Probability Random Variables Stochastic Process

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2 2 S)LJD VKRXOGUHDG -------------- e –( x – 4.5 ) ⁄ 4.5  3 2π 1 2 S)LJE VKRXOGUHDG ----------e – ( x – 3.0 ) ⁄ 4  4π S3UREOHPVKRXOGUHDG³QHWJDLQRUORVVH[FHHGV´ S3UREOHPVKRXOGUHDG³:HSLFNDWUDQGRP n ≤ N FRPSRQHQWV´

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Errata-1

EH J V\PEROVZKLFKZHUHGURSSHGIURPWKH)RXUWK(GLWLRQ 1RWHWKDW x 1 –x2 ⁄ 2 VRWKDW G ( x ) = ∫ J ( ξ ) dξ J ( x ) = ----------e 2π –∞ FRPSDUHZLWK(T DQGVHH1RWDWLRQDO3HFXODULWLHVRIWKH+LQWVVHFWLRQ  ‡ S7DEOHQRWHWKDWWKH7H[WGHILQLWLRQRI erf ( x ) LVQRQVWDQGDUG7KHUHODWLRQ VKLSLV 1 x 1 x erf ( x ) = --- erf  ------- = --- Φ  ------- 2 2 2 2 ZKHUHWKHILUVWWHUPLVWKH7H[WGHILQLWLRQWKHVHFRQGWHUPLVWKHVWDQGDUGGHILQLWLRQ DQGWKHWKLUGWHUPLVWKH³*UDGVKWH\Q 5\]KLN´QRWDWLRQ ‡ SILUVWOLQHDIWHU(T WKHUHVWULFWLRQ x > 0 LVQRWQHHGHG

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S3UREOHPVKRXOGUHDG³8VLQJ7DEOH´DQGLQSDUWE ³ x LV N ( η, σ ) ´ S3UREOHPVKRXOGUHDG³ x LV N ( 10, 1 ) ´ S3UREOHPVKRXOGUHDG³ x LV N ( 0, 4 ) ´ S3UREOHPVKRXOGUHDG³ x LV N ( 1000, 400 ) ´ S3UREOHPWKHUHIHUHQFHWR(T LVQRWFRUUHFW7KHDFWXDOHTXDWLRQ LQWHQGHGZDVUHPRYHGLQJRLQJWRWKH)RXUWK(GLWLRQ(T LVDFORVHHTXLYDOHQW WRZKDWZDVLQWHQGHG S3UREOHPVKRXOGUHDG³WKHQ F ( x ) = 1 IRU x ≥ b ´ S3UREOHPVKRXOGUHDG³$V\VWHPKDVFRPSRQHQWV´ S3UREOHPWKHODVW³!´VKRXOGEHD³³LQWKH+LQW S3UREOHPVKRXOGVD\LQDGGLWLRQ³$VVXPHDOVR k 1, k 2 « n, k 3 ´

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SHTXDWLRQDIWHU(T WKHODVWWHUPVKRXOGEH U ( x ) 

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S([DPSOHQH[WWRODVWOLQHVKRXOGUHDG³WKDW E { ( x – η ) 2 } = σ 2 DQG´ SHTXDWLRQIROORZLQJ(T VKRXOGUHDG P{ x – η ≥ ε} =

η–ε

∫–∞

f ( x ) dx + … 2

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S([DPSOHVKRXOGUHDG³IXQFWLRQRIDQ N ( η, σ ) UDQGRPYDULDEOH´ SWZROLQHVDIWHU(T WKHULJKWKDQGHTXDWLRQVKRXOGKDYHDGRXEOHSULPH RQWKHSKLRQWKHOHIWKDQGVLGH S3UREOHPVKRXOGUHDG N ( 5, 4 ) 

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S3UREOHPVKRXOGUHDG y =

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S3UREOHPD VKRXOGUHDG Φ x ( ω ) = ( 1 – jωβ )

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Φ x ( ω ) = ( 1 – j2ω ) ‡

–n ⁄ 2

x –α

DQGE VKRXOGUHDG



S3UREOHPE WKHHTXDWLRQOLQHVKRXOGHQG k ≤ min ( M, n ) 

Errata-2

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S(T ORZHUOLPLWRQWKHILUVWLQWHJUDOVKRXOGUHDG y = – ∞  S(T VHH6SHFLDO1RWHLQWKH+LQWVVHFWLRQRIWKLVYROXPH S([DPSOHOLQHVKRXOGEHJLQ³ φ ( s 1, s 2 ) = e A ´

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S3UREOHPVHFRQGFHQWHUHGHTXDWLRQOHIWVLGHVKRXOGUHDG f z ( z ) =  S3UREOHPVHH6SHFLDO1RWHLQWKH+LQWVVHFWLRQRIWKLVYROXPH,WLVHDVL HVWLILWUHDGV³H[FHHGV 2 ⁄ λ ´DQG³RULJLQDOFRPSRQHQWE\ 1 ⁄ λ "´ S3UREOHPVKRXOGDVNWRVKRZWKDW w LVDQH[SRQHQWLDOUDQGRPYDULDEOH S3UREOHPKDV³H[FHVV´LQIRUPDWLRQ6HHWKHKLQWIRUWKLVSUREOHP S3UREOHPWKHILUVWGLVSOD\HTXDWLRQVKRXOGUHDG

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 E{z} =

∑ pn E { g ( xn, y ) xn } n

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S3UREOHPVKRXOGUHDG³ β y ( t ) = f y ( t ( y > t ) ) DQG´

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S(T VKRXOGQRWKDYHFRPPDVLQWKHGHQRPLQDWRURIWKHIUDFWLRQ SOLQHVKRXOGUHDG³DVLQ([DPSOH´ SVHFRQGOLQHRIWKHVHFWLRQ³(UJRGLFLW\´VKRXOGUHIHUWR6HF S([DPSOHVKRXOGUHDG³LQWKHLQWHUYDO ( 0, T ) ´ S(T VKRXOGUHDG ∞

∫–∞ x α fi ( x ) dx < K < ∞ IRUDOO i

S(T VKRXOGUHDG E { xi 3 } ≤ cσ i2 DOO i S([DPSOHLQWZRSODFHVWKHUHIHUHQFHWR VKRXOGEHWR  S3UREOHPVKRXOGEHJLQ³DUHQRUPDOXQFRUUHODWHGZLWK]HURPHDQ´

