LTI Discrt-Tim Systms i Trasform Domai Frqucy Rspos Trasfr Fuctio Itroductio to Filtrs Taia Stathai 811b t.stathai@imprial.ac.u
Frqucy Rspos of a LTI Discrt-Tim Systm Th wll ow covolutio sum dscriptio of a LTI discrt-tim systm is giv by y[ ] h [ ] x[ ] Taig th DTFT of both sids w obtai Y ( y[ ] h[ ] x[ ]
Frqucy Rspos of a LTI Discrt-Tim Systm Or, Y ( h[ ] l x[ l] ( l+ h[ ] l x[l] l X (
Frqucy Rspos of a Frqucy Rspos of a LTI Discrt LTI Discrt-Tim Systm Tim Systm Hc, w ca writ Th abov quatio rlats th iput ad th output of a LTI systm i th frqucy domai It follows that ( ( ( ] [ ( X H X h Y ( / ( ( X Y H
Frqucy Rspos of a LTI Discrt-Tim Systm Th fuctio H ( is calld th frqucy rspos of th LTI discrt-tim systm H ( provids a frqucy-domai dscriptio of th systm H ( is prcisly th DTFT of th impuls rspos {h[]} of th systm
Frqucy Rspos of a LTI Discrt-Tim Systm H (, i gral, is a complx fuctio ofwith a priod 2π It ca b xprssd i trms of its ral ad imagiary part H ( H ( + H ( or, i trms of its magitud ad phas, whr H ( θ( r H ( argh ( im θ(
Frqucy Rspos of a LTI Discrt-Tim Systm Not: Magitud ad phas fuctios ar ral fuctios of, whras th frqucy rspos is a complx fuctio of If th impuls rsposh[] is ral th it is prov that th magitud fuctio is a v fuctio of: H ( H ( ad th phas fuctio is a odd fuctio of: θ( θ( Liwis, for a ral impuls rsposh[], is v ad H im( is odd H r(
Magitud ad Phas Rspos Th fuctio H ( is calld th magitud rspos ad th fuctio θ( is calld th phas rspos of a LTI discrt-tim systm Dsig spcificatios for a LTI discrttim systm, i may applicatios, ar giv i trms of th magitud rspos or th phas rspos or both
Gai Fuctio Attuatio/Loss Fuctio I som cass, th magitud fuctio is spcifid i dcibls as whr G( G( 20log 10 H ( db is calld th gai fuctio Th gativ of th gai fuctio A( G( is calld th attuatio or loss fuctio
Eigfuctio Eigfuctio of a Systm of a Systm If th iput of a LTI systm is a sigl complx xpotial fuctio th it follows that th output is giv by Lt < < x o, ] [ o o o h h y ( ] [ ] [ ] [ h H ] [ (
Eigfuctio of a Systm Thrfor, w ca writ y[ ] H ( o Thus for a complx xpotial iput sigal o, th output of a LTI discrt-tim systm is also a complx xpotial sigal of th sam frqucy multiplid by a complx costath ( o o is calld igfuctio of th systm o
Frqucy Rspos of a Systm Dscribd by a Diffrc Equatio Cosidr a LTI discrt-tim systm charactrizd by a diffrc quatio N M 0 d y[ ] 0px[ ] Its frqucy rspos is obtaid by taig th DTFT of both sids of th abov quatio H ( M 0 N 0 p d
Exampl 1 Frqucy Rspos of a Movig Avrag (MA Filtr Cosidr th M-poit movig avrag filtr with a impuls rspos giv by h[] 1 / M, 0 M 1 0, othrwis Its frqucy rspos is th giv by H ( 1 M M 1 0
Exampl 1 - Movig Avrag Filtr Or, H 1 M 1 1 M ( 1 si( M/ 2 ( M 1 / 2 M si( / 2
Exampl 1 - Movig Avrag Filtr Thus, th magitud rspos of thmpoit movig avrag filtr is giv by 1 si( M / 2 H ( M si( / 2 ad th phas rspos is giv by θ ( M/ 2 ( M 1 + π µ ( 2 2π 1 M
