The Z-Trsform DSP eture - Adres Ss ss@su.edu 6 Coyrght 6 Adres Ss -- Poles d Zeros of I geerl the trsfer futo s rtol; t hs umertor d deomtor olyoml. The roots of the umertor d deomtor olyomls re lled the eros d the oles resetvely. Pole-ero deomostos of re qute useful d rovde tuto sgl lyss d flter desg.... G G... 6 Coyrght 6 Adres Ss -- where G s g ftor
Exmle: Poles d Zeros of..8 the ero O of ths trsfer futo s t = the ole of ths trsfer futo s t =.8 Imgry rt.5 O -.5 - - -.5.5 Rel rt 6 Coyrght 6 Adres Ss --3 Exmle: Poles-Zeros of Seod Order System.3435.45.95.55 j45 j45. 95e. 95e o o j7. 34 j7. 34. 746e. 746e o lm o oles eros Re Note tht the flter oeffets re rel vlued d therefore oles d eros our omlex ojugte rs. 6 Coyrght 6 Adres Ss --4
Poles d Zeros d Stblty The loto of the oles determes the stblty of the flter. I ft, for stblty ll the oles of usl flter must be sde the ut rle, tht s for ll =,,..., IIR flters my be ll-ole or ole-ero d stblty s lwys oer. FIR or ll-ero * flters re lwys stble. * deedg o the wy the trsfer futo s ormled the term ll-ero for FIR systems my be mrese beuse there my be oles loted t ero FIR system - smlrly ll-ole IIR my hve eros loted t ero 6 Coyrght 6 Adres Ss --5 The Frequey Resose Futo The trsfer futo s b b... b... by evlutg o the ut rle,.e. for e j e j b b e j j e... be... e j j 6 Coyrght 6 Adres Ss --6
The Frequey Resose Futo Cot. The frequey resose futo d s omlex d erod Wth erod. The ormled frequees re ssoted to the smlg frequees f s by Smlg erod T f f s rd rd/s Smlg frequey where f s s the smlg frequey d f s y frequey of terest. I rte, oe detemes the frequey resose u to hlf the smlg frequey fold-over frequey. 6 Coyrght 6 Adres Ss --7 The Frequey Resose Futo Cot. Foldover Frequey fs/ fs The frequey resose s usully lotted w.r.t. ormled frequees The frequey resose s erod wth erod f s Se frequees of terest re u to the bdwdth of the log sgl the setrum s usully lotted u to f s /, the foldover frequey 6 Coyrght 6 Adres Ss --8
The Frequey Resose d Poles d Zeros The mgtude frequey resose futo e j G The hse frequey resose futo e e j j rg e j rg e j rg e j 6 Coyrght 6 Adres Ss --9 Remrks o Effets of Poles d Zeros o e j Poles ted to rete eks the mgtude frequey resose Zeros ted to rete vlleys the mgtude frequey resose Seletve flters re desged effetly by lg oles lose to the ut rle Shr othes re heved effetly wth eros led very lose to the ut rle f restrted to ll-ole flter shr oth the frequey resose wll requre hgh order desg my oles. f restrted to ll-ero FIR flter shr hgh Q ek the frequey resose wll requre my eros log mulse resose or hgh order FIR. 6 Coyrght 6 Adres Ss --
Z le d Frequey gtude Resose lm oles eros Re gtude..5. gtude Resose.5. 3 64 96 8 Frequey Idex Thet=*PI*Idex/8 FFT dex Foldover Frequey fs/ fs 6 Coyrght 6 Adres Ss -- Zero otos d Frequey Resose o o o o foldover ovg eros towrds the ut rle retes shrer vlleys 6 Coyrght 6 Adres Ss --
6 Coyrght 6 Adres Ss --3 Pole otos d Frequey Resoses As the oles move towrds the ut rle we get shrer eks d f the oles re led o the ut rle we get oslltor. 6 Coyrght 6 Adres Ss --4 Comutg Flter Resoses Usg the Iverse Z-Trsform Prtl Frtos: Gve the trsfer futo l b b b...... for dstt oles wrte s:...
DSP EEE 47/59 Adres Ss eture 6 Coyrght 6 Adres Ss --5 Iverse Z-Trsform The oeffets j re foud usg rtl frtos, d gve the trsform rs:,,,, s well s the ROC, oe fd the sequee orresodg to. 6 Coyrght 6 Adres Ss --6
6 Coyrght 6 Adres Ss --7 Prtl Frtos Gve rtol futo: b b b...... for dstt oles we wrte s:... If >= where re ostts to be determed 6 Coyrght 6 Adres Ss --8 Prtl Frtos Exmle Gve the seod order futo Use rtl frtos to wrte s : 6 6 5 3 - - 9 8
Prtl Frtos Exmle Cot. d gve the ROC the h 9 8, 3 If the ROC s defed dfferetly the the tme dom sequees wll be dfferet. 6 Coyrght 6 Adres Ss --9 Prtl Frtos Exmle Cot. For exmle, ROC: 3 yelds h 8, 3 h 9, 6 Coyrght 6 Adres Ss --
Prtl Frtos Exmle Cot. For the ROC: 3 yelds h, 8 9 3 6 Coyrght 6 Adres Ss -- Iverse Z-Trsform - Reeted Poles Gve rtol futo: the the tme dom sequee s For ROC: h, 6 6 6 6 h, 6 6 Coyrght 6 Adres Ss --
DSP EEE 47/59 Adres Ss eture 6 Coyrght 6 Adres Ss --3 Iverse Z-Trsform usg the Resdue Theorem We wll restrt our dsusso to usl sequees. Exteso to ousl s strght-forwrd. Gve the -trsform: x Cuhy s Itegrl Theorem: k k d x d 6 Coyrght 6 Adres Ss --4
Iverse Z-Trsform usg the Resdue Theorem Cot. If the tegrto s outerlokwse o otour whh s wth the ROC d ludes the ut rle, the k k d x d If the th eloses the org the k d j k therefore x d j Note the smlrty wth the verse DTFT whh s sel se of the trsform 6 Coyrght 6 Adres Ss --5 Iverse Z-Trsform usg the Resdue Theorem Cot. Cuhy s resdue theorem sttes tht for rtol olyomls the tegrl bove be omuted s sum of resdues, tht s gve: A... x the res [ ] 6 Coyrght 6 Adres Ss --6
6 Coyrght 6 Adres Ss --7 Iverse Z-Trsform usg the Resdue Theorem Cot. For oles of multlty m m m m d d m res! ] [ For sgle ole res ] [ ] [ Exmles: Sgle ole 6 Coyrght 6 Adres Ss --8 Iverse Z-Trsform usg the Resdue Theorem Cot. res ] [ Pole of multlty d d res ] [
6 Coyrght 6 Adres Ss --9 Exmle: Stedy-Stte d Trset Resose of Dgtl Flters Usg the verse -trsform Cosder the frst order IIR flter.7 3 Y 3 y.7 x.7 3 6 Coyrght 6 Adres Ss --3 Exmle: Stedy-Stte d Trset Resose Cot., 7.7.7.7 3.7 3 ] [ ] [.7.7 y y Y res Y res y Stedy-stte ole of / trset ole of flter
Exmle:Stedy-Stte d Trset Resose y 7.7, 9 8 7 6 5 4 3 4 6 8 4 6 8 6 Coyrght 6 Adres Ss --3