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S([DPSOHUHVXOWVDUHYDOLGIRUSRVLWLYHWLPHVRQO\7KHULJKWKDQGWHUPRI WKHHTXDWLRQIROORZLQJ(T LVQRWFRUUHFWLI t 1 = t 2 = t ZKHUHLWSURGXFHV 2λt  &RUUHFWO\ C ( t, t ) = λt  S HTXDWLRQVIROORZLQJ(T DUHRQO\WUXHLI t 2 ≤ t 1 ,QFDVH t 1 ≤ t 2 DOO WKH t 2 ¶VLQWKHVHHTXDWLRQVVKRXOGEHUHSODFHGE\ t 1 ¶V 2

b

∫a …

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S(TVKRXOGUHDG E { s – η s } =

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SILUVWOLQHRI³3URRI´QHDUSDJHERWWRPVKRXOGUHDG³>6HH @´ S([DPSOHVKRXOGDVVXPHD666SURFHVV SHTXDWLRQDIWHU(T VKRXOGKDYH g ( x ) = +c ZKHQ x > c DQG g ( x ) = – c 

Errata-3

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ZKHQ x < – c  S(TV DQG DUHRQO\WUXHIRUDUHDOV\VWHP:KHQWKHV\VWHPLV FRPSOH[WKHFRQMXJDWHV L 2* DQG h * VKRXOGDSSHDU

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SHTXDWLRQEHIRUH([DPSOHVKRXOGUHDG³ h ( α )h * ( β ) ´

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S(T VKRXOGEHJLQ³ a n y ( n ) ( t ) + … ´

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SQH[WWRODVWOLQHVKRXOGUHDG³FDVHVRI DQG ´1RWHWKDW H ( ω ) LV XVHGKHUHZLWKRXWDQ\LQWURGXFWLRQ,WLVWKHV\VWHPIXQFWLRQZKHUH ∞ 1 ∞ H ( ω ) = ∫ h ( t )e –jωt dt h ( t ) = ------ ∫ H ( ω )e jωt dω 2π –∞ –∞

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S(T XVHV ρ ( τ ) ZLWKRXWFDOOLQJLWWKHGHWHUPLQLVWLFDXWRFRUUHODWLRQRI h ( t ) DQGWKHQH[WOLQHRQS VKRXOGUHDG³ZKLWHQRLVHZLWKLQWHQVLW\ q  SWZROLQHVEHIRUH(T VKRXOGUHDG³,WLVWKXVDVLPSOHORZSDVVILOWHU´ S(T VKRXOGUHDG ( 2n ) ( τ ) S yy ( ω ) = jω 2n S xx ( ω ) R yy ( τ ) = ( – 1 ) n R xx

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1 S([DPSOHOLQHVKRXOGUHDG³HTXDOV ----- ´ πt

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S(T VKRXOGUHDG³ E { x ( t + τ 1 ) – x ( t ) 2 } = 0 ´

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S(T WKHVXPVKRXOGEHJLQDW m = 1  S([DPSOHQRWHWKDW a LVSUHVXPHGUHDODQGOLQHVKRXOGUHDG q R yy [ m ] = -------------2- a m 1–a S3UREOHPVKRXOGUHDG

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E{ w2( t) } = ‡

t

∫0 ( t – τ ) 2 q ( τ ) dτ

S3UREOHPVKRXOGUHDG 1 S y ( ω ) = 2πRx2 ( 0 )δ ( ω ) + --- S x ( ω ) * S x ( ω ) π

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S3UREOHPVKRXOGUHDG³ R [ 0 ]R [ 2 ] ≥ 2R 2 [ 1 ] – R 2 [ 0 ] ´

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S3UREOHPVKRXOGUHDG³ x [ n ] = Ae FDQQRWPDNHEROGJUHHNV\PEROV

jnω

´ ω LVDUDQGRPYDULDEOHKHUH,

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SOLQHDIWHU(T VKRXOGUHDG³ k = 1.37 × 10 – 23 -RXOHVGHJUHH.´ SILUVWZRUGVKRXOGUHDG³LQHUWLDKDVWKH´ SIRXUOLQHVDIWHU(T VKRXOGUHDG³DQG E { n 2 ( t ) } = λ 2 t 2 + λt ´ S(T VHFRQGSDUW VKRXOGUHDG³ Rxxˆ ( – τ ) = – R xxˆ ( τ ) ´

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SOLQHWKUHHVKRXOGUHDG³WKDW S zz ( ω ) LVVSHFLILHG´

Errata-4

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SWZROLQHVEHIRUH(T VKRXOGUHDG³66&6SURFHVVZLWKSHULRG T DQG S(T VKRXOGUHDG 1 S x ( ω ) = --- 6 c ( e jωT ) H ( ω ) 2 ´ T ∞

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S(T LVLPSURSHU7RIL[WKLVGHILQH w ( t ) =

∑ cn U ( t – nT ) IRU t ≥ 0  n=0

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ZLWKDQDSSURSULDWHDOWHUQDWHGHILQLWLRQIRU t < 0 1RZ w ( 0 - ) = 0  SODVWHTXDWLRQVKRXOGKDYHWKHWHUP R c [ n – r ] QRW R c ( n – r )  S(T VKRXOGUHDG ∞

∑

… =

∞

Rc [ m ]

m = –∞

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∑

δ [ t + τ – ( m + r )T ]δ ( t – rT )

r = –∞

SHTXDWLRQDIWHU(T VKRXOGUHDG ∞

T

∑ ∫0 δ [ t + τ – ( m + r )T ]δ ( t – rT ) dt

= δ ( τ – mT )

r = –∞

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S(T VKRXOGUHDG 1 S z ( ω ) = --T

∞

∑

m = –∞

1 R c [ m ]e – jmωT = --- S c ( ωT ) T

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S(T VKRXOGLQGLFDWHHTXDOLW\LQPHDQVTXDUH S(T SUHVXPHVWKHSURFHVVLVUHDO SILYHOLQHVEHIRUH(T VKRXOGUHDG³ T 0 ≤ π ⁄ σ ´

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SOLQHEHIRUH(T$ VKRXOGUHDG³IRUDQ\ c > 0 ´ SOLQHLVLQFRUUHFW6HHWKHVROXWLRQWR3UREOHP

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S3UREOHPHTXDWLRQVKRXOGUHDG³ S x ( ω ) = 2π -----2T

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S3UREOHPQHHGVWKHDVVXPSWLRQWKDW x ( t ) LVLQGHSHQGHQWRIDOO t i DQG 1 VKRXOGUHDG³ X c ( ω ) = --λ