Exampl 1 - Movig Avrag Filtr Frqucy Rspos Computatio Usig MATLAB Th fuctiofrqz(h,w ca b usd to dtrmi th valus of th frqucy rspos vctorhat a st of giv frqucy poitsw Fromh, th ral ad imagiary parts ca b computd usig th fuctiosral ad imag, ad th magitud ad phas fuctios usig th fuctiosabs ad agl
Exampl 1 - Movig Avrag Filtr Frqucy Rspos Computatio Usig MATLAB Th magitud ad phas rsposs of am-poit movig avrag filtr ar show blow Th umps i th phas fuctio occur at th zros of th frqucy rspos, whr th sic fuctio chags sig. 1 0.8 M5 M14 100 50 Magitud 0.6 0.4 Phas, dgrs 0-50 -100 0.2-150 M5 M14 0 0 0.2 0.4 0.6 0.8 1 /π -200 0 0.2 0.4 0.6 0.8 1 /π
Exampl 1 - Movig Avrag Filtr Frqucy Rspos Computatio Usig MATLAB Th phas rspos of a discrt-tim systm wh dtrmid by a computr may xhibit umps by a amout 2π causd by th way th arctagt fuctio is computd Th phas rspos ca b mad a cotiuous fuctio ofby uwrappig th phas rspos across th umps
Exampl 1 - Movig Avrag Filtr Frqucy Rspos Computatio Usig MATLAB To this d th fuctiouwrap ca b usd, providd th computd phas is i radias Th umps by th amout of 2π should ot b cofusd with th umps causd by th zros of th frqucy rspos as idicatd i th phas rspos of th movig avrag filtr
Exampl 2 Stady-Stat Rspos Dtrmi th stady-stat outputy[] of a ral cofficit LTI discrt-tim systm with a frqucy rspos H ( for a iput x[ ] Acos( o +φ, < < Not that th frqucy rspos dtrmis th stady-stat rspos of a LTI discrt-tim systm to a siusoidal iput
Exampl 2 Stady Stat Rspos W ca xprss th iputx[] as x [ ] g[ ] + g *[ ] whr g[ ] 2 1 A Now th output of th systm for a iput o is simply H ( o φ o o
Exampl 2 Stady Stat Rspos Bcaus of liarity, th rsposv[] to a iputg[] is giv by v[ ] 1 2 φ A H ( o o Liwis, th outputv*[] to th iputg*[] is v*[ ] 1 2 φ A H ( o o
Exampl 2 Stady Stat Rspos Combiig th last two quatios w gt o o o o H A H A ( 2 1 ( 2 1 + φ φ ] *[ ] [ ] [ v v y + { } o o o o o AH ( ( ( 2 1 + φ θ φ θ ( cos( ( +θ +φ o o H A o
Exampl 2 Stady Stat Rspos Thus, th outputy[] has th sam siusoidal wavform as th iput with two diffrcs: (1 th amplitud is multiplid by ( o, th valu of th magitud fuctio at o (2 th output has a phas lag rlativ to th iput by a amout θ( o, th valu of th phas fuctio at o H
Exampl 3 Rspos to a Causal Expotial Squc Th xprssio for th stady-stat rspos dvlopd arlir assums that th systm is iitially rlaxd bfor th applicatio of th iput I practic, xcitatio (iput to a discrttim systm is usually a right-sidd squc applid at som sampl idx o W dvlop th xprssio for th output for such a iput
Exampl 3 Rspos to a Causal Expotial Squc Without ay loss of grality, assum for < 0 From th iput-output rlatio y [ ] h[ ] x[ ] w obsrv that for a iput x[ ] [ ] th output is giv by µ y[ ] h[ ] 0 ( µ [ ] x[ ] 0
Exampl 3 Rspos to a Causal Expotial Squc Or, Th output for< 0 isy[] 0 Th output for is giv by ] [ ] [ ] [ 0 h y µ h h + 1 0 ] [ ] [ 0 h y 0 ] [ ] [
Exampl 3 Rspos to a Causal Expotial Squc Or, Th first trm o th RHS is th sam as that obtaid wh th iput is applid at 0 to a iitially rlaxd systm ad is th stady-stat rspos: h H y + 1 ] [ ( ] [ sr H y ( ] [
Exampl 3 Rspos to a Causal Expotial Squc Th scod trm o th RHS is calld th trasit rspos: To dtrmi th ffct of th abov trm o th total output rspos, w obsrv tr h y + 1 ] [ ] [ + + 0 1 1 ( ] [ ] [ ] [ ] [ tr h h h y
