Th output y[ of a frquncy-sctiv LTI iscrt-tim systm with a frquncy rspons H ( xhibits som ay rativ to th input caus by th nonro phas rspons θ( ω arg{ H ( } of th systm For an input A cos( ωo n + φ, < n < th output is y[ AH ( o cos( ω n + θ( ω + φ Thus, th output ags in phas by θ( ωo raians Rwriting th abov quation w gt θ( ω o y[ AH ( o cos ωo n+ + φ ωo o o 3 This xprssion inicats a tim ay, nown as phas ay, at ω ω o givn by θ( ωo τ p( ωo ω ow consir th cas whn th input signa contains many sinusoia componnts with iffrnt frquncis that ar not harmonicay rat o 4 In this cas, ach componnt of th input wi go through iffrnt phas ays whn procss by a frquncy-sctiv LTI iscrt-tim systm Thn, th output signa, in gnra, wi not oo i th input signa Th signa ay now is fin using a iffrnt paramtr To vop th ncssary xprssion, consir a iscrt-tim signa obtain by a oub-siban supprss carrir (DSB-SC mouation with a carrir frquncy ω c of a ow-frquncy sinusoia signa of frquncy : ω o Acos( ω o ncos( ω n c Th input can b rwrittn as Acos( ω n + A cos( ω n whr ω ωc ω o an ω u ω c + ω o Lt th abov input b procss by an LTI iscrt-tim systm with a frquncy rspons H ( satisfying th conition H ( u for ω ω ω u 5 6
Th output y[ is thn givn by y[ A cos( ω n + θ( ω + Acos( ω n + θ( ω θ( ω + θ( ω θ( ω θ( ω Acos ω + u cos ω + u cn on ot: Th output is aso in th form of a mouat carrir signa with th sam carrir frquncy ω c an th sam mouation frquncy as th input ω o u u Howvr, th two componnts hav iffrnt phas ags rativ to thir corrsponing componnts in th input ow consir th cas whn th mouat input is a narrowban signa with th frquncis ω an ωu vry cos to th carrir frquncy ω, i.. ω is vry sma c o 7 8 9 In th nighborhoo of ω c w can xprss th unwrapp phas rspons θ c (ω as θc( ω θc( ω θc ( ωc + ( ω ωc ω ωωc by maing a Tayor s sris xpansion an ping ony th first two trms Using th abov formua w now vauat th tim ays of th carrir an th mouating componnts In th cas of th carrir signa w hav θc ( ωu + θc ( ω θc ω ( c ωc ωc which is sn to b th sam as th phas ay if ony th carrir signa is pass through th systm In th cas of th mouating componnt w hav θc ( ωu θc( ω θ c ( ω ωu ω ω ωωc Th paramtr θ c ( ω τg( ωc ω ωωc is ca th group ay or nvop ay caus by th systm at ω ω c Th group ay is a masur of th inarity of th phas function as a function of th frquncy It is th tim ay btwn th wavforms of unrying continuous-tim signas whos samp vrsions, samp at t nt, ar prcisy th input an th output iscrt-tim signas
If th phas function an th anguar frquncy ω ar in raians pr scon, thn th group ay is in scons Figur bow iustrats th vauation of th phas ay an th group ay Figur bow shows th wavform of an ampitu-mouat input an th output gnrat by an LTI systm 3 4 5 ot: Th carrir componnt at th output is ay by th phas ay an th nvop of th output is ay by th group ay rativ to th wavform of th unrying continuous-tim input signa Th wavform of th unrying continuoustim output shows istortion whn th group ay of th LTI systm is not constant ovr th banwith of th mouat signa 6 If th istortion is unaccptab, a ay quair is usuay casca with th LTI systm so that th ovra group ay of th casca is approximaty inar ovr th ban of intrst To p th magnitu rspons of th parnt LTI systm unchang th quair must hav a constant magnitu rspons at a frquncis 7 Examp - Th phas function of th FIR fitr y[ α + β n ] + α n ] is θ ( ω ω Hnc its group ay is givn by τ g ( ω vrifying th rsut obtain arir by simuation 8 Examp - For th -point moving-avrag fitr /, n h[, othrwis th phas function is / ( ω π θ( ω + π µ ω Hnc its group ay is g( ω τ 3
9 Frquncy Rspons of th LTI Discrt-Tim Systm Th convoution sum scription of th LTI iscrt-tim systm is givn by y[ h[ ] n ] Taing th DTFT of both sis w obtain Y( y[ n n n ] j n h [ ] x [ ω n Frquncy Rspons of th LTI Discrt-Tim Systm Or, Y( h[ ] ] [] h[ ] x X( ( + Frquncy Rspons of th LTI Discrt-Tim Systm Hnc, w can writ ω ( ω ω ω j j j j Y h[ ] X ( H ( X( In th abov H ( is th frquncy rsponsof th LTI systm Th abov quation rats th input an th output of an LTI systm in th frquncy omain Frquncy Rspons of th LTI Discrt-Tim Systm It foows from th prvious quation j ω H ( Y( / X( For an LTI systm scrib by a inar constant cofficint iffrnc quation of th form w hav H ( p 3 A gnraiation of th frquncy rspons function Th convoution sum scription of an LTI iscrt-tim systm with an impus rsponsh[ is givn by h y [ [ ] n ] 4 Taing th -transforms of both sis w gt n Y( y[ h[ ] n ] n n n h [ ] n ] n ( + h[ ] ] n 4