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T

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x ( t i )e – jωti ´,WLVQRWQHHGHGWKDW E { x ( t ) } = 0 

ti < c

S3UREOHPVKRXOGUHDG³ y ( t ) = B cos ( ω 0 t + ϕ ) + y n ( t ) ´

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SWZROLQHVDIWHU(T VKRXOGUHDG³V\VWHPRI)LJLV´ SOLQHDIWHU(T VKRXOGUHDG³ α i = γ i / ( 1 ⁄ z i ) ´,FDQQRWGXSOLFDWHWKH 7H[WIRQWVEXWWKHUHLVDSUREOHPRIFRQVLVWHQF\KHUH SOLQHEHIRUH(T VKRXOGUHIHUWR([DPSOH

Errata-5

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SOLQHEHIRUH(T VKRXOGUHDG³ B ( – ω ) = – B ( ω ) ´ SOLQHDIWHU(TD VKRXOGUHIHUWR(T QRW  S3UREOHPVKRXOGUHDG –1 ⁄ 2 λn –1 ⁄ 2 n’ a + λ β n =  a + ----- β ’ = -----n   2 2

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S3UREOHPVKRXOGUHDG³ E { x n x k∗ } = … ´

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S(T VKRXOGUHDG³ ∫ C ( τ ) dτ < ∞ ´ 0

Errata-6

Gary Matchett

Hints — 9/5/02

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Hints — 9/5/02

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Gary Matchett

Hints — 9/5/02

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Gary Matchett

Hints — 9/5/02

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H-4

Gary Matchett

Hints — 9/5/02

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H-5

Gary Matchett

Hints — 9/5/02

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H-6

Gary Matchett

Hints — 9/5/02

n

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H-7

Gary Matchett

Hints — 9/5/02

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H-8

Gary Matchett

Hints — 9/5/02

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H-9

Gary Matchett

n

(p + q) =

Hints — 9/5/02

n

n

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H-10

Gary Matchett

Hints — 9/5/02

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H-11

Gary Matchett

Hints — 9/5/02

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H-12

Gary Matchett

Hints — 9/5/02

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H-13

Gary Matchett

Hints — 9/5/02

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H-14

Gary Matchett

Hints — 9/5/02

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H-15

Gary Matchett

Hints — 9/5/02

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H-16

Gary Matchett

Hints — 9/5/02

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H-17

Gary Matchett

Hints — 9/5/02

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H-18

Gary Matchett

Hints — 9/5/02

x i ≥ 0 LVQRWQHHGHG 0/HWHYHQW An EHWKDWKHDGVILUVWDSSHDUVRQWKH n ¶WKWRVVLQJ1RWHWKDW [ A 1, A 2, … ] LVDSDUWLWLRQ8VHWRWDOSUREDELOLW\WRILQG E { x1 } 1H[WHVWDEOLVK E { x m x m – 1 } E\DVLPLODUDUJXPHQWZKHUHHYHQW B n LVWKDWWKHILUVWDSSHDUDQFHRI KHDGVDIWHUWRVV x m – 1 LVRQWRVV ( x m – 1 + n ) )LQDOO\XVH E { x m } = E { E { x m x m – 1 } }  jωm

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H-19

Gary Matchett

Hints — 9/5/02

2

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H-20

Gary Matchett

Hints — 9/5/02

/HW f n ( y ) EHWKHGHQVLW\RI y n = y n – 1 + x n 8VH(TWRUHODWH f n ( ) WR f n – 1 ( )  DQGWR f x ( ) 6ROYHIRUWKHILUVWIHZGHQVLWLHVFRPSDUHZLWKWKH(UODQJGHQVLW\(T  DQGJXHVVWKHJHQHUDOUHVXOW7KHQFRQILUPWKHJHQHUDOUHVXOWE\LQGXFWLRQ 8VH3UREOHPDQGWKHFHQWUDOOLPLWWKHRUHP 1RKLQW 8VH3UREOHPEWRVKRZWKDWWKHVXPRI&DXFK\UDQGRPYDULDEOHVLV&DXFK\ DQGQHYHUEHFRPHVJDXVVLDQ 1RWHWKDW x DQG y DUHSUHVXPHGWREHQRUPDO8VH*RRGPDQ¶VWKHRUHP(T HYHQWKRXJK x DQG y DUHVFDODUV

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H-21

Gary Matchett

Hints — 9/5/02

x 1 x 2 SODQHZKHUH x 1 – x 2 ≥ a DQGLQWHJUDWHWKHVHFRQGRUGHUGHQVLW\RYHU D x  1RKLQW 'RQRWIRUJHWWKHFRPSOH[FRQMXJDWHLQ(T 8VH(T %,07KH7H[WGRHVQRWIDLUO\SUHSDUH\RXWRVROYHWKLVSUREOHP,QWKHGLVFXV VLRQRIOLQHDUFRQVWDQWFRHIILFLHQWGLIIHUHQWLDOHTXDWLRQVEHJLQQLQJRQSDJH WKH7H[WQRWHVWKDWVXFKHTXDWLRQVDUHQRWXQLTXHO\VROYDEOHZLWKRXWLQLWLDOFRQGL WLRQV7RDVVXUHDXQLTXHVROXWLRQDQGWRDVVXUHWKHOLQHDULW\FRQGLWLRQ(T LV VDWLVILHGWKH7H[WVD\VWKDWLWZLOOSUHVXPHDVROXWLRQZLWK³]HUR´LQLWLDOFRQGLWLRQVDW t = 0 7KHWURXEOHZLWKWKLVDSSURDFKLVWKDWWKHGLIIHUHQWLDOHTXDWLRQWKHQRQO\ KROGVIRU t ≥ 0 DQGWKHUHLVQRZD\WKDW y ( t ) FDQEH:66%\FOHDULPSOLFDWLRQRIWKLV SUREOHPWKHHTXDWLRQLVWRKROGIRUDOOWLPHDQG y ( t ) ZLOOEH:66 7KHZD\WRDFFRPSOLVKWKHVHJRDOVWRJHWKHULVWULFN\6XSSRVHZHVHWVRPHDUELWUDU\ LQLWLDOFRQGLWLRQVDW t = t 0 DQGSUHVXPHWKDWWKHGLIIHUHQWLDOHTXDWLRQKROGVIRU t ≥ t 0 1RZZHOHW t 0 → – ∞ 7KLVDSSURDFKDFFRPSOLVKHVDOOWKHREMHFWLYHVSURYLGHG WKDWWKHGLIIHUHQWLDOHTXDWLRQKDVDXQLTXH VROXWLRQXQGHUVXFKDVVXPSWLRQVDQG WKHHIIHFWRIWKHLQLWLDOFRQGLWLRQVDW t 0 RQWKHVROXWLRQDWVRPHIL[HG t GHFOLQHVWR]HUR DV t 0 → – ∞  7KLVLVWUXHIRUVWDEOHGLIIHUHQWLDOHTXDWLRQV7KHGLIIHUHQWLDOHTXDWLRQ an y