Exampl 3 Rspos to a Causal Expotial Squc For a causal, stabl LTI IIR discrt-tim systm,h[] is absolutly summabl As a rsult, th trasit rspos y tr [] is a boudd squc Morovr, as, +1h[ ] 0 ad hc, th trasit rspos dcays to zro asgts vry larg
Exampl 3 Rspos to a Causal Expotial Squc For a causal FIR LTI discrt-tim systm with a impuls rspos of lgthn+ 1, h[] 0 for> N Hc, y tr [ ] 0 for>n 1 Hr th output rachs th stady-stat valu y [ ] H ( at N sr
Th Trasfr Fuctio Th trasfr fuctio is a gralizatio of th frqucy rspos fuctio Th covolutio sum dscriptio of a LTI discrt-tim systm with a impuls rsposh[] is giv by y [ ] h[ ] x[ ]
Th Trasfr Fuctio Th Trasfr Fuctio Taig th z-trasform of both sids w gt z x h z y z Y ] [ ] [ ] [ ( z x h ] [ ] [ + z x h l l l ( ] [ ] [
Th Trasfr Fuctio Or, Y( z Thrfor, Thus, h l x z [ ] [l] l z X (z Y( z h[ ] z X ( z H(z Y(z H(zX(z
Th Trasfr Fuctio Hc, H ( z Y( z/ X ( z Th fuctioh(z, which is th z-trasform of th impuls rsposh[] of th LTI systm, is calld th trasfr fuctio or th systm fuctio Th ivrs z-trasform of th trasfr fuctioh(z yilds th impuls rsposh[]
Th Trasfr Fuctio Cosidr a LTI discrt-tim systm charactrizd by a diffrc quatio N M 0 d y[ ] 0px[ ] Its trasfr fuctio is obtaid by taig th z-trasform of both sids of th abov quatio H( z M 0 N 0 p d z z
Th Trasfr Fuctio Th Trasfr Fuctio Or, quivaltly A altrat form of th trasfr fuctio is giv by N N M M M N z d z p z z H 0 0 ( ( N M z z d p z H 1 1 1 1 0 0 (1 (1 ( λ ξ
Th Trasfr Fuctio Or, quivaltly H( z p d 0 0 z ( N M M 1 N 1 ξ λ ξ 1, ξ2,..., ξ M ar th fiit zros, ad λ 1, λ2,..., λ N ar th fiit pols of H(z IfN> M, thr ar additioal ( N M zros at z 0 IfN< M, thr ar additioal ( M N pols atz 0 ( z ( z
Th Trasfr Fuctio If M N thh(z ca b r-xprssd as M N P1 ( z H ( z z l η l + D( l 0 z whr th dgr of P 1 (z is lss than Th ratioal fuctio P 1 ( z/ D( z is calld a propr fractio
Th Trasfr Fuctio Simpl Pols: I most practical cass, th ratioal z-trasform of itrsth(z is a propr fractio with simpl pols Lt th pols ofh(z b at z λ, 1 N A partial-fractio xpasio ofh(z is th of th form N ρ l H ( z 1 l 1 1 λlz
Th Trasfr Fuctio ρ l Th costats i th partial-fractio xpasio ar calld th rsidus ad ar giv by λ 1 (1 z H ( z ρ l l z λ Each trm of th sum i partial-fractio xpasio has a ROC giv by z > λ l ad, thus has a ivrs trasform of th form ( λ µ [ ] ρl l l
Th Trasfr Fuctio - Stability Thrfor, th impuls rspos h[] is of ifiit duratio (IIR ad is giv by h[ ] l 1 Thus, th ROC is giv by N ρ ( l λl µ Furthrmor, for stability,, which mas that all pols must li isid th uit circl λ z [ ] > maxλ <1
Th Trasfr Fuctio - Stability Thrfor, for a stabl ad causal digital filtr for whichh[] is a right-sidd squc, th ROC will iclud th uit circl ad th tir z-pla icludig th poitz O th othr had, FIR digital filtrs with boudd impuls rspos ar always stabl Problm: Us th abov approach to dtrmi th ivrs of a ratioal z-trasform of a ocausal squc
