隨機(jī)信號(hào)課件過(guò)程的譜分析kejian

3.1、隨機(jī)3.1.13.1.1、簡(jiǎn)單回顧（確定信號(hào)

x2t3.1、隨機(jī)3.1.13.1.1、簡(jiǎn)單回顧（確定信號(hào) lim1

x(t)dt 222222

2T

Ax

xtt3.1、隨機(jī)3.1.13.1.1、簡(jiǎn)單回顧（確定信號(hào)X() x(t)e3.1、隨機(jī)3.1.13.1.1、簡(jiǎn)單回顧（確定信號(hào)x(t)

x2(t)dt

x(t)1X()ejtd

1

x(t)ejtdt

X()X()d

|X()|23.1、隨機(jī)3.1.13.1.1、簡(jiǎn)單回顧（確定信號(hào)P

1

x2(t)dt

1

|X(T,)|2TtTtT

T Tx2(t)dt

x2

xT(t)XT(

1

|X()|2TT T

P

1

x2(t)dt

1

|X()|2Tlim dTT

T 3.1、隨機(jī)3.1.23.1.2、隨機(jī)信號(hào)的（平均）功率譜密度PowerSpectraDensity: t3.1、隨機(jī)3.1.23.1.2、隨機(jī)信號(hào)的（平均）功率譜密度PowerSpectraDensity:確定信號(hào)功率譜密度S

|X()|2 T

隨機(jī)信號(hào)平均功率譜密 |X(T,)|2 E[|X(T,)|2S()Elim

T

隨機(jī)信隨機(jī)tT3.1、隨機(jī)T3.1.23.1.2、隨機(jī)信號(hào)的（平均）功率譜密度PowerSpectraDensity:TElimT

X2(t)dt

lim

2T

T2TQ 1AE[X2Q 1 |X(T,)|2 E lim d

1

E|X(T,)|2 lim T S2 3.1、隨機(jī)3.1.23.1.2、隨機(jī)信號(hào)的（平均）功率譜密度PowerSpectraDensity:E[X2(t)]1 S2 、隨機(jī)ExEx.3.1：平穩(wěn)隨機(jī)信號(hào)X(t)X(t)abcos(0t其中a，b和0均為常數(shù)，：U[0,2)，求X(t)解XX(T,)

T0[abcos(t)]eT0

bejsin[(0

ejsin[(0XE[|X(T,)|2]X

sin2

b2

sin2[()T]sin2[(000 ( ( 、隨機(jī)S()lim1E[|X(T,)|2 T sin(T)lim2a2T T

b2Tsin[( b2Tsin[()T]2

(0

(0

sin(T)下面求lim

（1）當(dāng)0時(shí)，由于sin(T1，所sin(T)lim

0,3.1、隨機(jī)（2）當(dāng)0時(shí)，由于sin(T1sin(T)綜上

lim

,sin(T)lim

又因2T[sin(T)]2x(t)，其中x(t)為三角波，如下圖所1010Tt

、隨機(jī)所

Tsin(T)

dx(0)2 2

K()dK最終可以得到S()2a2()b2()b2( 3.1.3、功率譜密度與復(fù)頻 斯變換 cex(t)X(s):s x2(t)dt

X(s)estds

x(t)estdt1

XT(t)XXSX(s)lim EXX(T,s)XX(T,T

s

SX系統(tǒng)分析\求平均功率QAE[X2系統(tǒng)分析\求平均功率

SXEx.3.2:WSS.X(t),PSD:S),E[X(t)]2X429解：SXsSX|js

(s2 (s2)(s10s2 (s1)(s3)(s1)(sQE[X2(t)]

S

2j r3 r3 3mm (s1)(s3)(s3)

(s2)(s (s1)(s3)(s3) E[X2(t)]35 3.2.1、PropertiesofWSS

S()

E[|XX(T,)|2] 2T

S()S* SX()SX()onlyforrealRP:X(T,)X*(T,)|X(T,)|2|X(T,)

S()d:E[X2(t)]XX XX1

X2Xsin

| 28642 ,e(1)

63234 j621422 6323.2.2、有理譜分解定rationalaakqSX() :(P4)pqqkbk

X S(s)a2(sa1 (sa2p),aX (sb1 (sb2q Bydefinition:SX(s)SX RealX(t):SX(s)SX(sS(s)

a

S (s1(sp (s* (s* (s1(sp (s* (s*

(s* (s*) S SX極點(diǎn)全在s左平零點(diǎn)在s左平面或虛軸

s右平零點(diǎn)在s右平面或虛軸 維納- RX()SX

R()1 SX X

SX()

()eW.S.S維納- 理的證證明：若X(t)為寬平穩(wěn)隨E[|X(T,)|2 SX()lim lim E[XX(T,)XX(T, Tlim1E

TX(t)ejt1

T

1 2lim1 TE[X(t)X(t)]ej(tt)dt T2TT

lim1 TR

t)ej(t2t1)dtSX()RX(SX()RX(R()XS(f)ej2fdf1X2SX

W.S.S維納- 理的證t2t1tt1t1tt2tdt1dtdt2 t

tS()lim

R

t)ej(t2t1)dt

2T

lim

R()e

2T lim1

T2T

lim1 (2T ()ejd

2T(2T ()eT2T

1||R()ejdR()eX

2T 維納- 理（續(xù) 200-201頁(yè)證明：無(wú)論隨機(jī)過(guò)程平穩(wěn)與否，總存在功率譜密度SX()