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H-22

Gary Matchett

Hints — 9/5/02

1 ∞ jωt h ( t ) ↔ H ( ω )  h ( t ) = ------ ∫ H ( ω )e dω 2π –∞ $OVRLQWKLVSUREOHPDVVXPH E { v ( t ) } = 0 7RPDNHWKLQJVPRUHPDQDJHDEOHGHILQH WKH]HURPHDQSURFHVV z ( t ) = y ( t ) – 2 DQGEHJLQZRUNZLWKLW -XVWQRWHWKDW f ( t 1 )g ( t 2 )δ ( t 1 – t 2 ) = f ( t 1 )g ( t 1 )δ ( t 1 – t 2 ) = f ( t 2 )g ( t 2 )δ ( t 1 – t 2 )  0,I ϕ ( τ ) LVWKHSKDVHDQJOHRI R xy ( τ ) FRQVLGHU E { x ( t + τ ) ± e

jϕ ( τ )

2

y( t) } 

6KRZWKDW x ( t ) = y ( t ) LQWKHPHDQVTXDUHVHQVH7KHQXVHWKH6FKZDU]LQHTXDOLW\ (TWZLFHWRVKRZ R xx ( τ ) = Rxy ( τ ) DQG R yy ( τ ) = Rxy ( τ )  1RKLQW 3UHVXPH ϕ WREHUHDO8VH Φ ( 1 ) = 0 WRHVWDEOLVK E { cos ϕ } = E { sin ϕ } = 0 DQG XVH Φ ( 2 ) = 0 WRHVWDEOLVK E { cos 2ϕ } = E { sin 2ϕ } = 0 7KHQVKRZ η ( t ) = 0 DQG 1 R ( t 1, t 2 ) = --- cos [ ω ( t 1 – t 2 ) ]  2 D %7KHUHLVDPLVVLQJGHILQLWLRQKHUH7KHVWRFKDVWLFSURFHVV x ( t ) KDVRUWKRJR QDOLQFUHPHQWVLIIRU t a ≤ t b ≤ t c ≤ t d  E { [ x ( t d ) – x ( t c ) ] [ x ( t b ) – x ( t a ) ] } = 0 8VLQJWKLV GHILQLWLRQVXEVWLWXWH t d = t 2  t c = t b = t 1 DQG t a = 0 E )
∞

∫–∞ fy ε ( y1, …, y n ; t1, …, tn ε )fε ( ε ) dε DQGWKDW

f y ε ( y 1, …, y n ; t 1, …, t n ε ) = f x ( y 1, …, y n ; t 1 – ε, …, t n – ε )  8VH(TVDQG E 1RWHWKDW f x ( x, t ) = f x ( x ) 1RWHWKDW z DQG w DUHMRLQWO\QRUPDO8VH(TWR ILQGWKHLUFRUUHODWLRQFRHIILFLHQW )
H-23

Gary Matchett

Hints — 9/5/02

VWDQWPHDQ 7KHGLVFXVVLRQIROORZLQJ(TDVVXUHVWKDW x’( t ) LVQRUPDO(T 2

ZLOOKHOSWRILQG σ x’  06HHWKHKLQWLQWKHSUREOHPZKLFKLV8VH(TDQGHVWDEOLVKWKH)RXULHU ∞

VHULHV sin

–1

z =

1

∑ --n- [ J0 ( nπ ) – ( –1 )

n

]sin nπz 

n=1

3UHVXPHDVLVLPSOLHGE\WKHSUREOHPWKDW x ( t ) LV:667KHUHDUHWZRZD\VWR SURFHHG2QHZD\LVGLUHFWEXWDOJHEUDLFDOO\GHPDQGLQJ7KHRWKHUZD\LVOHVVGLUHFW EXWHDVLHU7KHGLUHFWPHWKRGVLPSO\UHOLHVRQWKHUHODWLRQVKLSV E{ g( x( t ) ) } = E { g ( x ( t 1 ), x ( t 2 ) ) } =

∞

∫–∞ g ( x )f ( x, t ) dx ∞

∞

∫–∞ ∫–∞ g ( x1, x2 )f ( x1, x2 ; t1, t2 ) dx1 dx2

ZKLFKDUHREYLRXVH[WHQVLRQVRI(TVDQG7KHILUVWDQGVHFRQGRUGHUGHQ VLW\IXQFWLRQDUHREWDLQHGIURPWKHIDFWWKDW x ( t ) LVQRUPDO7KHLQWHJUDWLRQLVGLIIL FXOWEXWGRDEOH7KHOHVVGLUHFWPHWKRGUHOLHVRQWKHFORVHFRQQHFWLRQEHWZHHQWKH GHVLUHGH[SHFWHGYDOXHVDQGWKHILUVWDQGVHFRQGRUGHUFKDUDFWHULVWLFIXQFWLRQVRI x ( t ) ZKLFKDUHIXOO\GHYHORSHGIRUDQRUPDOSURFHVVLQ(TVDQG,Q 2

HLWKHUFDVHRQHPXVWQRWHWKDW σ ( t ) = R x ( 0 ) DQG r ( t + τ, t ) = r ( τ ) = R x ( τ ) ⁄ R x ( 0 )  E )$VVXPH τR x ( τ ) → 0 DV τ → 0 QRWMXVW R x ( τ ) → 0 DVLQWKHSUREOHPVWDWH PHQW  :ULWH y ( t 1 )y ( t 2 ) DVDGRXEOHLQWHJUDODQGWDNHH[SHFWHGYDOXHV 2

D :ULWH y ( t ) DVDGRXEOHLQWHJUDODQGWDNHH[SHFWHGYDOXHVE ,0)LQGWKH JHQHUDOIRUPDOVROXWLRQWRWKLVGLIIHUHQWLDOHTXDWLRQDQHTXDWLRQZKLFKLVQRWWLPH LQYDULDQW 7KHQSURFHHGDVLQSDUWD  D :ULWHDIRUPDOVROXWLRQIRU y ( t ) WRILQGWKHLPSXOVHUHVSRQVHIXQFWLRQ h ( t )  7KHQXVH(T 1RKLQW )$VVXPHWKDW x ( t ) LV:668VHDPHWKRGDQDORJRXVWRWKDWXVHGWRGHYHORS(T  D 8VHWKHVROXWLRQWR3UREOHPRURWKHUZLVHILQGWKHLPSXOVHUHVSRQVHIXQF H-24

