DTFT Proprtis Exampl-Dtrmin th DTFT Y ( of n y[ ( n + α µ [, α < n Lt α µ [, α < W an thrfor writ y [ n + From Tabl 3.3, th DTFT of is givn by X ( α DTFT Proprtis Using th diffrntiation proprty of th DTFT givn in Tabl 3., w obsrv that th DTFT of n is givn by ( dx d α α ( α Nxt using th linarity proprty of th DTFT givn in Tabl 3.4 w arriv at α Y ( + ( α α ( α 3 DTFT Proprtis Exampl-Dtrmin th DTFT V ( of th squn v[ dfind by dv[ + dv[ n ] pδ[ + pδ [ n ] From Tabl 3.3, th DTFT of δ [ is Using th tim-shifting proprty of th DTFT givn in Tabl 3.4 w obsrv that th DTFT of δ[ n ] is and th DTFT of v[ n ] is V ( 4 DTFT Proprtis Using th linarity proprty of Tabl 3.4 w thn obtain th frquny-domain rprsntation of dv[ + dv[ n ] pδ[ + pδ [ n ] as d V ( + d V ( p + p Solving th abov quation w gt p + p V ( d + d Enrgy Dnsity Sptrum Enrgy Dnsity Sptrum 5 Th total nrgy of a finit-nrgy squn g[ is givn by From Parsval s rlation givn in Tabl 3.4 w obsrv that E g E g n g[ g[ G( n π π π 6 Th quantity S gg ( G( is alld th nrgy dnsity sptrum Th ara undr this urv in th rang π π dividd by π is th nrgy of th squn
Band-limitd Disrt-tim tim Signals Band-limitd Disrt-tim tim Signals Sin th sptrum of a disrt-tim signal is a priodi funtion of with a priod π, a full-band signal has a sptrum oupying th frquny rang π π A band-limitd disrt-tim signal has a sptrum that is limitd to a portion of th frquny rang π π An idal band-limitd signal has a sptrum that is zro outsid a frquny rang < a b < π, that is, < X ( a, b < < π An idal band-limitd disrt-tim signal annot b gnratd in prati 7 8 Band-limitd Disrt-tim tim Signals Band-limitd Disrt-tim tim Signals 9 A lassifiation of a band-limitd disrttim signal is basd on th frquny rang whr most of th signal s nrgy is onntratd A lowpass disrt-tim ral signal has a sptrum oupying th frquny rang < π and has a bandwidth of < p p A highpass disrt-tim ral signal has a sptrum oupying th frquny rang < p < π and has a bandwidth of π p A bandpass disrt-tim ral signal has a sptrum oupying th frquny rang < L H < π and has a bandwidth of H L Band-limitd Disrt-tim tim Signals Exampl Considr th squn (. 5 n µ [ Its DTFT is givn blow on th lft along with its magnitud sptrum shown blow X( on th right X (. 5 Magnitud.5.5..4.6.8 Band-limitd Disrt-tim tim Signals It an b shown that 8% of th nrgy of this lowpass signal is ontaind in th frquny rang. 58π Hn, w an dfin th 8% bandwidth to b.58π radians
3 Enrgy Dnsity Sptrum Exampl-Comput th nrgy of th squn sin n h n LP [ ] πn, < n < Hr π hlp[ H LP( π n whr H LP ( π,, < π 4 Enrgy Dnsity Sptrum Thrfor h LP n [ < π π Hn, h LP [ is a finit-nrgy lowpass squn 5 DTFT Computation Using MATLAB Th funtion frqz an b usd to omput th valus of th DTFT of a squn, dsribd as a rational funtion in in th form of M p + p +... + pm X ( N d + d +... + d at a prsribd st of disrt frquny points l N 6 DTFT Computation Using MATLAB For xampl, th statmnt H frqz(num,dn,w rturns th frquny rspons valus as a vtor H of a DTFT dfind in trms of th vtors num and dn ontaining th offiints { p i } and { d i }, rsptivly at a prsribd st of frqunis btwn and π givn by th vtor w 7 DTFT Computation Using MATLAB Thr ar svral othr forms of th funtion frqz Program 3_.m in th txt an b usd to omput th valus of th DTFT of a ral squn It omputs th ral and imaginary parts, and th magnitud and phas of th DTFT 8 DTFT Computation Using MATLAB Exampl-Plots of th ral and imaginary parts, and th magnitud and phas of th DTFT.8.33 +.5 X ( +.6 3.33 +.37 3 ar shown on th nxt slid +.8 +.7 +.4 4 4 3