Th Trasfr Fuctio - Stability O th othr had, a IIR filtr may b ustabl if ot dsigd proprly I additio, a origially stabl IIR filtr charactrizd by ifiit prcisio cofficits may bcom ustabl wh cofficits gt quatizd du to implmtatio
Exampl 4 Effcts of Quatizatio Cosidr th causal IIR trasfr fuctio H( z 1 1. 845z 1 1 + 0. 850586z 2 Th plot of th impuls rspos cofficits is show o th xt slid
Exampl 4 Effcts of Quatizatio 6 Amplitud 4 2 h[] 0 0 10 20 30 40 50 60 70 Tim idx As ca b s from th abov plot, th impuls rspos cofficith[] dcays rapidly to zro valu asicrass
Exampl 4 Effcts of Quatizatio Th absolut summability coditio ofh[] is satisfid Hc,H(z is a stabl trasfr fuctio Now, cosidr th cas wh th trasfr fuctio cofficits ar roudd to valus with 2 digits aftr th dcimal poit: ^ 1 H( z 1 2 1 1. 85z + 0. 85z
Exampl 4 Effcts of Quatizatio A plot of th impuls rspos of h^[] is show blow 6 h^[] Amplitud 4 2 0 0 10 20 30 40 50 60 70 Tim idx
Exampl 4 Effcts of Quatizatio I this cas, th impuls rspos cofficit h^[] icrass rapidly to a costat valu as icrass Hc, th absolut summability coditio of is violatd Thus, (z is a ustabl trasfr fuctio H^
Exampl 5 Trasfr Fuctio of a Movig Avrag Filtr Cosidr th M-poit movig-avrag FIR filtr with a impuls rspos 1/ M, 0 M 1 h[] 0, othrwis Its trasfr fuctio is th giv by 1 H ( z M M 1 0 z 1 z M (1 M z M z 1 1 M[ zm ( z 1]
Exampl 5 Movig Avrag Filtr Th trasfr fuctio has M zros o th uit circl at z 2π / M, 0 M 1 Thr ar M 1 pols atz 0 ad a sigl pol atz 1 M 8 Th pol atz 1 xactly cacls th zro at z 1 Th ROC is th tir z-pla xcptz 0 Imagiary Part 1 0.5 0-0.5-1 -1-0.5 0 0.5 1 Ral Part 7
Exampl 6 A IIR Filtr A causal LTI IIR digital filtr is dscribd by a costat cofficit diffrc quatio giv by y[ ] x[ 1] 1.2x[ 2] + x[ 3] + 1.3y[ 1] 1.4y[ 2] + 0.222y[ 3] Its trasfr fuctio is thrfor giv by H ( z z 1 1.3z 1 1 1.2z + 1.04z 2 2 3 + z 0.222z 3
Exampl 6 A IIR Filtr Altrat forms: z2 1.2z+ 1 H ( z z3 1.3z2+ 1.04z 0.222 ( z 0.6+ 0.8( z 0.6 0.8 ( z 0.3( z 0.5+ 0.7( z 0.5 0.7 Not: Pols farthst from z 0 hav a magitud 0.74 ROC: z > 0.74 Imagiary Part 1 0.5 0-0.5-1 -1-0.5 0 0.5 1 Ral Part
Frqucy Rspos from Trasfr Fuctio If th ROC of th trasfr fuctio H(z icluds th uit circl, th th frqucy rspos H ( of th LTI digital filtr ca b obtaid simply as follows: H ( H ( z z For a ral cofficit trasfr fuctioh(z it ca b show that 2 H ( H ( H *( H ( H ( H ( z H ( z 1 z
Frqucy Rspos from Frqucy Rspos from Trasfr Fuctio Trasfr Fuctio For a stabl ratioal trasfr fuctio i th form th factord form of th frqucy rspos is giv by N M M N z z z d p z H 1 1 0 0 ( ( ( ( λ ξ λ ξ N M M N d p H 1 1 ( 0 0 ( ( (
Frqucy Rspos from Trasfr Fuctio It is covit to visualiz th cotributios of th zro factor ( z ξ ad th pol factor ( z λ from th factord form of th frqucy rspos Th magitud fuctio is giv by H ( p d 0 0 ( N M M 1 N 1 ξ λ
Frqucy Rspos from Trasfr Fuctio which rducs to M ξ p 1 H ( 0 d N 0 1 λ Th phas rspos for a ratioal trasfr fuctio is of th form argh ( arg( p0 / d0 +( N M + M 1 N arg( ξ arg( 1 λ
Frqucy Rspos from Trasfr Fuctio Th magitud-squard fuctio of a ralcofficit trasfr fuctio ca b computd usig H ( 2 p d 0 0 2 M 1 N 1 ( ( ξ λ ( ( ξ λ * *