Gary Matchett

Hints — 9/5/02

WLRQ8VH(TVDQGE 8VH([DPSOH D 06HH([DPSOH E 8VH cos ω 0 τ cos ωτ = [ cos ( ω 0 – ω )τ + cos ( ω 0 + ω )τ ] ⁄ 2 ZLWKSDUWD  $VVXPH x ( t ) LVUHDO 8VHWKHLGHQWLWLHV e

j2aω

+e

– j 2aω

2

= 2 cos 2aω DQG 1 – cos 2aω = 2sin 2aω 

1RKLQW 06KRZ η y = I 8VH(TIURP([DPSOHWRILQG R y WKHQXVH7DEOHWR ILQG Sy  jωτ 2 1 ∞ 6LQFH S ( ω ) ≥ 0 FOHDUO\ A = ------ ∫ S ( ω ) Σ i a i e i dω ≥ 0  2π –∞

D 08VH&DXFK\¶VUHVLGXHWKHRUHPWRILQG R ( τ ) YLDFRQWRXULQWHJUDWLRQDVLQWKH VROXWLRQWR3UREOHPE 08VHWKHVDPHDSSURDFK7KHSROHKHUHLVRIRUGHU d 2 WZRDQGWKHUHVLGXHLVJLYHQE\ r = ------- [ ( ω – ωp ) h ( ω ) ] ZKHUH ω p LVWKHSROH dω ω = ωp DQG h ( ω ) LVWKHLQWHJUDQG ' H ( ω ) LVDFRPSOH[IXQFWLRQRIWKHUHDOYDULDEOH ω  + ( s ) LVDFRPSOH[IXQFWLRQ RIWKHFRPSOH[YDULDEOH s 7RWDNHWKHFRPSOH[FRQMXJDWHRI + ( s ) \RXPXVWFRQMXJDWH ERWKWKHIXQFWLRQ +( ) DQGWKHYDULDEOH s 7RDVVLVWLQWKHQRWDWLRQGHILQHLQWKH ILUVWDQGVHFRQGSDUWVRIWKHSUREOHP  : ( s ) = +* ( –s* ) =

∞

∫–∞ h* ( t )e

st

dt  : ( z ) = + * ( 1 ⁄ z * ) = Σ n h * [ n ]z

n

%7RVROYHWKLVSUREOHPLWLVEHVWWRXVHWKHJHQHUDOFRQYROXWLRQWKHRUHP7KHUH DUHWZRYHUVLRQVDQGERWKDVVXPHWKUHHSDLUVRI)RXULHUWUDQVIRUPV f ( τ ) ↔ F ( ω )  g ( τ ) ↔ G ( ω ) DQG h ( τ ) ↔ H ( ω )  , ,I H ( ω ) = F ( ω )G ( ω ) WKHQ h ( τ ) = f ( τ ) * g ( τ ) 1 ,, ,I h ( τ ) = f ( τ )g ( τ ) WKHQ H ( ω ) = ------ F ( ω ) * G ( ω ) 2π 1RWLFHWKHIDFWRURIWZRSLLQWKHVHFRQGYHUVLRQRIWKHWKHRUHP$SSDUHQWO\WKLVIDF

H-25

Gary Matchett

Hints — 9/5/02

WRUZDVQHJOHFWHGLQWKHVHWWLQJRIWKHSUREOHPOHDGLQJWRWKHW\SR7KHFRUUHFWUHVXOW 1 2 LV S y ( ω ) = 2πR x ( 0 )δ ( ω ) + --- Sx ( ω ) * S x ( ω )  π $OVR\RXQHHGWRNQRZZKDWLGHDO/3ORZSDVV DQG%3EDQGSDVV VSHFWUDDUH 7KH\DUHVLPSO\XQLWLQWHQVLW\ZKLWHQRLVHSURFHVVHVSXWWKURXJKDQLGHDOL]HGILOWHU 7KH\DUHLOOXVWUDWHGEHORZ Sx ( ω )

Sx ( ω ) 1

1

ω

ω –ω 2

ωc

–ωc

– ω1

/RZ3DVV

ω1

ω2

%DQG3DVV

8VH(TWRILQG Rxx’( τ ) DQG R x’x’( τ ) 1RWHWKDW R xx’( τ ) LVGLVFRQWLQXRXVDWWKH RULJLQOHDGLQJWRD δ ( τ ) WHUPLQ R x’x’( τ ) 8VH7DEOHWRFRQYHUWIURP R yy ( τ ) WR S yy ( ω )  D 8VH([DPSOHDE 8VH(T 1RWHWKDW R ( 0 ) PXVWEHUHDODQGQRQQHJDWLYHVRWKDWLWPXVWEHWKDW jϕ

R ( τ 1 ) = R ( 0 )e IRUVRPHSKDVHDQJOH ϕ 'HILQH ω = ϕ ⁄ τ 1  D 07KHSUHVXPSWLRQLVWKDW x ( t ) LVUHDO([SUHVV E { x ( t )xˆ ( t ) } LQWHUPVRI S xx ( ω ) XVLQI(T$WWHPSWLQJWRXVH R xx ( τ ) ZLOOIDLO E '([SUHVV x˜ ( t ) P\V\PEROIRUGRXEOHXSVLGHGRZQKDWVZKLFK,ODFN LQWHUPV 2

RI x ( t ) DQG h ( t ) YLDDQLWHUDWHGFRQYROXWLRQ6LQFH H ( ω ) = – 1 LWIROORZVWKDW ρ ( t ) = – δ ( t ) $QLQQRYDWLYHDOWHUQDWHDSSURDFKLQYROYHVWKH)RXULHUWUDQVIRUPRI x ( t ) LWVHOIEXWWKLVPHWKRGODFNVJHQHUDOLW\ )LQGVRPH G ( ω ) VXFKWKDW S yy ( ω ) = S xx ( ω )G ( ω ) 7KHQILQGWKDW ω = ± ω 0 PD[L PL]HV G ( ω ) DQGSXWDOOWKHDYDLODEOHHQHUJ\RI S xx ( ω ) DWWKHVHIUHTXHQFLHV 8VHLQHTXDOLW\WRVKRZWKDW S xy ( ω ) = 0 IRU ω ≠ ω 0 'HGXFHWKDW

