Errata and Hints for Problems

Errata, Hints, and Problem Solutions WRDFFRPSDQ\WKHWH[W

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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

H-33

Gary Matchett

Hints — 9/5/02

H-34