The smple flers dscussed so far are characered eher by a sngle passband and/or a sngle sopband There are applcaons where flers wh mulple passbands and sopbands are requred Thecomb fler s an example of such flers In s mos general form a comb fler has a frequency response ha s a perodc of ω wh a perod π/l where L s a posve neger IfH s a fler wh a sngle passband and/or a sngle sopband a comb fler can be easly generaed from by replacng each delay n s realaon wh L delays resulng n a srucure wh a ransfer L gven by G H If H e exhbs a peak a ω p hen G e wll exhb L peaks a ω p k / L k L n he frequency range ω < π Lkewse f H e has a noch a ω o hen G e wll have L noches a ω o k / L k L n he frequency range ω < π comb fler can be generaed from eher an FIR or an IIR prooype fler 4 For example he comb fler generaed from he prooype lowpass FIR fler H has a ransfer L L G H Comb fler from lowpass prooype G e has L noches.8 a ω kπ/l and L.6 peaks a ω π k/l.4 k L n he. frequency range ω < π agnude.5.5 ω/π 5 For example he comb fler generaed from he prooype hghpass FIR fler H has a ransfer L L G H Comb fler from hghpass prooype G e has L peaks.8 a ω kπ/l and L.6 noches a ω π k/l.4 k L n he. frequency range ω < π agnude.5.5 ω/π 6 Dependng on applcaons comb flers wh oher ypes of perodc magnude responses can be easly generaed by appropraely choosng he prooype fler For example he -pon movng average fler H has been used as a prooype
Ths fler has a peak magnude a ω and noches a ω πl / l The correspondng comb fler has a ransfer L G L whose magnude has L peaks a ω πk/l k L and L noches a ω πk/l k L se of dgal ransfer s wh complemenary characerscs ofen fnds useful applcaons n pracce Four useful complemenary relaons are descrbed nex along wh some applcaons 7 8 9 Delay- se of L ransfer s { H } L s defned o be delaycomplemenary of each oher f he sum of her ransfer s s equal o some neger mulple of un delays.e. where L n o H β no β s a nonnegave neger delay-complemenary par { H H } can be readly desgned f one of he pars s a known Type FIR ransfer of odd lengh Le H be a Type FIR ransfer of lengh K Then s delay-complemenary ransfer s gven by H K H Le he magnude response of H be equal o ± δ p n he passband and less han or equal o δs n he sopband where δ p and δ s are very small numbers Now he frequency response of H can be expressed as j ω jkω H e e H where ω s he amplude response Is delay-complemenary ransfer H has a frequency response gven by j ω jkω jkω H e e H ω e [ H ω] Now n he passband δ p H ω δ p and n he sopband δs H ω δ s I follows from he above equaon ha n he sopband δ p H ω δ p and n he passband H ω δs H ω δ s
s a resul H has a complemenary magnude response characersc o ha of H wh a sopband exacly dencal o he passband of H and a passband ha s exacly dencal o he sopband of H Thus f H s a lowpass fler H wll be a hghpass fler and vce versa The frequency ω o a whch H ωo H ωo.5 he gan responses of boh flers are 6 db below her maxmum values The frequency ω o s hus called he 6-dB crossover frequency 4 5 Example-Consder he Type bandsop ransfer H 4 4 5 4 5 8 4 BS 64 Is delay-complemenary Type bandpass ransfer s gven by H BP HBS 4 4 5 4 5 8 4 64 6 Plos of he magnude responses of H BS and H BP are shown below agnude H BS.8.6.4. H BP..4.6.8 ω/π 7 llpass Complemenary Flers se of dgal ransfer s { H } s defned o be allpasscomplemenary of each oher f he sum of her ransfer s s equal o an allpass.e. H 8 Power- se of dgal ransfer s { H } s defned o be powercomplemenary of each oher f he sum of her square-magnude responses s equal o a consan K for all values of ω.e. H e K for all ω