Amplitud.5 -.5 DTFT Computation Using MATLAB Ral part -..4.6.8 Magnitud Sptrum Amplitud.5.5..4.6.8 /π 4 Imaginary part Phas Sptrum DTFT Computation Using MATLAB Not: Th phas sptrum displays a disontinuity of π at.7 This disontinuity an b rmovd using th funtion unwrap as indiatd blow - Unwrappd Phas Sptrum Magnitud.8.6.4 Phas, radians Phas, radians - -3-4 -5. - -6 9..4.6.8-4..4.6.8 Copyright 5, S. K. Mitra -7..4.6.8 Linar Convolution Using DTFT An important proprty of th DTFT is givn by th onvolution thorm in Tabl 3.4 It stats that if y[ * h[, thn th DTFT Y ( of y[ is givn by ( Y X ( H ( An impliation of this rsult is that th linar onvolution y[ of th squns and h[ an b prformd as follows: Linar Convolution Using DTFT Comput th DTFTs X ( and H ( of th squns and h[, rsptivly Form th DTFT ( Y X ( H ( 3 Comput th IDFT y[ of Y ( h[ DTFT DTFT X ( Y ( H ( IDTFT y[ 3 Th Unwrappd Phas Funtion In numrial omputation, whn th omputd phas funtion is outsid th rang [ π, π], th phas is omputd modulo π, to bring th omputd valu to this rang Thus. th phas funtions of som squns xhibit disontinuitis of π radians in th plot 4 Th Unwrappd Phas Funtion For xampl, thr is a disontinuity of π at.7 in th phas rspons blow 4 8 33 + 5 33 3.... +. 8 X ( 3 4 +. 37 +. 7 +. 6 +. 4 Phas, radians 4 - Phas Sptrum -4..4.6.8 4
5 Th Unwrappd Phas Funtion In suh ass, oftn an altrnat typ of phas funtion that is ontinuous funtion of is drivd from th original phas funtion by rmoving th disontinuitis of π Pross of disontinuity rmoval is alld unwrapping th phas Th unwrappd phas funtion will b dnotd as ( θ 6 Th Unwrappd Phas Funtion In MATLAB, th unwrapping an b implmntd using th M-fil unwrap Th unwrappd phas funtion of th DTFT of prvious pag is shown blow Phas, radians - -4-6 Unwrappd Phas Funtion -8..4.6.8 Th Unwrappd Phas Funtion Th onditions undr whih th phas funtion will b a ontinuous funtion of is nxt drivd Now ln X ( X ( + θ( whr θ( arg{ H ( } Th Unwrappd Phas Funtion If ln X ( xits, thn its drivativ with rspt to also xists and is givn by d ln X ( X ( X ( dx r( dx ( + dx ( im 7 8 9 Th Unwrappd Phas Funtion From ln X ( X ( + θ(, d ln X ( / is also givn by d ln X ( d X ( + d θ( 3 Th Unwrappd Phas Funtion Thus, d θ( / is givn by th imaginary part of dx r( dx + im( X ( Hn, dθ( dx [ ( X ( im r X ( ( im ( dx r X ] 5
Th Unwrappd Phas Funtion Th phas funtion an thus b dfind unquivoally by its drivativ d θ( / : dθ( η θ( [ ] dη, with th onstraint dη θ ( Th Unwrappd Phas Funtion Th phas funtion dfind by θ( dθ( η dη [ ] dη is alld th unwrappd phas funtion of X ( and it is a ontinuous funtion of ln X ( xists 3 3 33 Th Unwrappd Phas Funtion Morovr, th phas funtion will b an odd funtion of if π d θ( η π [ ] dη dη If th abov onstraint is not satisfid, thn th omputd phas funtion will xhibit absolut umps gratr than π 34 Th Frquny Rspons Most disrt-tim signals nountrd in prati an b rprsntd as a linar ombination of a vry larg, mayb infinit, numbr of sinusoidal disrt-tim signals of diffrnt angular frqunis Thus, knowing th rspons of th LTI systm to a singl sinusoidal signal, w an dtrmin its rspons to mor ompliatd signals by making us of th suprposition proprty Th Frquny Rspons Th Frquny Rspons 35 An important proprty of an LTI systm is that for rtain typs of input signals, alld ign funtions, th output signal is th input signal multiplid by a omplx onstant W onsidr hr on suh ign funtion as th input 36 Considr th LTI disrt-tim systm with an impuls rspons {h[} shown blow h[ y[ Its input-output rlationship in th timdomain is givn by th onvolution sum h k y [ [ n 6
