Problem Session (3) for Chapter 4 Signal Modeling

Transcription

1 Pobm Sssio fo Cht Sig Modig Soutios to Pobms d... Fid th Pdé oimtio of scod-od to sig tht is giv by [... ] T i.. d so o. I oth wods usig oimtio of th fom b b b H fid th cofficits b b b d. Soutio Giv [... ] T b b b fid H usig Pdé oimtio. Fom H it foows tht d q so tht th q5 cofficits b b b d to b foud. Bsd o d q w c fom th foowig qutios i mti fom to fid th 5 cofficits b b b Fom th st qutios fo d istig w hv > > Fom th fist q qutios fo b w hv b b b Th dsid mod is H.. A thid-od -o Pdé oimtio to sig hs b foud to b H Wht ifomtio bout c b dtmid fom this mod? Soutio Giv H obtid fom Pdé oimtio fid fom H.

2 Fom giv H w hv b d. Fom th Pdé oimtio mthod w hv b Fom th fist qutios w hv b ; ; > ; > ; 7 Th mod cotis th ifomtio bout fo d s foows [ ] T W wt to mod sig usig -o mod of th fom b H Fo m with th mod is b H Div th om qutios tht dfi th cofficits tht miimi th Poy o wh d div ssio fo th miimum o. Soutio Poy's mthod to mod usig th o mod: b H i div th om qutios fo fidig tht miimis wh ii div ssio fo th miimum o. To miimi th squ o w my st ; q which ds to th othogoity ici

3 Istig i th bov qutio w hv o quivty Dfiig w obti th om qutios Usig i th Poy s o bcoms Usig th othogoity ici i.. fo i Eq. w my fid th miimum o Usig th miimum o c b ssd s.5. Suos tht th fist fiv vus i th utocotio squc fo th ocss [ ] T... 9 / 9 /6 9 / 8 9 / Us th modifid Yu-W qutio mthod to fid ARMA mod fo. b A th giv vus i th utocotio squc cosistt with th mod tht you foud i t?

4 Soutio Giv ocss with [ 9 / 9 / 8 9 /6 9 /...] T us th MYWE to fid ARMA mod fo i.. fid b b d i th td systm fuctio b b H fom which w s d q. b ˆ?? Soutio fo Fom d q th Yu-W qutios of th fom c c Th modifid Yu-W qutio th st qutio givs us 9 9 ; i.. 8 Fom th fist two qutios d otig tht is -vud w hv c c > c c 9 Fom c d c w my hv 5 A / d [ C ] c c 8 which giv 5 [ C ] A / Tig th cus t of [ ] A / [ P ] [ Cq ] A/ ] C w obti y which is th cus t of P y B B /. Fom th symmty Py Py / w my hv P y Sct fctoitio of P y ds to P y B B / Thus w hv B B Th dsid ARMA mod is H A Soutio fo b Th mod ˆ is ARMA ocss which c b dscibd by th Yu-W qutios

5 c ; ˆ ˆ ; 5 wh c d c 8 5 : ˆ ˆ ; 8.5ˆ 5 8 ˆ ; > ; > : ˆ ˆ ;.5 ˆ ˆ 9 9 : ˆ ˆ ; > ˆ ˆ 8 : ˆ ˆ ; > ˆ ˆ 9 ˆ ˆ 6 9 ˆ ˆ Cocusio: th giv vus of of th ocss qu to th vus of ˆ fom th ARMA mod foud i. Th ARMA mod ˆ mtchs th giv ocss fcty. Thus th giv ocss is ARMA ocss... If. 5 d. 75 fid th vus of d b i th foowig AR mod fo ˆ ˆ ˆ b w wh w is uit vic whit ois. Soutio Giv. 5 d. 75 fid d b i th mod fo ˆ ˆ ˆ b w w w Th systm fuctio fo th AR mod c b ssd s b H otig tht d q th Yu-W qutios th foowig fom c.5.75 c > Th modifid Yu-W qutios giv us >

6 Fom th fist qutio fo c w hv 5 c Sic c b th Th dsid AR mod is b ˆ ˆ ˆ w.. Us th mthod of sct fctoitio to fid movig vg mod of od fo ocss whos utocotio squc is [.5 ] T Soutio Giv ocss with [.5 ] T fid MA mod fo. Th MA mod c b witt s H b b b Th ow sctum is P H H / b b b b b b otig is [ ][ ] [ b b b b ] b b b b b Fom [ ] T.5 i... 5 d w hv.5 P Comig Eq. with Eq. w hv b b b b b b b.5 b b 5 Istig b i Eq. 5 ito Eqs. d yids sctivy b b b b 6.5b b 7 b Sovig Eqs. 6 d 7 fo b d b givs b d b Fom Eq. 5 d b it foows tht b Th dsid MA mod is H 6