R(t,t)eXR(t,t)X

SX**平均功

2 S

E[X2(t)] S()dR2

2 **利用自相關(guān)函數(shù)和功率譜密度皆為偶函數(shù)的性質(zhì)，又可將 S() R()cosd,

() S

維納- 理（續(xù)Ex.3.3：隨機(jī)信號(hào)X(t)X(t)acos(0t式中a和0均為常數(shù)，~U(0/2)，求X(t)的平）通過(guò)均方值的時(shí)間平均值求平均E[X2(t)]E[a2cos2(t0 a2 22cos(20t2) 2 /2222

cos(20t

sin(2 維納- 理（續(xù)顯然，這個(gè)隨機(jī)號(hào)的平均功率QAE[X2

1

aE[X2(t)]dtaT （2）利用維納- 理，即通過(guò)自相關(guān)函數(shù)時(shí)間平均值的傅葉變換獲得信號(hào)功率譜密度，再對(duì)頻率積分RX(t,t)a2E{cos(0t)cos[0(t)2acos()a2E{cos[(2t)2 a2

/222cos(0)2

2

cos[0(2t)acos()a2sin[(2t 維納- 理（續(xù)所ARX(t,t 2于S()a2()a2(X平均功

Q S()d維納- 理（續(xù)Ex.Ex.3.4：若平穩(wěn)信號(hào)X(t)的功率譜密度為SX()，又Y(t)式中a和0均為常數(shù)，求Y(t)的功解：RY(tt2RX){cos(0cos[0(2t)]}ARY(t,t 2RX于SX SY/2

/2

~U(0,2~U(0,2]，證明 () ()的概率密度函數(shù)為f()f()，兩獨(dú)立的Ex.3.5：已知X(tacos(t)，其中a為實(shí)常數(shù)，和XXRX(t,t)E{X(t)X(t1a2E{cos(2t2)}E(cos)1a2E(cos)1

cosf1

1[ejej]f2

2 2 d

f

2IFT{f所以SX()a ()RX()SXea| 2/ a 1|t|/T,|t| 4sin2(T/2)/(T2T ea||cos(0 1/0 RX()SXsin00// 10 1 1 0 、平X(nX(tT均勻采樣后得到的廣義關(guān)函數(shù)序列是X(t)自相關(guān)函數(shù)的采樣序列，即X(n)X(nT)X(t)RX(m)E[X(n)X(nRX()

二、離散維納- R(m)

S

E[|X(n)|2]R(0)

SS()S()XRX

、譜 z zXmR(m)zm, (m)1XX2jS(z)zm1dzX

R(m)R(m),S(z)S(z1

S(z)B(z)B(z1X(z (z

(z1 (z1B(z)C (z1) (zM

B(z1) (z1 (z1極點(diǎn)在零點(diǎn)在單位圓

極點(diǎn)在單位圓零點(diǎn)在單位圓、聯(lián)合平穩(wěn)隨機(jī)過(guò)程的互譜密、X(t和Y(t互功率的

Elim2TTX(t)Y(t)dtElim2TXT(t)YT(t)dtT TlimT

E{XRXY(t,

QElimQElimTY

Y(t)X(t)dt

2T

2T

ttT1 (t,t)T1 注意：對(duì)于實(shí)隨機(jī)信號(hào)QXY3.5、聯(lián)合平穩(wěn)隨機(jī)過(guò)程的互譜密TX(t)Y(t)dtX(t)YTX(t)Y(t)dtX(t)YTTXX(t)Y(t)TT1212X(T,X)Y(T,)12X(T,X)XY(T, X(T,)XXY(T,TX(t)Y(t)dt1X(T,)X(T,XY1 X(T,)X(T,Y El EliX(t)Y(t)dtEli

2T

2T2 X(T,)XX(T,)X(T,

2

E{X(T, (T,)}

3.5、聯(lián)合平穩(wěn)隨機(jī)過(guò)程的互譜密、互功率譜密度的第一種TT

Y(t)X(t)dt

Y(t)X Y(t)X(t)1X(T,)X(T,)

X(T,)X(T,

X(T,)X(T,

X (T,T

(t)X(t)dt

XX(T,)X(T, El EliY(t)X(t)dtEli

2T

2T2

2

E{X(T, (T,)}

3.5、聯(lián)合平穩(wěn)隨機(jī)過(guò)程的互譜密、互功率譜密度的第二種 ()lim1E{X(T, T X(T, (T,)X()Y X(t)Y(

X(t)Y(T E{X(T,

E{X(t)Y(t)}dteT1

RXY(t,t)1

T

T2T () E{X(T,)X(T,)}

RXY RXY(t,t)eT

(t,t)dtelim

T2T 3.5、聯(lián)合平穩(wěn)隨機(jī)過(guò)程的互譜密、互功率譜密度的第二種 ()lim1E{X(T, T X(T, (T,)Y()X Y(t)X(

Y(t)X(T E{X(T,

E{Y(t)X(t)}dteT1

RYX(t,t)1

T

T2T () E{X(T, (T,)}

TT

(t,t)dte11

(t,t

T

3.5、聯(lián)合平穩(wěn)隨機(jī)過(guò)程的互譜密若兩隨機(jī)過(guò)程聯(lián) (t,t)

()

()e

(t,t)

SYX()

()eQ

(t,t)(t,t)

XY(0)(0)

1

SXY YX

() (),

SXY()dQYX

SYXS對(duì)于復(fù)隨機(jī)信S () (),

()d

3.5、聯(lián)合平穩(wěn)隨機(jī)過(guò)程的互譜密Ex.Ex.3.6：已知X(t)的自相關(guān)函數(shù)為RX)，SXX()RXX(t,t)E{X(t)X(t X(tT)X(t)EX(t)l

l