￥ =N0d =N0d*h=N0ht) TX(t-t)Y(t延遲延遲0T1T

T

2t WirelessSupportPeople例.已知Xt的自相關(guān)函數(shù)RX求Xt和Xt的相關(guān)函數(shù)RXXtX(t)

*X(t

H(f)=-jsgn(f t *1=

RX偶函數(shù)fi?X奇函數(shù)fiRXt與Xt

t奇函數(shù)fi

0tXt 過程;Xt Xt與Xt

t WirelessSupportPeople,+ttYt =-￥-

thuhvdudv

t=￥[ ￥ t+u-vhuhvdudv]e-j2pftdt-￥-￥- ￥￥￥ hu)hv) t+-￥￥￥ ￥￥￥￥￥￥

huej2pfu ve-

a-j2pfa X=H*(f)H(fX

X=H(f)2 X WirelessSupportPeople ?Y(f)=PX

(f)=

(f WirelessSupportPeople P(f)=N0 -￥<f< R(t)=N0 orR(t-t)

N0d(t-t

WirelessSupportPeople 3.6 T若X= TTN0TN=0f2=0f2 證：EXEn(t)f(t)dtTE{n(t)}f(t)dt DX=En(t1)f(t1)dt1iTn(t2)f(t2)dt2 =0

= T

-t

)dt0

Tf2(t2

Tf2(t)dt0

XX

X WirelessSupportPeopleX T 3.6 TTX1T

n(t)f1(t)dt,X2=

n(t)f2

為確定函數(shù)，則E(XX)= 1212

2

T0 f(t)f(t)T0 E{XX}E{XX}=

1

Tn(tTn(t)f(tTn(t)f(t=0

= TN0 -t)f(t)f(t)dt0

2 WirelessSupportPeople 3.6 / nP(f)= n ￥￥

P(f)ej2pftdfnn

ej2pfHt-e-j2p= 0

ej2pftdf= -

=N02sin2pfHt=

R =

=s2Sapf

fHtkn2

n(t)與n(t2

) H WirelessSupportPeople WirelessSupportPeople WirelessSupportPeople=a(t)cosf(t)coswct-a(t)sinf(t)sinwctcoswctsinwcXct=a(tcosf(t):同相分量，Xst=a(tsinf(ta(t)=X

t+

f(tarctan XtX(t)=Re

Xct=ReX(t)ejwct LZ(t)=X(t)+jX(t)=X(t)ejwctLXL(t)=Xc(t)+jXs(t)=at WirelessSupportPeopleZt的自相關(guān)函數(shù)和功率譜密度RZ=E*tZt

R?

t

)

t?

t=u

R t? WirelessSupportPeopleXLt =tXLt L=EZ*tej2pfctZt+te-j2pfct+t}

e-j2p =L (fLPZ(fPX(fPX(f?s(t)=?(t)coswct-X(t)sinw)]若X(t) WirelessSupportPeoplet (0)= (0)=R(0)=s RX

(0)=-RX

所以PX(wPX(w WirelessSupportPeoplePX(f),

(fPX(f之間關(guān)系 (f)=P(f)*1[d(f-f)+d(f+f +P(f)[-jsgn(f)]*1[d(f-f)-d(f+f 2 =1P(f-f)+P(f+f -1P(f-f)sgn(f-f)-P(f+f)sgn(f+f =PX(f-fc)u(fc-f)+PX(f+fc)u(f+fc WirelessSupportPeoplePX(fAu(f+fcfc+fPX(f+fc-20fPX(f-fcu(fc-f02fPPX(f)=PX(f 0fWirelessSupportPeople (f)= (f)

PX(f-fc)+PX(f+fc

-fc￡f￡

PXffcc

s

cos WirelessSupportPeople WirelessSupportPeoplepX

(xc,xs)=c1

(xc)s

(xsx

x 2s2 2s2 x

+x2 s WirelessSupportPeoplea(t和f(tXc(t)=a(t)cosf(t)Xs(t)=a(t)sinpaf(a,f)=pXX[Xc(a,f),Xs(a,f)] J

=cosf(t)

WirelessSupportPeople

2=aexp(-

),a?

pf(f)=-￥paf(a,f)da= =1,0￡f￡

pa(apf(fpaf(a,,fa(t)f(t) WirelessSupportPeople令X(t)=Acoswct+ n(t)=nc(t)coswct+ns(t)sinwctX(t)=[A+nc(t)]coswct+ns(t)sin令c(t)Anc(t(t) 過程，同一時(shí)刻獨(dú)立 (t) ns(t)均值為

n(tn(t的方差均為s WirelessSupportPeople(t)ns(t)的聯(lián)合概率密度（同一時(shí)刻 (n,

exp[- s

X(t)=R(t)coswct+q(t)=R(t)coswctcosq(t)-R(t)coswctc：n(t)cns(t)=R(t)

R(t)n2(t)+n2n2(t)+n2csn(t) WirelessSupportPeopleX(t)R(tcos[wct+q(t)]R(t:包絡(luò)，q(t): pnn[n(r,q), J,

r(r,q(r,q(r,q(r,q2

+A2-2rAcosq r?0,0￡q￡ WirelessSupportPeople

r

2p

pR(r)=

s

p1BesselI0x)p1

2pR(r)=s2

)I0

),r?——萊斯分布/廣義瑞利分布vraA

2

)I0 WirelessSupportPeople匹配濾波器：某一特定時(shí)刻的輸出信噪比)tt+)tt+t=tt0 WirelessSupportPeople tttst)S(f

ttt

tttn(t) ttt200

sot

n2

￥￥

H

eEn2(t)

P(f)df

￥

2 -￥

-￥￥輸出信噪比￥st

2？H(fS(fej2pft0？=ro=

En2(t

Wireless WirelessSupportPeople

22￥X￥

2￥2￥ WirelessSupportPeople￥2￥2Se￥2￥2

N0

N0

222￡ 222￡

￥￡-

222

=2ENo

E￥￥

o\o

-0 W