Gomtric Itrprtatio of Frqucy Rspos Computatio Th factord form of th frqucy rspos M p ξ N M H 0 ( 1( ( d N 0 1( λ is covit to dvlop a gomtric itrprtatio of th frqucy rspos computatio from th pol-zro plot as varis from 0 to 2π o th uit circl
Gomtric Itrprtatio of Frqucy Rspos Computatio Th gomtric itrprtatio ca b usd to obtai a stch of th rspos as a fuctio of th frqucy A typical factor i th factord form of th frqucy rspos is giv by ρ ( φ whr ρ φ is a zro if it is zro factor or is a pol if it is a pol factor
Gomtric Itrprtatio of Frqucy Rspos Computatio As show blow i thz-pla th factor ( ρ φ rprsts a vctor startig at th poit z ρφ ad dig o th uit circl atz
Gomtric Itrprtatio of Frqucy Rspos Computatio As is varid from 0 to 2π, th tip of th vctor movs coutrclocis from th poit z 1 tracig th uit circl ad bac to th poit z 1
Gomtric Itrprtatio of Frqucy Rspos Computatio As idicatd by p H ( d 0 0 M 1 N 1 th magitud rspos H ( at a spcific valu of is giv by th product of th magituds of all zro vctors dividd by th product of th magituds of all pol vctors ξ λ
Gomtric Itrprtatio of Frqucy Rspos Computatio Liwis, from argh ( arg( p0 / d0 +( N M M N + arg( ξ 1arg( λ w obsrv that th phas rspos at a spcific valu ofis obtaid by addig th phas of th trm p 0 / d 0 ad th liar-phas trm ( N M to th sum of th agls of th zro vctors mius th agls of th pol vctors 1
Gomtric Itrprtatio of Frqucy Rspos Computatio Thus, a approximat plot of th magitud ad phas rsposs of th trasfr fuctio of a LTI digital filtr ca b dvlopd by xamiig th pol ad zro locatios Now, a zro (pol vctor has th smallst magitud wh φ
Gomtric Itrprtatio of Frqucy Rspos Computatio To highly attuat sigal compots i a spcifid frqucy rag, w d to plac zros vry clos to or o th uit circl i this rag Liwis, to highly mphasiz sigal compots i a spcifid frqucy rag, w d to plac pols vry clos to or o th uit circl i this rag
Th Cocpt of Filtrig O applicatio of a LTI discrt-tim systm is to pass crtai frqucy compots of a iput squc without ay distortio (if possibl ad to bloc othr frqucy compots Such systms ar calld digital filtrs ad o of th mai subcts of discussio i this cours
Th Cocpt of Filtrig Th Cocpt of Filtrig Lt us cosidr th IDTFT This trasform xprsss a arbitrary systm output sigal as a liar wightd sum of a ifiit umbr of xpotial squcs π π π ( ( 2 1 ] [ d X H y
Th Cocpt of Filtrig Thus, by appropriatly choosig th valus of th magitud fuctio H ( of th LTI digital filtr, som of th iput frqucy compots ca b slctivly havily attuatd or filtrd with rspct to th othrs
Th Cocpt of Filtrig To udrstad th mchaism bhid th dsig of frqucy-slctiv filtrs, cosidr a ral-cofficit LTI discrt-tim systm charactrizd by a magitud fuctio H ( 1, 0, c < π c
Th Cocpt of Filtrig W apply a iput x [ ] Acos 1+ B cos2, 0< 1 < c < 2 < π to this systm Bcaus of liarity, th output of this systm is of th form y[ ] AH ( 1 cos 1 +θ( 1 ( + H ( 2 cos 2+θ( 2 B (
Th Cocpt of Filtrig As H ( 2 th output rducs to 1 1, H ( y[ ] AH ( 1 cos( ( 1+θ 1 Thus, th systm acts li a lowpass filtr I th followig xampl, w cosidr th dsig of a vry simpl digital filtr 0