Or, X ( Thrfor, Y( h[ ] X ( Thus, Y( h[ ] ] H ( Y( H(X( Hnc, H ( Y( / X ( Th function H(, which is th -transform of th impus rspons h[ of th LTI systm, is ca th transfr function or th systm function Th invrs -transform of th transfr function H( yis th impus rspons h[ 5 6 7 Consir an LTI iscrt-tim systm charactri by a iffrnc quation y[ n ] p n ] Its transfr function is obtain by taing th -transform of both sis of th abov quation Thus p H ( 8 Or, quivanty as ( p H ( An atrnat form of th transfr function is givn by p H ( ( ξ ( λ 9 Or, quivanty as p ( ( ξ H ( ( λ ξ, ξ,..., ξ ar th finit ros, an λ, λ,..., λ ar th finit pos of H( If >, thr ar aitiona ( ros at If <, thr ar aitiona ( pos at 3 For a causa IIR igita fitr, th impus rspons is a causa squnc Th ROC of th causa transfr function p ( ( ξ H ( ( λ is thus xtrior to a circ going through th po furthst from th origin Thus th ROC is givn by > maxλ 5
3 Examp - Consir th -point movingavrag FIR fitr with an impus rspons h[ /, n, othrwis Its transfr function is thn givn by H ( n n ( [ ( ] 3 Th transfr function has ros on th unit circ at j π /, Thr ar pos at an a sing po at 8 Th po at.5 xacty cancs th 7 ro at -.5 Th ROC is th ntir - -pan xcpt - -.5.5 Imaginary Part Ra Part 33 Examp - A causa LTI IIR igita fitr is scrib by a constant cofficint iffrnc quation givn by y[ n ]. n ] + n 3] +.3 y[ n ].4 y[ n ] +. y[ n 3] Its transfr function is thrfor givn by H (.3. +.4 3 +. 3 34 Atrnat forms:. + H ( 3.3 +.4. (.6+ j.8(.6 j.8 (.3(.5 + j.7(.5 j.7 ot: Pos farthst from hav a magnitu.74 ROC: >.74 Imaginary Part.5 -.5 - - -.5.5 Ra Part 35 Frquncy Rspons from Transfr Function If th ROC of th transfr function H( incus th unit circ, thn th frquncy rspons H ( of th LTI igita fitr can b obtain simpy as foows: H ( j ω H ( For a ra cofficint transfr function H( it can b shown that H ( H( H *( H ( H ( H( H ( 36 Frquncy Rspons from Transfr Function For a stab rationa transfr function in th form p H ( ( ξ ( ( λ th factor form of th frquncy rspons is givn by ω ω ξ j p ( j ( H ( ( λ 6
37 Frquncy Rspons from Transfr Function It is convnint to visuai th contributions of th ro factor ( ξ an th po factor ( λ from th factor form of th frquncy rspons Th magnitu function is givn by ξ p ( H ( λ 38 Frquncy Rspons from Transfr Function which rucs to ξ p H ( λ Th phas rspons for a rationa transfr function is of th form arg H ( j ω arg( p / + ω( + ω arg( j ξ arg( λ 39 Frquncy Rspons from Transfr Function Th magnitu-squar function of a racofficint transfr function can b comput using H ( p ( ( ξ λ ( ( ξ λ * * 4 Th factor form of th frquncy rspons ω ω ξ j p ( j ( H ( ( λ is convnint to vop a gomtric intrprtation of th frquncy rspons computation from th po-ro pot as ω varis from to π on th unit circ 4 Th gomtric intrprtation can b us to obtain a stch of th rspons as a function of th frquncy A typica factor in th factor form of th frquncy rspons is givn by j ω jφ ( ρ whr ρ jφ is a ro if it is ro factor or is a po if it is a po factor 4 As shown bow in th -pan th factor ( ρj φ rprsnts a vctor starting at th point ρ jφ an ning on th unit circ at 7
43 As ω is vari from to π, th tip of th vctor movs countrcocis from th point tracing th unit circ an bac to th point 44 As inicat by ξ p H ( λ th magnitu rspons H( at a spcific vau of ω is givn by th prouct of th magnitus of a ro vctors ivi by th prouct of th magnitus of a po vctors 45 Liwis, from arg H ( j ω arg( p / + ω( j + arg( ξ arg( λ w obsrv that th phas rspons at a spcific vau of ω is obtain by aing th phas of th trm p / an th inar-phas trm ω( to th sum of th angs of th ro vctors minus th angs of th po vctors ω 46 Thus, an approximat pot of th magnitu an phas rsponss of th transfr function of an LTI igita fitr can b vop by xamining th po an ro ocations ow, a ro (po vctor has th smast magnitu whn ω φ To highy attnuat signa componnts in a spcifi frquncy rang, w n to pac ros vry cos to or on th unit circ in this rang Liwis, to highy mphasi signa componnts in a spcifi frquncy rang, w n to pac pos vry cos to or on th unit circ in this rang 47 8