H-26

Gary Matchett

Hints — 9/5/02

S xy ( ω ) = 2πBδ ( ω – ω 0 )  D 8VHFRQYROXWLRQLQWHJUDOVGLUHFWO\1RWH R yx LQVWHDGRI R xy  3UHVXPHWKDWRYHUVXIILFLHQWO\VPDOOLQWHUYDOVRIWKHIUHTXHQF\D[LVWKDWWKHIXQF WLRQV S xy ( ω )  S xx ( ω ) DQG S yy ( ω ) DUHHVVHQWLDOO\FRQVWDQW8VH(T 3UHVXPHWKDW x ( t ) LVUHDO8VHWKHFRVLQHLQHTXDOLW\(T 3URFHHGDVLQ3UREOHP1RWHWKHW\SRWKHJUHDWHUWKDQVLJQVKRXOGEHD JUHDWHUWKDQRUHTXDOVLJQ ([SUHVV R [ m ] LQWHUPVRI f ( ω ) &RPSDUHZLWK(T D ,0)LQGWKHVROXWLRQVWRWKHKRPRJHQHRXVGLIIHUHQWLDOHTXDWLRQ8VHWKH YDULDWLRQRISDUDPHWHUVPHWKRGWRILQGWKHVROXWLRQWRWKHQRQKRPRJHQHRXVGLIIHU 2

HQWLDOHTXDWLRQ:ULWH y ( t ) DVDGRXEOHLQWHJUDODQGWDNHH[SHFWHGYDOXHV E ,'8VHWKHVDPHDSSURDFKDVDERYHIRUWKHGLIIHUHQFHHTXDWLRQ D ,)LQGWKHJHQHUDOVROXWLRQWRWKHGLIIHUHQFHHTXDWLRQ E ,0RGLI\WKHGHYHORSPHQWRISDUWD 

 &KDSWHU D 8VH(TE 8VH(TDQG([DPSOH 8VH(TIRU f x ( x, t ) DQG f y ( y, t ) 7KHQXVH(TIRU f z ( z, t )  ,)LQGWKHGLIIHUHQWLDOHTXDWLRQUHODWLQJ v ( t ) DQG n e ( t ) 8VH/DSODFHWUDQVIRUPVWR 2

ILQG + ( s ) DQGWKHQVROYHIRU H ( ω ) 8VH(TWRILQG S v ( ω ) $VLPLODUSURFHVV ZRUNVIRUWKHFXUUHQWFDVH 1RWHWKDW e

– 2αt

1 U ( t ) ↔ ------------------------ DQGLI h ( t ) ↔ H ( ω ) WKHQ h’( t ) ↔ jωH ( ω )  ( jω + 2α )

8VH([DPSOHDQG(T 3UHVXPHWKDWWKH:LHQHUSURFHVV w ( t ) LVDQRUPDOVWRFKDVWLFSURFHVVZLWK]HUR PHDQ8VH(TWRILQG R y DQG3UREOHPWRILQG R z  8VH&DPSEHOO¶VWKHRUHP(T1RWHWKDW s ( 7 ) FDQEH]HURRQO\LIQR3RLVVRQ

H-27

Gary Matchett

Hints — 9/5/02

SRLQWVDUULYHLQWKHLQWHUYDOSULRUWR t 0 = 7 ZKHQ h ( 7 – t ) LVQRW]HUR 6KRZWKDW S xy ( ω ) LVSXUHO\LPDJLQDU\7KHQXVH(TVDQGWRFRQILQH H( ω)  '$EUXWHIRUFHDSSURDFKZRUNVKHUHEXWLWLVERWKOHQJWK\DQGGHPDQGLQJ )$VWKH7H[WSRLQWVRXWRQWKHERWWRPRISDJHJLYHQMXVW x ( t ) WKHUHLVQR XQDPELJXRXVFRPSOH[HQYHORSH
T

∫0 f ( t )e

2 – jωt

dt

∞

∑

m = –∞

2πm δ  ω – ----------- T

%HJLQE\VKRZLQJWKDWWKHVWRFKDVWLFSURFHVV y ( t ) = f ( t ) LV66&67KHQVKRZ x ( t ) LV 666DQGXVH(TIRU R xx ( τ ) )LQG S xx ( ω ) GLUHFWO\IURP(TWKHQVSOLWXS WKHLQILQLWHWLPHLQWHJUDOLQWRDVXPRYHUFRQVHFXWLYHVSDQVRIOHQJWK T )LQDOO\XVH (T$WRJHWWKHGHVLUHGUHVXOW 00LPLF(TDQGGHILQH N

yN ( t ) = x ( t + τ ) –

∑ n = –N

sin σ ( τ – nT ) x ( t + nT ) --------------------------------  T = π ⁄ σ σ ( τ – nT )

1RWHWKDW ε N ( t ) = y N ( 0 )  $VVXPH x ( t ) LVUHDODQGXVHWKHSURRIDWWKHERWWRPRISDJH 8VH(T(TZLWK ω = 0 DQG7DEOH 6HHKLQWDERYH

H-28

Gary Matchett

Hints — 9/5/02

π 1RWHWKDWLQWKHUDQJH τ < ------ DQG ω ≤ σ WKHQ cos ωτ ≥ cos στ $OVRDVVXPHWKDW 2σ x ( t ) LVUHDODVXVXDOLQ%/SUREOHPV 07KLVLVDVWUDLJKWIRUZDUGDSSOLFDWLRQRIWKH³3DSRXOLV6DPSOLQJ([SDQVLRQ´RQ 7H[WSDJHEXWGLIILFXOWEHFDXVHLWLQYROYHVFRQVLGHUDEOHPDQLSXODWLRQ)RUVRPH ω jστ 1 jστ KHOSQRWHWKDW P 1 ( ω, τ ) = 1 – ---- ( e – 1 )  P2 ( ω, τ ) = ----- ( e – 1 ) DQG σ jσ 0 jωτ jωτ 1 0 ω jστ 1 jστ p 1 ( τ ) = --- ∫ 1 – ---- ( e – 1 ) e dω  p 2 ( τ ) = -------2- ( e – 1 ) ∫ e dω 8VH(T σ –σ σ –σ jσ

ZLWK t 0 = 0 QRWLQJWKDW y 1 ( nT ) = y 1 ( n∆ ) = x ( n∆ )  y 2 ( nT ) = y 2 ( n∆ ) = x’( n∆ )  8VHWKHPHWKRGRI(TZLWK f ( t ) = cos ω 0 t cos ωt DQGZLWKWKHLQWHJUDO OLPLWVIURP – a WR a *HWHTXLYDOHQWIRUPXODVWRWKRVHIROORZLQJ(T 1 ))LUVWFRUUHFWWKHW\SR7KHVXPVKRXOGEH X c ( ω ) = --λ