9 By analyc connuaon he above propery s equal o H H K for all ω for real coeffcen H Usually by scalng he ransfer s he power-complemenary propery s defned for K For a par of power-complemenary ransfer s H and H he frequency where H e o H e o.5 s called he cross-over frequency hs frequency he gan responses of boh flers are -db below her maxmum values s a resul ω o s called he -db crossover frequency ω o Example-Consder he wo ransfer s H and H gven by H [ ] H [ where and are sable allpass ransfer s Noe ha H H Hence H and H are allpass complemenary ] I can be shown ha H and H are also power-complemenary oreover H and H are boundedreal ransfer s Doubly- se of ransfer s sasfyng boh he allpass complemenary and he powercomplemenary properes s known as a doubly-complemenary se 4 par of doubly-complemenary IIR ransfer s H and H wh a sum of allpass decomposon can be smply realed as ndcaed below / H H 4
5 Example-The frs-order lowpass ransfer α H LP α can be expressed as H where α [ ] α LP α α 6 Is power-complemenary hghpass ransfer s hus gven by α H HP [ ] α α α The above expresson s precsely he frsorder hghpass ransfer descrbed earler Fgure below demonsraes he allpass complemenary propery and he power complemenary propery of H LP and H HP agnude H e H e LP HP.8 H e HP.6 H e.4 LP agnude H e H e LP HP.8 H e HP.6.4 Power-Symmerc Flers real-coeffcen causal dgal fler wh a ransfer H s sad o be a powersymmerc fler f sasfes he condon H H H H K where K > s a consan.. H LP e 7..4.6.8 ω/π..4.6.8 ω/π 8 9 I can be shown ha he gan Gω of a power-symmerc ransfer a ω π s gven by log K db If we defne G H hen follows from he defnon of he power-symmerc fler ha H and G are powercomplemenary as H H G G a consan Conjugae Quadrac Fler If a power-symmerc fler has an FIR ransfer H of order N hen he FIR dgal fler wh a ransfer N G H s called a conjugae quadrac fler of H and vce-versa 5
I follows from he defnon ha G s also a power-symmerc causal fler I also can be seen ha a par of conjugae quadrac flers H and G are also power-complemenary Example-Le H 6 We form H H H H 6 6 H s a power-symmerc ransfer 6 6 4 5 4 4 5 4 Dgal Two-Pars The LTI dscree-me sysems consdered so far are sngle-npu sngle-oupu srucures characered by a ransfer Ofen such a sysem can be effcenly realed by nerconnecng wo-npu wooupu srucures more commonly called wo-pars 4 Dgal Two-Pars Fgures below show wo commonly used block dagram represenaons of a wo-par Here and denoe he wo oupus and and denoe he wo npus where he dependences on he varable has been omed for smplcy Dgal Two-Pars Dgal Two-Pars 5 The npu-oupu relaon of a dgal wopar s gven by In he above relaon he marx τ gven by τ s called he ransfer marx of he wo-par 6 I follows from he npu-oupu relaon ha he ransfer parameers can be found as follows: 6
Dgal Two-Pars n alernae characeraon of he wo-par s n erms of s chan parameers as C D B where he marx Γ gven by Γ C D B - - s called he chan marx of he wo-par Dgal Two-Pars The relaon beween he ransfer parameers and he chan parameers are gven by C D BC C B C D 7 8 9 Two-Par Inerconnecon Schemes Cascade Connecon - Γ-cascade Here B C D - - B C C C - B D B D D - 4 Two-Par Inerconnecon Schemes Bu from fgure and Subsung he above relaons n he frs equaon on he prevous slde and combnng he wo equaons we ge C Hence C D B C B D C B D C B D B D 4 Two-Par Inerconnecon Schemes Cascade Connecon - τ-cascade - - - - Here 4 Two-Par Inerconnecon Schemes Bu from fgure and Subsung he above relaons n he frs equaon on he prevous slde and combnng he wo equaons we ge Hence 7
