隨機信號通過線性系統(tǒng)kejian

y(t) LTI 同的時移，即y(t)L[x(t)]L[ax1(t)bx2(t)]aL[x1(t)]bL[x2(t)]UnitUnitImpulseResponse&Frequencyy(t)x(t)h(t)h(u)x(tY()X()H();Y(s)X(s)If If If If|Y(|2|X(|2|H(|2 h(t) )h(t))exp),jt000~CR~CR

h(t)exp2(tt)2jH()

1

0h(t)(t)

et/(L~RH(L~R2h(t)2

RReRt/

ejt0(0),H()

/2

/2 h(t)2

sin(tt0

eR~LH(R~L

RjLR

(tt01RH()~1RH()~C11*et/(20

eRt/LX(t,r1)X(t,r2)x2

y(t,r1)0h()x1(ty(t,r2)0h()x2(tX(t,rm)xm

y(t,r) (t Y(t)X(t)h(t)h(u)X(th(t)X(t)X(u)h(t Y(t)0h(u)X(tu)duX(u)h(tmY(t)Eh(u)X(tu)du 0h(u)EX(tu)du0

0

(tmX(t)h(t) (t,t)E[X(t)Y(t)]EX(t

RX(t1,t2)h(t2)RYX(t1,t2)RX(t1,t2)h(t1)

h(u)RX(t1,

RY(t1,t2)E[Y(t1)Y(t2E

0h(u)h(v)E[X(t1u)X(t2 0h(u)h(v)RX(t1u,t2RX(t1,t2)h(t1)h(t2RXY(t1,t2)RX(t1,t2)h(t2RYX(t1,t2)RX(t1,t2)h(t1RY(t1,t2)RX(t1,t2)h(t1)h(t2RYX(t1,t2)h(t2)RXY(t1,t2)h(t1E[Y(t1)Y(t2 Y(tnE[X(t1)X(t2 X(tn)]h(t1)h(t2) h(tn

mX(t) mY(t)

mX(t)mXconst. mY(t)mY

RX(t,t)

X(t)

Y(t)RY(t,

)

R(0)E[X2(t)]

R(0)E[Y2(t)]

X(t):SSSY(t):

X(t)

Y(t)

X(t)X(t) ()

mY(t)mX0h()d RY(t,t) 0h(u)h(v)RX(vh()h()RX()RYRXY(t,t)0h(u)RX(u)duRX()h()RXY()RYX(t,t)RX()h()RYX()RY()RXY()h()RYX()h() 0h(u)h(v)RX(uv)dudv 0|h(u)||h(v)||RX(uv)|dudv RX 0|h(u)||h(v)|dudvRX(0)0|h(u)|du0|h(v)|)統(tǒng)，若輸入信號是均值為mX和相關函數(shù)為RX()的寬平穩(wěn)信號，則連RXY()RX()

()

()h()

YYRRYXRY()RX()h()h(RXY()RYX()h()

RYX

Y(t)li

T TTTlim1h(u)X(tu)duT Tlim1X(tu)dtT 0mXh(u)duY(t)Y(tlim1

Y(t)Y(tT TTlim1h(u)X(tu)duh(v)X(tv)dvT

li

TX(tu)X(tv)dt 0h(u)h(v)RX(uv)dudvRYK tY(t)X(t)h(t)0h(u)X(t h(u)X(tu)U(tu)du0h(u)X(t0mY(t)E{Y(t)}th(u)X(t0t h(u)E{X(tu)}dut

th(u)dut

R

RY()00RX(12)h(1)h(2R()

h()

N0()h()d h()h(

2

2

e1e(1)U( :例題：設有白噪聲電壓X(t)，其自相關函數(shù)為X(例題：設有白噪聲電壓X(t)，其自相關函數(shù)為X(N0()，將它加入2

1 當0

N RY()

1

由于RY(R()N0e||, 輸出功率為RY(0)N04

mY(t)mX0h()d

H()

0

h()dmY

RY()RX()h()h()SY()SX|H()|2S()