Th Frquny Rspons Th Frquny Rspons 37 If th input is of th form n, < n < thn it follows that th output is givn by [ n k n ] h [ k ] ( k y h [ k ] k Lt H ( k h[ k k n 38 Thn w an writ n y[ H ( Thus for a omplx xponntial input signal n, th output of an LTI disrt-tim systm is also a omplx xponntial signal of th sam frquny multiplid by a omplx onstant H ( n Thus is an ign funtion of th systm 39 Th Frquny Rspons Th quantity H ( is alld th frquny rspons of th LTI disrt-tim systm H ( provids a frquny-domain dsription of th systm H ( is prisly th DTFT of th impuls rspons {h[} of th systm 4 Th Frquny Rspons H (, in gnral, is a omplx funtion of with a priod π It an b xprssd in trms of its ral and imaginary parts H ( Hr( + Him( or, in trms of its magnitud and phas, ( ( θ H H ( whr θ( arg H ( Th Frquny Rspons Th Frquny Rspons 4 Th funtion H ( is alld th magnitud rspons and th funtion θ( is alld th phas rspons of th LTI disrt-tim systm Dsign spifiations for th LTI disrttim systm, in many appliations, ar givn in trms of th magnitud rspons or th phas rspons or both 4 In som ass, th magnitud funtion is spifid in dibls as H G log ( db ( whr G( is alld th gain funtion Th ngativ of th gain funtion A ( G( is alld th attnuation or loss funtion 7
43 Th Frquny Rspons Not:Magnitud and phas funtions ar ral funtions of, whras th frquny rspons is a omplx funtion of If th impuls rspons h[ is ral thn it follows from Tabl 3. that th magnitud funtion is an vn funtion of : ( H H ( and th phas funtion is an odd funtion of : θ( θ( 44 Th Frquny Rspons Likwis, for a ral impuls rspons h[, H r( is vn and H ( im is odd Exampl-Considr th M-point moving avrag filtr with an impuls rspons givn by h[ / M, n M, othrwis Its frquny rspons is thn givn by H ( M M n n 45 Th Frquny Rspons Likwis, for a ral impuls rspons h[, H r( is vn and H ( im is odd Exampl-Considr th M-point moving avrag filtr with an impuls rspons givn by h[ / M, n M, othrwis Its frquny rspons is thn givn by H ( M M n n 46 Or, Th Frquny Rspons n n H ( M n n M M n M ( M M n sin( M/ ( M / M sin( / 47 Th Frquny Rspons Thus, th magnitud rspons of th M-point moving avrag filtr is givn by sin( M/ H ( M sin( / and th phas rspons is givn by M/ ( M θ( + π µ k πk M ( 48 Frquny Rspons Computation Using MATLAB Th funtion frqz(h,,w an b usd to dtrmin th valus of th frquny rspons vtor h at a st of givn frquny points w Fromh, th ral and imaginary parts an b omputd using th funtions ral and imag, and th magnitud and phas funtions using th funtions abs and angl 8