∑

x ( t i )e

– jωt i

1H[W\RX

i ti < a

PXVWDVVXPHWKDW x ( t ) LVLQGHSHQGHQWRIWKHSURFHVV z ( t ) =

∑ δ ( t – ti ) 7KHLQWHJUDO i

WUDQVIRUPRI(TYDOLGIRUDQ\IXQFWLRQ C ( τ ) LVDOVRXVHIXO 1RKLQW n

&RQVLGHU I =

n

∑ ∑

2

a i b *j – a j b *i ≥ 0 

i=1 j=1

D '7KHWUHDWPHQWRIGLVFUHWHSURFHVVHVLQWKH7H[WLVWRREULHIWRHYHQJLYHD XVHIXOKLQWKHUHH[FHSWWRPLPLFWKHPDWFKHGILOWHUGLVFXVVLRQIRUDFRQWLQXRXVSUR FHVV$VVXPHUHDOSURFHVVHV 2

yf [ 0 ] E '1RWHWKDW v [ n ] LVQRWSUHVXPHGWREHZKLWHKHUH7RPD[LPL]H r = ---------------------- 2 E { yv [ n ] } \RXVKRXOGPLQLPL]HWKHGHQRPLQDWRUZKLOHKROGLQJWKHQXPHUDWRUFRQVWDQW8VH WKHPHWKRGRI/DJUDQJHPXOWLSOLHUV 0)LQGWKHLPSXOVHUHVSRQVHIXQFWLRQ h ( t ) VHH([DPSOH )URPWKHGHWHU PLQLVWLFLQSXW f ( t ) = A cos ω 0 t ILQGWKHGHWHUPLQLVWLFRXWSXWLQWKHIRUP y f ( t ) = B cos ( ω 0 t + ϕ ) WRUHODWH B WR A  α DQG ω 0 )LQGWKHVSHFWUXPRIWKHQRLVH

H-29

Gary Matchett

Hints — 9/5/02

RXWSXWDQGIURPWKDWILQGWKHDXWRFRUUHODWLRQRIWKHQRLVHRXWSXWXVLQJ7DEOH 8VH(TWRILQGWKHDYHUDJHSRZHURIWKHQRLVHRXWSXW T

0D ,WKHOSVWRGHILQHWKHYHFWRUVDQGPDWULFHV a = ( a 0, a 1, …, a m )  R 00 R 01 … R 0m R 10 R 11 … R 1m

T

f = ( f 0, f 1, …, f m ) = ( f ( t 0 ), f ( t 0 – T ), …, f ( t 0 – mT ) ) DQG R =

… … … R m0 R m1 … R mm



ZKHUH R ij = R v ( iT – jT ) = E { v ( t 0 – iT )v ( t 0 – jT ) } 7KHQVKRZ m

yf = yf ( t0 ) =

∑ aifi

T

T

2

= a f DQG E = E { y v ( t 0 ) } = a Ra 0D[LPL]H r E\PLQLPL]LQJ

i=0

E VXEMHFWWRWKHFRQVWUDLQWRIDJLYHQYDOXHRI y f > 0 E\WKHPHWKRGRI/DJUDQJHPXO WLSOLHUV T –1

E 7KHPD[LPXPYDOXHRI r LV f R f =

yf ⁄ k 

7KLVIROORZVGLUHFWO\IURP(TXVLQJUHSHDWHGO\ WKHLGHQWLW\ ∞

∫–∞ e

– jωτ

dτ = 2πδ ( ω ) 

06HYHUDOIDFWVDUHXVHIXOKHUH,I t 1 < t 2 < t 3 WKHQ x˜ ( t 3 ) = x˜ ( t 1 ) + [ x˜ ( t 2 ) – x˜ ( t 1 ) ] + [ x˜ ( t 3 ) – x˜ ( t 2 ) ] DQGWKHWKUHHUDQGRPYDULDEOHV x˜ ( t 1 )  x˜ ( t 2 ) – x˜ ( t 1 ) DQG x˜ ( t 3 ) – x˜ ( t 2 ) DUHPHDQ]HURDQGLQGHSHQGHQWEHFDXVHWKH\FRXQW 3RLVVRQSRLQWVLQQRQRYHUODSSLQJLQWHUYDOV)URPWKHGLVFXVVLRQRI3RLVVRQUDQGRP 3

YDULDEOHVRQSDJHRIWKH7H[WLWIROORZVWKDW E { x˜ ( t 1 ) } = λt 1 )LQDOO\LJQRUHWKH KLQWLQWKHSUREOHPDQGXVHLQVWHDGWKHWKUHHLGHQWLWLHV ∂min ( t a, t b ) min ( t 1, t 2, t 3 ) = min ( t 1, min ( t 2, t 3 ) )  ----------------------------- = U ( t b – t a ) ∂t a U ( min ( t 2, t 3 ) – t 1 ) = U ( t 2 – t 1 )U ( t 3 – t 1 ) )ROORZWKHRXWOLQHLQWKHSUREOHP

 &KDSWHU ')LQGLQJWKHZKLWHQLQJILOWHULVHDV\HVSHFLDOO\LI\RXQRWHWKDW

H-30

Gary Matchett

Hints — 9/5/02

2

–2

cos 2ω = ( z + z ) ⁄ 2 )LQGLQJWKHDXWRFRUUHODWLRQVHTXHQFH R xx [ m ] LVKDUG²VR KDUGWKDWQRKLQWLVJLYHQKHUHVHHWKHVROXWLRQ N ( s )N ( – s ) N( s) )DFWRU 6 ( s ) = --------------------------- WKHQVHW / ( s ) = -----------  D ( s )D ( – s ) D(s) ∞

8VHWKHFRQYROXWLRQVXP s [ n ] =

∑ ls [ k ]i [ n – k ]  k=0

D ,1RWHWKDW R yx’( τ ) = E { y’( t + τ )x * ( t ) } HWF E ,08VHWKHGLVFXVVLRQIROORZLQJ(T1RWHWKDW +

-

+

6 yx ( s ) = 6 yx ( s ) = q ⁄ D ( s ) WRILQG R yx ( τ ) IRU τ > 0 1RWHWKDW 6 yy ( s ) = 6 yy ( – s ) DQG + q 6 yy ( s ) = --------------------------- WRILQG 6 yy ( s ) DQGKHQFH R yy ( τ ) IRU τ > 0  D ( s )D ( – s )