SY(s)SX

|H()|

SYSX

RXY()RX()h()RYX()RX()SXY()SX()H()SXY()/SXSXY()/SYX()H()/ej2H

()R()1 S

1

RY(0)2SY

2X1 X1

2S

jH(s)H(s)SXrh()h()h()

y(n)LTILTI y(nmL[x(nm)]L[ax1(n)bx2(n)]aL[x1(n)]bL[x2(n)]UnitUnitImpulseResponse&Frequencyy(n)x(n)h(n) h(k)x(nk)Y(ej)X(ej)H(ej);Y(z)X(z)If If

If If

Y(n)X(n)h(n)h(k)X(nk)

mY(n)E[Y(n)]h(k)E[X(nh(k)mX(nk)

RXY(n,nm)E[X(n)Y(nm)]EX(n)h(k)X(nmk) h(k)E[X(n)X(nmh(k)RX(n,nmk)RYX(n,nm)h(k)RX(nk,nRY(n,nm)E[Y(n)Y(n

h(k)X(nk)

h(v)X(nmv)Ek0

h(k)h(v)E[X(nk)X(nm k0 h(k)h(v)RX(nk,nmk0

mY(n)h(k)mX(nk)mXh(k) RXY(n,nm)h(k)RX(n,nmk)h(m)RX(m)RXYRYX(n,nm)h(k)RX(nk,nh(m)RX(m)RYX

RY(n,nm)h(k)h(v)RX(nk,nmk0 h(k)h(v)RX(km k0h(m)h(m)RX(m)RY RY(0)h(k)h(v)RX(k k0

mYmx

h(k)mx

h(k)z

k

k0 RXY(m)h(m)RX(m) SXY(z) H(z)SX SXY )SX RYX(m)h(m)RX(m)

SYX ???

RY(m)h(m)h(m)RX(m)SY(z)

)SX

SY )

SX R(0) S(z)z1dz |H(ej)|2S(ej 2j

2 SY(s)SY

SY()|H()|2 SY ) S()S(s)S(s)S(s) yH(s)S y S(ej)S(z)Si(z)So(z) yH(z)Si ySY(s)H(s)H(s)SX SY()|H()|2SXSY(z) )SX

SY )

SX(e SX()SX(s)S(s)S(s)S(s)S XX

H(s)[S SX(ej)SX(z)Si(z)So(z)Si(z)Si(z XX

H(z)[Si white

band-passBBRY(m)E[Y(n)Y(n

n

nm EX2B

NfY(y1,y2,...,yN;n1,n2,...,nN)fY(yn

2(2NB)N/

exp2NByn例題：信號X(t)例題：信號X(t)示，其中W2B。按每秒2BY(n){X[n/(2B)],n0,12,...}。求該序列的自相關函數(shù)以及N維概率密度函數(shù)。

10SBS(f S(feq 0XX(S(fS S0XXRXS(fXSX2SXSX 1 1

SX() 2SXRX 1S

Beqc1/c cRX 2RX

K

E[Y2(t)]

N0/

N 0eK2d N E[Y2(t)]

|H()0

eK2dN0K

21

|H()|2d E[Y2(t)]N0e|H() RCC|H()|2|H(j)|H(j)

r(t)h(t)h(t)|H()e

|H(0) |H(0) e

RC Y(t)Y(t)0h()X(ttX()h(tlimX(k)h(tkN0tc<<k<<

kkΔK硬限制：g(x線性半波檢波：x0,g(x0;x0,g(x)線性全波檢波：g(x)|x平方器：g(x二次失真：g(xx 度函數(shù)fX(x1,x2;t1,t2)；fY(y)fX[xh(y)]|dh(y)dy|fY1Y2(y1,y2)|J|fX1X2[x1h1(y1,y2),x2h2(y1,y2)]

ey/fY(y)

2

fX(y)

2yfX y)

,yy fX(x1,x2;t1,t2)；輸出均值：mY(tE{Y(tg(x)fX

(t,t)

)g(x)f(x,x;t,

)dx

例：例：隨機信號X(t)Acos0tBsin0t，其中A和B是兩個 存在兩個時刻，X(t)AB的相同

g(x)

DRY()E[Y(t)Y(t

E1

2

G(v)ejvX(t)dv4

G(u)G(v)E{ej[uX(t)vX(t)]R()

Y

R() G(s)G(s)C(s,s;)ds 42

kC(u, (1)k(uv)kC(u,v;) YkR() Y

G(u)G(v)(1)k(uv)