49 Frquny Rspons Computation Using MATLAB Exampl- Program 3_.m an b usd to gnrat th magnitud and gain rsponss of an M-point moving avrag filtr as shown blow Magnitud.8.6.4. M5 M4..4.6.8 Phas, dgrs 5-5 - -5 M5 M4 -..4.6.8 Copyright 5, S. K. Mitra 5 Frquny Rspons Computation Using MATLAB Th phas rspons of a disrt-tim systm whn dtrmind by a omputr may xhibit umps by an amount π ausd by th way th artangnt funtion is omputd Th phas rspons an b mad a ontinuous funtion of by unwrapping th phas rspons aross th umps 5 Frquny Rspons Computation Using MATLAB To this nd th funtion unwrap an b usd, providd th omputd phas is in radians Th umps by th amount of π should not b onfusd with th umps ausd by th zros of th frquny rspons as indiatd in th phas rspons of th moving avrag filtr 5 Stady-Stat Stat Rspons Not that th frquny rspons also dtrmins th stady-stat rspons of an LTI disrt-tim systm to a sinusoidal input Exampl-Dtrmin th stady-stat output y[ of a ral offiint LTI disrt-tim systm with a frquny rspons H ( for an input Aos( o n + φ, < n < Stady-Stat Stat Rspons Stady-Stat Stat Rspons W an xprss th input as x [ g[ + g *[ whr g[ A φ on Now th output of th systm for an input o n is simply H ( o on Baus of linarity, th rspons v[ to an input g[ is givn by φ o v[ A H ( on Likwis, th output v*[ to th input g*[ is φ o v*[ A H ( on 53 54 9
Stady-Stat Stat Rspons Stady-Stat Stat Rspons Combining th last two quations w gt y [ v[ + v*[ A φ H ( o on + A φh ( o on A H ( o θ( o φ o n + θ( o φ o n { } A H( o os( on + θ( o + φ Thus, th output y[ has th sam sinusoidal wavform as th input with two diffrns: ( th amplitud is multiplid by H ( o, th valu of th magnitud funtion at o ( th output has a phas lag rlativ to th input by an amount θ( o, th valu of th phas funtion at o 55 56 57 Rspons to a Causal Exponntial Squn Th xprssion for th stady-stat rspons dvlopd arlir assums that th systm is initially rlaxd bfor th appliation of th input In prati, xitation to a disrt-tim systm is usually a right-sidd squn applid at som sampl indx n n o W dvlop th xprssion for th output for suh an input 58 Rspons to a Causal Exponntial Squn Without any loss of gnrality, assum for n < From th input-output rlation y [ k h[ n w obsrv that for an input nµ [ th output is givn by n y[ h[ k ( n k [ n ] µ 59 Or, Rspons to a Causal Exponntial Squn y[ n h[ k n µ k [ Th output for n < is y[ Th output for n is givn by n y n h k k [ ] n [ ] k h k k n [ ] h[ k k n+ k n 6 Or, y[ Rspons to a Causal Exponntial Squn H n ( h[ k n+ k Th first trm on th RHS is th sam as that obtaind whn th input is applid at n to an initially rlaxd systm and is th stady-stat rspons: sr y [ H ( n n
6 Rspons to a Causal Exponntial Squn Th sond trm on th RHS is alld th transint rspons: y k n tr n h k [ ] [ ] k n+ To dtrmin th fft of th abov trm on th total output rspons, w obsrv y [ tr k n+ h[ ( k n h[ h[ k ] k n+ k 6 Rspons to a Causal Exponntial Squn For a ausal, stabl LTI IIR disrt-tim systm, h[ is absolutly summabl As a rsult, th transint rspons y tr [ is a boundd squn Morovr, as n, kn+ h[ and hn, th transint rspons days to zro as n gts vry larg 63 Rspons to a Causal Exponntial Squn For a ausal FIR LTI disrt-tim systm with an impuls rspons h[ of lngth N +, h[ for n > N Hn, y tr [ for n > N Hr th output rahs th stady-stat valu y [ H ( n at n N sr 64 Th Conpt of Filtring On appliation of an LTI disrt-tim systm is to pass rtain frquny omponnts in an input squn without any distortion (if possibl and to blok othr frquny omponnts Suh systms ar alld digital filtrs and on of th main subts of disussion in this ours Th Conpt of Filtring Th ky to th filtring pross is π π π X ( n It xprsss an arbitrary input as a linar wightd sum of an infinit numbr of xponntial squns, or quivalntly, as a linar wightd sum of sinusoidal squns Th Conpt of Filtring Thus, by appropriatly hoosing th valus of th magnitud funtion H ( of th LTI digital filtr at frqunis orrsponding to th frqunis of th sinusoidal omponnts of th input, som of ths omponnts an b sltivly havily attnuatd or filtrd with rspt to th othrs 65 66