4 Two-Par Inerconnecon Schemes Consraned Two-Par G H I can be shown ha C D G H B G G G 44 We have shown ha he BIBO sably of a causal raonal ransfer requres ha all s poles be nsde he un crcle For very hgh-order ransfer s s very dffcul o deermne he pole locaons analycally Roo locaons can of course be deermned on a compuer by some ype of roo fndng algorhms 45 We now oulne a smple algebrac es ha does no requre he deermnaon of pole locaons The Sably Trangle For a nd-order ransfer he sably can be easly checked by examnng s denomnaor coeffcens 46 Le D d d denoe he denomnaor of he ransfer In erms of s poles D can be expressed as D λ λ λ λ λλ Comparng he las wo equaons we ge d λ λ d λλ 47 The poles are nsde he un crcle f λ < λ < Now he coeffcen d s gven by he produc of he poles Hence we mus have d < I can be shown ha he second coeffcen condon s gven by d < d 48 The regon n he d d -plane where he wo coeffcen condon are sasfed called he sably rangle s shown below Sably regon 8
49 Example-Consder he wo nd-order bandpass ransfer s desgned earler: H BP. 889. 7444. 768 H BP. 67. 55. 76545 5 In he case of H BP we observe ha d.7444 d. 7689 Snce here d > H BP s unsable On he oher hand n he case of H BP we observe ha d.5598 d.765458 Here d < and d < d and hence s BIBO sable H BP 5 General Sably Tes Procedure Le D denoe he denomnaor of an -h order causal IIR ransfer H: D d where we assume d for smplcy Defne an -h order allpass ransfer : D D 5 Or equvalenly d d d d... d d... d d If we express D λ hen follows ha λ d 5 Now for sably we mus have λ < whch mples he condon d < Defne k d Then a necessary condon for sably of and hence he ransfer H s gven by < k 54 ssume he above condon holds We now form a new k d k d Subsung he raonal form of n he above equaon we ge d... d d d... d d 9
55 where d d d d d Hence s an allpass of order Now he poles λ o of are gven by he roos of he equaon λ o k 56 By assumpon k < Hence λ o > If s a sable allpass hen < for > for > for < Thus f s a sable allpass hen he condon λ o > holds only f < λ o 57 Or n oher words s a sable allpass Thus f s a sable allpass and k < hen s also a sable allpass of one order lower We now prove he converse.e. f s a sable allpass and k < hen s also a sable allpass 58 To hs end we express n erms of arrvng a k k ζ o If s a pole of hen ζ By assumpon o ζo k < k holds 59 > Therefore ζo ζo.e. ζ > ζ o o The above condon mples ζ o > f ζ o ssume s a sable allpass Then for Thus for ζ o we should have ζ o 6 Thus here s a conradcon On he oher hand f ζ o < hen from > for < we have ζ o > The above condon does no volae he condon ζ > ζ o o
6 Thus f k < and f s a sable allpass hen s also a sable allpass Summarng a necessary and suffcen se of condons for he causal allpass o be sable s herefore: k < and The allpass s sable 6 Thus once we have checked he condon k < we es nex for he sably of he lower-order allpass The process s hen repeaed generang a se of coeffcens: k k... k k and a se of allpass s of decreasng order:... 6 The allpass s sable f and only f k < for Example-Tes he sably of H 4 4 FromH we generae a 4-h order allpass 4 4 4 4 4 d4 d d 4 4 4 d 4 4 4 d Noe: k d 4 4 4 < 4 d d d4 64 Usng d d4d d 4 d 4 we deermne he coeffcens { d} of he hrd-order allpass from he coeffcens { d } of 4 : d d d 5 5 5 d d d 5 5 5 65 Noe: k d < 5 Followng he above procedure we derve he nex wo lower-order allpass s: 79 59 4 4 59 79 4 4 5 5 66 Noe: 79 4 5 < k < k Snce all of he sably condons are sasfed 4 and hence H are sable Noe:I s no necessary o derve snce can be esed for sably usng he coeffcen condons