6KRZ R xx [ m ] = R ss [ m ] + R vv [ m ] DQG R vv [ m ] = qδ [ m ] 'HGXFHWKDW 1 6 ss ( z ) = ------------------------------- ZKHUH D ( z ) KDVDOOLWVURRWV z i ZLWK z i < 1 &RQFOXGHWKDW D ( z )D ( 1 ⁄ z ) 6 xx ( z ) LVDOVRUDWLRQDOZLWKWKHVDPHSROHVDV 6 ss ( z ) DQGWKDW 6 xx ( z ) = 6 xx ( 1 ⁄ z )  n

1 'HILQH s ( t ) = --n

∑ x ( t + kT ) DQGUHJDUG s ( t ) DVWKHRXWSXWRIDOLQHDUV\VWHPZLWK k=1

LQSXW x ( t ) )LQGWKHLPSXOVHUHVSRQVHIXQFWLRQWKHV\VWHPIXQFWLRQDQGXVH(T  ')3UHVXPH x ( t ) LV:66VRWKHRULJLQRIWKHWLPHVFDOHPD\EHVKLIWHGVRWKDWWKH LQWHUYDO ( 0, T ) LQWKHROGWLPHVFDOHFRUUHVSRQGVWRWKHLQWHUYDO ( – a, a ) LQWKHQHZ WLPHVFDOH3DUWLFXODUL]HWKHLQWHJUDOHTXDWLRQ'LIIHUHQWLDWHWKHLQWHJUDOHTXD WLRQWZLFHWRREWDLQDGLIIHUHQWLDOHTXDWLRQ)LQGWKHJHQHUDOVROXWLRQVWRWKHGLIIHUHQ WLDOHTXDWLRQDQGSXWWKHPEDFNLQWRWKHLQWHJUDOHTXDWLRQWROHDUQPRUHDERXWWKHP 8VHWKHQRUPDOL]DWLRQHTXDWLRQ(TWRVFDOHWKH ϕ ( t ) IXQFWLRQV1RWHWKDWWKH λ n – 1 ⁄ 2 λ n’ –1 ⁄ 2   ---UHVXOWVVKRXOGUHDG β n =  a +  DQG β n’ =  a + ------ QRWWKHUHVXOWVJLYHQ 2 2 LQWKH7H[W 2

:ULWH E { X ( ω ) } DVDGRXEOHLQWHJUDO7UDQVIRUPWRDVLQJOHLQWHJUDODVLQ(T 'LIIHUHQWLDWHZLWKUHVSHFWWR T 

H-31

Gary Matchett

Hints — 9/5/02

1RKLQW 1RKLQW 08VH(TD 6KRZ  E { x ( t )x* ( t ) } = E { x ( t )xˆ * ( t ) } = E { xˆ ( t )x * ( t ) } = E { xˆ ( t )xˆ * ( t ) } = R ( 0 ) ,QRUGHUWRGRSDUWD LWLVXVHIXOWRGRSDUWE F 6XEVWLWXWH s = t – α LQWKH β n ( α )  H[SUHVVLRQ $VLVXVXDOLQWKHVHFDVHVVKRZWKDW E { x ( t ) } = 0 DQGWKDW E { x ( t )x* ( s ) } LVDIXQF WLRQRI t – s  'HGXFHWKDWLI A DQG B VDWLVI\(TVWKHQLWPXVWEHWKDW E { A ( u )A ( v ) } = E { B ( u )B ( v ) } = Q ( u )δ ( u – v ) DQG E { A ( u )B ( v ) } = 0  T

01RWHWKDW ∫ f ( t )e

– jωt

∞

dt =

–T

∫–∞ f ( t )pT ( t )e

– jωt

dt ZKHUH –T < t < T

1 pT ( t ) = U ( t – T ) – U ( T – t ) =  0 8VHWKHIUHTXHQF\FRQYROXWLRQWKHRUHPLI F1 ( ω ) = F2 ( ω ) =

∞

∫–∞ f2 ( t )e

– jωt

otherwise ∞

∫–∞ f1 ( t )e

– jωt

dt DQG

dt WKHQ

 F 12 ( ω ) =

∞

∫–∞

f 1 ( t )f 2 ( t )e

– jωt

1 ∞ dt = ------ ∫ F 1 ( y )F 2 ( ω – y ) dy 2π –∞

∞ sin 2 αT

- dα = Tπ ------  8VH(T1RWHWKDW ∫ ---------------2 2 0 α

 &KDSWHU 1RKLQW 2

,'1RWHWKDW x ( t ) LV:668VH(TWRGHGXFHWKDW f ( x, x ; τ ) → f ( x ) DV τ → ∞ )LQGERWKVLGHVRIWKLVOLPLWH[SOLFLWO\DQGVKRZWKDWWKHOLPLWUHTXLUHV

H-32

Gary Matchett

Hints — 9/5/02

r ( τ ) → 0 8VH(T 8VH(T 6KRZWKDW C zz ( τ ) GRHVQRWGHSHQGRQ τ VRWKDW(TFDQQRWEHVDWLVILHG 2

,5HFDOOWKDW lim R T = R xy ( λ ) LIDQGRQO\LIERWK E { R T } = R xy ( λ ) DQG σ RT → 0  T→∞

)',7KHFRQGLWLRQLQWKLVSUREOHPVKRXOGUHDG R ( t + τ, t ) → η ( t + τ )η ( t ) DV 2

τ → ∞ XQLIRUPO\LQ t $OVR\RXPXVWDVVXPH C ( t, t ) < σ IRUVRPH σ DQGDOO t  1 T ,JQRUHWKHKLQWLQWKH7H[W'HILQHWKHDYHUDJHPHDQDV η = --- ∫ η ( t ) dt 6KRZWKDW T 0 2 1 c lim ------ ∫ η ( t ) dt = η 8VHWKLVUHVXOWWRVKRZWKDW lim E { ( η c – η ) } = 0 LVHTXLYD c → ∞ 2c – c c→∞ OHQWWRWKHUHYLVHGFRQGLWLRQDERYH'R3UREOHPILUVWWRXQGHUVWDQGWKLVODWWHU UHVXOW 2

,'$VVXPHWKDW C ( t, t ) < σ IRUVRPH σ DQGDOO t 8VH(TWRJHW 1 T T 2 σ T = --------2- ∫ ∫ C ( t 1, t 2 ) dt 1 dt 2 &KDQJHYDULDEOHVWR t = t 2  τ = t 1 – t 2 %UHDNXSWKH 4T –T –T LQWHJUDOLQWKH tτ SODQHLQWRSRUWLRQVZKHUH τ ≥ T 2 DQG τ < T 2 DQGILQGZD\VWR OLPLWERWKSDUWV

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Gary Matchett

Hints — 9/5/02

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