67 Th Conpt of Filtring To undrstand th mhanism bhind th dsign of frquny-sltiv filtrs, onsidr a ral-offiint LTI disrt-tim systm haratrizd by a magnitud funtion H (,, < π 68 Th Conpt of Filtring W apply an input Aosn + Bosn, < < < to this systm Baus of linarity, th output of this systm is of th form ( + θ( y[ A H ( os n ( + θ( + B H ( os n < π 69 Th Conpt of Filtring As H (, H ( th output rdus to y[ A H ( os( n + θ( Thus, th systm ats lik a lowpass filtr In th following xampl, w onsidr th dsign of a vry simpl digital filtr 7 Th Conpt of Filtring Exampl-Th input onsists of a sum of two sinusoidal squns of angular frqunis. rad/sampl and.4 rad/sampl W nd to dsign a highpass filtr that will pass th high-frquny omponnt of th input but blok th low-frquny omponnt For simpliity, assum th filtr to b an FIR filtr of lngth 3 with an impuls rspons: h[] h[] α, h[] β 7 Th Conpt of Filtring Th onvolution sum dsription of this filtr is thn givn by y[ h[] + h[] n ] + h[] n ] α + β n ] + α n ] y[ and ar, rsptivly, th output and th input squns Dsign Obtiv: Choos suitabl valus of α and β so that th output is a sinusoidal squn with a frquny.4 rad/sampl 7 Th Conpt of Filtring Now, th frquny rspons of th FIR filtr is givn by H ( h[] + h[] + h[] α( + + β + α + β (αos + β
73 Th Conpt of Filtring Th magnitud and phas funtions ar H ( αos + β θ ( In ordr to blok th low-frquny omponnt, th magnitud funtion at. should b qual to zro Likwis, to pass th high-frquny omponnt, th magnitud funtion at.4 should b qual to on 74 Th Conpt of Filtring Thus, th two onditions that must b satisfid ar H (. αos(. + β. H ( 4 αos(.4 + β Solving th abov two quations w gt α 6.7695 β 3.456335 75 Th Conpt of Filtring Thus th output-input rlation of th FIR filtr is givn by y[ 6.7695 + n ] + 3.456335 n whr th input is x [ {os(.n + os(.4n} µ [ Program 3_3.m an b usd to vrify th filtring ation of th abov systm ( ] 76 Th Conpt of Filtring Figur blow shows th plots gnratd by running this program Amplitud 4 3 y[ x [ x [ - 4 6 8 Tim indx n 77 Th Conpt of Filtring Th first svn sampls of th output ar shown blow 78 Th Conpt of Filtring From this tabl, it an b sn that, nglting th last signifiant digit, y[ os(.4( n for n Computation of th prsnt valu of th output rquirs th knowldg of th prsnt and two prvious input sampls Hn, th first two output sampls, y[] and y[], ar th rsult of assumd zro input sampl valus at n and n 3
79 Th Conpt of Filtring Thrfor, first two output sampls onstitut th transint part of th output Sin th impuls rspons is of lngth 3, th stady-stat is rahd at n N Not also that th output is dlayd vrsion of th high-frquny omponnt os(.4n of th input, and th dlay is on sampl priod 8 Phas Dlay If th input to an LTI systm H ( o is a sinusoidal signal of frquny o : Aos( o n + φ, < n < Thn, th output y[ is also a sinusoidal signal of th sam frquny o but lagging in phas by θ( o radians: y[ A H ( o os( on + θ( o + φ, < n < 8 Phas Dlay W an rwrit th output xprssion as y[ A H ( o os( o ( n τ p( o + φ whr θ( o τ p( o o is alld th phas dlay Th minus