Exampl 7 - Th Cocpt of Filtrig A iput sigal cosists of a sum of two siusoidal squcs of agular frqucis 0.1 rad/sampl ad 0.4 rad/sampl W d to dsig a highpass filtr that will pass th high-frqucy compots of th iput but bloc th low-frqucy compots For simplicity, assum th filtr to b a FIR filtr of lgth 3 with a impuls rspos: h[0] h[2] α,h[1] β
Exampl 7 - Th Cocpt of Filtrig Th covolutio sum dscriptio of this filtr is th giv by y[ ] h[0] x[ ] + h[1] x[ 1] + h[2] x[ 2] αx[ ] +βx[ 1] +αx[ 2] y[] adx[] ar, rspctivly, th output ad th iput squcs Dsig Obctiv: Choos suitabl valus of α ad β so that th output is a siusoidal squc with a frqucy 0.4 rad/sampl
Exampl 7 - Th Cocpt of Filtrig Now, th frqucy rspos of th FIR filtr is giv by H ( h[0] + h[1] + h[2] 2 α( 1+ 2 + β + 2α +β 2 (2αcos+β
Exampl 7 - Th Cocpt of Filtrig Th magitud ad phas fuctios ar H ( 2αcos+β θ ( I ordr to bloc th low-frqucy compot, th magitud fuctio at 0.1 should b qual to zro Liwis, to pass th high-frqucy compot, th magitud fuctio at 0.4 should b qual to o
Exampl 7 - Th Cocpt of Filtrig Thus, th two coditios that must b satisfid ar H ( 0. 1 2αcos(0.1 +β 0 0. H ( 4 2αcos(0.4 +β 1 Solvig th abov two quatios w gt α 6.76195 β13.456335
Exampl 7 - Th Cocpt of Filtrig Thus th output-iput rlatio of th FIR filtr is giv by y[ ] 6.76195x[ ] + x[ 2] + 13.456335x[ 1 ( ] whr th iput is x [ ] {cos(0.1 + cos(0.4} µ [ ]
Exampl 7 - Th Cocpt of Filtrig Figur blow shows th plots gratd by ruig this program 4 3 y[] x 2 [] x 1 [] Amplitud 2 1 0-1 0 20 40 60 80 100 Tim idx
Exampl 7 - Th Cocpt of Filtrig Th first sv sampls of th output ar show blow
Exampl 7 - Th Cocpt of Filtrig From this tabl, it ca b s that, glctig th last sigificat digit, y[ ] cos(0.4( 1 for 2 Computatio of th prst valu of th output rquirs th owldg of th prst ad two prvious iput sampls Hc, th first two output sampls,y[0] ady[1], ar th rsult of assumd zro iput sampl valus at 1 ad 2
Exampl 7 - Th Cocpt of Filtrig Thrfor, first two output sampls costitut th trasit part of th output Sic th impuls rspos is of lgth 3, th stady-stat is rachd at N 2 Not also that th output is dlayd vrsio of th high-frqucy compot cos(0.4 of th iput, ad th dlay is o sampl priod
Phas ad Group Dlays Th outputy[] of a frqucy-slctiv LTI discrt-tim systm with a frqucy rspos H ( xhibits som dlay rlativ to th iputx[] causd by th ozro phas rspos θ( arg{ H ( } of th systm For a iput [ ] Acos( + φ, < x o <
Phas ad Group Dlays th output is y[ ] AH ( o cos( +θ( + φ Thus, th output lags i phas by radias Rwritig th abov quatio w gt θ ( + o y[ ] AH ( o cos o +φ o o o θ( o
Phas ad Group Dlays This xprssio idicats a tim dlay, ow as phas dlay, at o giv by θ( o τp( o Now cosidr th cas wh th iput sigal cotais may siusoidal compots with diffrt frqucis that ar ot harmoically rlatd o
Phas ad Group Dlays I this cas, ach compot of th iput will go through diffrt phas dlays wh procssd by a frqucy-slctiv LTI discrt-tim systm Th, th output sigal, i gral, will ot loo li th iput sigal Th sigal dlay ow is dfid usig a diffrt paramtr