sign in front indiats phas lag 8 Phas Dlay Thus, th output y[ is a tim-dlayd vrsion of th input In gnral, y[ will not b dlayd rplia of unlss th phas dlay τ p ( o is an intgr Phas dlay has a physial maning only with rspt to th undrlying ontinuoustim funtions assoiatd with y[ and 83 Group Dlay Whn th input is omposd of many sinusoidal omponnts with diffrnt frqunis that ar not harmonially rlatd, ah omponnt will go through diffrnt phas dlays In this as, th signal dlay is dtrmind using th group dlay dfind by dθ( τg ( 84 Group Dlay In dfning th group dlay, it is assumd that th phas funtion is unwrappd so that its drivativs xist Group dlay also has a physial maning only with rspt to th undrlying ontinuous-tim funtions assoiatd with y[ and 4
Phas and Group Dlays A graphial omparison of th two typs of dlays ar indiatd blow Group dlay θ( o _τ g( o θ( Phas and Group Dlays Exampl-Th phas funtion of th FIR filtr y[ α + β n ] + α n ] is θ ( Hn its group dlay is givn by τ g ( vrifying th rsult obtaind arlir by simulation 85 Phas dlay _ τ p( o o 86 87 Phas and Group Dlays Exampl-For th M-point moving-avrag filtr / M, n M h[, othrwis th phas funtion is M/ ( M πk θ( + π µ k M Hn its group dlay is τ ( M g 88 Phas and Group Dlays Physial signifian of th two dlays ar bttr undrstood by xamining th ontinuous-tim as Considr an LTI ontinuous-tim systm with a frquny rspons θ ( Ω H ( Ω ( Ω a a Ha and xitd by a narrow-band amplitud modulatd ontinuous-tim signal x ( t a( tos( Ω t a Phas and Group Dlays a(t is a lowpass modulating signal with a band-limitd ontinuous-tim Fourir transform givn by A( Ω, Ω > Ωo and os( Ω t is th high-frquny arrir signal Phas and Group Dlays W assum that in th frquny rang Ω Ωo < Ω < Ω + Ωo th frquny rspons of th ontinuous-tim systm has a onstant magnitud and a linar phas: Ha( Ω Ha( Ω dθ a ( Ω θa( Ω θa( Ω ( Ω Ω dω ΩΩ Ω τ Ω + ( Ω Ω τ ( Ω p( g 89 9 5
Phas and Group Dlays Phas and Group Dlays 9 Now, th CTFT of x a (t is givn by ( A( [ Ω + Ω ] + A( [ Ω Ω ] X a( Ω Also, baus of th band-limiting onstraint X a ( Ω outsid th frquny rang Ω Ω < Ω < Ω + Ω o o 9 As a rsult, th output rspons y a (t of th LTI ontinuous-tim systm is givn by ya( t a( t τg ( Ω osω( t τ p( Ω assuming H a ( Ω As an b sn from th abov quation, th group dlay τ g ( Ω is prisly th dlay of th nvlop a(t of th input signal x a (t, whras, th phas dlay τ p ( Ω is th dlay of th arrir 93 Phas and Group Dlays Th figur blow illustrats th ffts of th two dlays on an amplitud modulatd sinusoidal signal Group dlay Amplitud τg a(tos(ω t a(t Tim t a(t _ τ g osω (t _ τ p a(t _ τ g Phas dlay 94 Phas and Group Dlays Th wavform of th undrlying ontinuous-tim output shows distortion whn th group dlay is not onstant ovr th bandwidth of th modulatd signal If th distortion is unaptabl, an allpass dlay qualizr is usually asadd with th LTI systm so that th ovrall group dlay is approximatly linar ovr th frquny rang of intrst whil kping th magnitud rspons of th original LTI systm unhangd Phas Dlay Computation Using MATLAB Phas dlay an b omputd using th funtion phasdlay Figur blow shows th phas dlay of th DTFT. 367( H (. 5335 +. 765 Group Dlay Computation Using MATLAB Group dlay an b omputd using th funtion grpdlay Figur blow shows th group dlay of th DTFT. 367( H (. 5335 +. 765 Phas dlay, sampls -5 - Group dlay, sampls 7 6 5 4 3 95-5..4.6.8 96..4.6.8 6