Phas ad Group Dlays To dvlop th cssary xprssio, cosidr a discrt-tim sigalx[] obtaid by a doubl-sidbad supprssd carrir (DSB-SC modulatio with a carrir frqucy c of a low-frqucy siusoidal sigal of frqucy : o x[ ] Acos( o cos( c
Phas ad Group Dlays Th iput ca b rwritt as x[ ] Acos( + Acos( 2 l l c o whr ad Lt th abov iput b procssd by a LTI discrt-tim systm with a frqucy rspos H ( satisfyig th coditio H ( 1 for 2 l u u c u + o
Phas ad Group Dlays Th outputy[] is th giv by A A 2 u +θ u y[ ] cos l +θ l + 2 θ( +θ( Acos + u l c cos 2 ( ( cos( ( θ( 2 Not: Th output is also i th form of a modulatd carrir sigal with th sam carrir frqucy c ad th sam modulatio frqucy as th iput o θ( + o u l
Phas ad Group Dlays Howvr, th two compots hav diffrt phas lags rlativ to thir corrspodig compots i th iput Now cosidr th cas wh th modulatd iput is a arrowbad sigal with th frqucis l ad u vry clos to th carrir frqucy, i.. is vry small c o
Phas ad Group Dlays I th ighborhood of c w ca xprss th uwrappd phas rspos θ c ( as dθ ( ( ( c θc θc c + ( c d by maig a Taylor s sris xpasio ad pig oly th first two trms Usig th abov formula w ow valuat th tim dlays of th carrir ad th modulatig compots c
Phas ad Group Dlays I th cas of th carrir sigal w hav θ +θ 2 c( u c( l c θ which is s to b th sam as th phas dlay if oly th carrir sigal is passd through th systm c ( c c
Phas ad Group Dlays I th cas of th modulatig compot w hav θc( u θc( dθ c l ( u l d c Th paramtr dθ c ( τg ( c d c is calld th group dlay or vlop dlay causd by th systm at c
Phas ad Group Dlays Th group dlay is a masur of th liarity of th phas fuctio as a fuctio of th frqucy It is th tim dlay btw th wavforms of udrlyig cotiuous-tim sigals whos sampld vrsios, sampld att T, ar prcisly th iput ad th output discrt-tim sigals
Phas ad Group Dlays If th phas fuctio ad th agular frqucyar i radias pr scod, th th group dlay is i scods Figur blow illustrats th valuatio of th phas dlay ad th group dlay
Phas ad Group Dlays Figur blow shows th wavform of a amplitud-modulatd iput ad th output gratd by a LTI systm
Phas ad Group Dlays Not: Th carrir compot at th output is dlayd by th phas dlay ad th vlop of th output is dlayd by th group dlay rlativ to th wavform of th udrlyig cotiuous-tim iput sigal Th wavform of th udrlyig cotiuoustim output shows distortio wh th group dlay of th LTI systm is ot costat ovr th badwidth of th modulatd sigal
Phas ad Group Dlays If th distortio is uaccptabl, a dlay qualizr is usually cascadd with th LTI systm so that th ovrall group dlay of th cascad is approximatly liar ovr th bad of itrst To p th magitud rspos of th part LTI systm uchagd th qualizr must hav a costat magitud rspos at all frqucis
Exampl 8 - Phas ad Group Dlay of a Scod Ordr FIR Filtr Th phas fuctio of th FIR filtr y[ ] αx[ ] + βx[ 1] + αx[ 2] with β > 2α is θ ( Hc its group dlay is giv by τ g ( 1
Exampl 8 - Phas ad Group Dlay of a M-ordr MA filtr For thm-poit movig-avrag filtr 1/ M, 0 M 1 h[] 0, othrwis th phas fuctio is θ ( ( 1 2 M M / 2 π + π 1 Hc its group dlay is 1 τ ( M g 2 µ 2 M