Discrete Fourier Transform. Discrete Fourier Transform. Discrete Fourier Transform. Discrete Fourier Transform. Discrete Fourier Transform

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Frank Jordan
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1 Discrt Fourir Trasform Dfiitio - T simplst rlatio btw a lt- squc x dfid for ω ad its DTFT X ( ) is ω obtaid by uiformly sampli X ( ) o t ω-axis btw ω < at ω From t dfiitio of t DTFT w tus av X X( ω ) ω x opyrit 5 S. K. itra Discrt Fourir Trasform ot: X is also a lt- squc i t frqucy domai T squcx is calld t discrt Fourir trasform (DFT) of t squc x Usi t otatio t DFT is usually xprssd as: X x opyrit 5 S. K. itra Discrt Fourir Trasform T ivrs discrt Fourir trasform (IDFT) is iv by x X To vrify t abov xprssio w multiply bot sids of t abov quatio by l ad sum t rsult from to opyrit 5 S. K. itra Discrt Fourir Trasform rsulti i x l X X X ( l) ( l) l opyrit 5 S. K. itra Discrt Fourir Trasform Discrt Fourir Trasform 5 ai us of t idtity ( l) for l r r a itr otrwis w obsrv tat t RS of t last quatio is qual to X l c l x X l opyrit 5 S. K. itra Exampl- osidr t lt- squc x Its -poit DFT is iv by X x x opyrit 5 S. K. itra

2 Discrt Fourir Trasform Discrt Fourir Trasform 7 Exampl- osidr t lt- squc y Its -poit DFT is iv by m m m m y y m m opyrit 5 S. K. itra 8 Exampl-osidr t lt- squc dfid for cos(r ) r Usi a trioomtric idtity w ca writ r r ( ) r r ( ) opyrit 5 S. K. itra Discrt Fourir Trasform Discrt Fourir Trasform 9 T-poit DFT of is tus iv by ( r) ( r ) opyrit 5 S. K. itra ai us of t idtity ( l) for l r r a itr otrwis w t for r for r otrwis opyrit 5 S. K. itra atrix Rlatios T DFT sampls dfid by ca b xprssd i matrix form as wr X X x x X D x X X... X T x x... x T opyrit 5 S. K. itra atrix Rlatios ad D is t DFT matrix iv by D ( ) ( ) O ( ) ( ) ( ) opyrit 5 S. K. itra

3 atrix Rlatios iwis t IDFT rlatio iv by x X ca b xprssd i matrix form as x D X wr D is t IDFT matrix opyrit 5 S. K. itra wr D ot: atrix Rlatios ( ) ( ) D D* O ( ) ( ) ( ) opyrit 5 S. K. itra 5 DFT omputatio Usi ATAB T fuctios to comput t DFT ad t IDFT ar fft ad ifft Ts fuctios ma us of FFT aloritms wic ar computatioally ily fficit compard to t dirct computatio Prorams 5_.m ad 5_.m illustrat t us of ts fuctios opyrit 5 S. K. itra DFT omputatio Usi ATAB Exampl-Proram 5_.m ca b usd to comput t DFT ad t DTFT of t squc x cos() 5 as sow blow aitud ormalizd aular frqucy idicats DFT sampls opyrit 5 S. K. itra 7 DTFT from DFT by Itrpolatio T-poit DFT X of a lt- squc x is simply t frqucy ω sampls of its DTFT X ( ) valuatd at uiformly spacd frqucy poits ω ω iv t -poit DFT X of a lt- ω squc x its DTFT X ( ) ca b uiquly dtrmid from X opyrit 5 S. K. itra 8 Tus X ( DTFT from DFT by Itrpolatio ω ) x ω X X ω ( ω ) S opyrit 5 S. K. itra

4 9 DTFT from DFT by Itrpolatio To dvlop a compact xprssio for t sum S lt ( ω ) r TS r From t abov rs r r r r r S r opyrit 5 S. K. itra DTFT from DFT by Itrpolatio Or quivaltly S rs ( r)s r c ( ω r S r ω( si ω ( si ω ) ) ω )( ) opyrit 5 S. K. itra Trfor X ( ω DTFT from DFT by Itrpolatio ) ω si X ω si ( ω )( ) Sampli t DTFT osidr a squc x wit a DTFT X ( ) ω sampl X ( ) at qually spacd poits ω dvlopi t ω frqucy sampls { X ( )} Tsfrqucy sampls ca b cosidrd as a -poit DFT wos - poit IDFT is a lt- squc y ω opyrit 5 S. K. itra opyrit 5 S. K. itra Sampli t DTFT Sampli t DTFT ow Tus ω X ( ) x l A IDFT of yilds y l ω ωl X ( ) X ( x l x l l l ) l l opyrit 5 S. K. itra i.. y l xl l ( l) x l l ai us of t idtity ( r) for r m otrwis opyrit 5 S. K. itra

5 5 Sampli t DTFT w arriv at t dsird rlatio y x m m Tus y is obtaid from x by addi a ifiit umbr of siftd rplicas of x wit ac rplica siftd by a itr multipl of sampli istats ad obsrvi t sum oly for t itrval opyrit 5 S. K. itra To apply Sampli t DTFT y x m m to fiit-lt squcs w assum tat t sampls outsid t spcifid ra ar zros Tus if x is a lt- squc wit t y x for opyrit 5 S. K. itra 7 Sampli t DTFT If > tr is a tim-domai aliasi of sampls of x i rati yadx caot b rcovrd from y Exampl-t { x } { 5} ω By sampli its DTFT X ( ) at ω ad t applyi a -poit IDFT to ts sampls w arriv at t squc y iv by opyrit 5 S. K. itra 8 Sampli t DTFT y x x x i.. { y } { } {x} caot b rcovrd from {y} opyrit 5 S. K. itra 9 umrical omputatio of t DTFT Usi t DFT A practical approac to t umrical computatio of t DTFT of a fiit-lt squc ω t X ( ) b t DTFT of a lt- squc x ω wis to valuat X ( ) at a ds rid of frqucis ω wr >> : opyrit 5 S. K. itra umrical omputatio of t DTFT Usi t DFT X ( ω ) x ω Dfi a w squc x x T X( ω ) x x opyrit 5 S. K. itra 5

6 umrical omputatio of t DTFT Usi t DFT ω Tus X ( ) is sstially a -poit DFT X of t lt- squc x T DFT X ca b computd vry fficitly usi t FFT aloritm if is a itr powr of T fuctio frqz mploys tis approac to valuat t frqucy rspos at a prscribd st of frqucis of a DTFT ω xprssd as a ratioal fuctio i opyrit 5 S. K. itra DFT Proprtis i t DTFT t DFT also satisfis a umbr of proprtis tat ar usful i sial procssi applicatios Som of ts proprtis ar sstially idtical to tos of t DTFT wil som otrs ar somwat diffrt A summary of t DFT proprtis ar iv i tabls i t followi slids opyrit 5 S. K. itra Tabl 5.: DFT Proprtis: Symmtry Rlatios Tabl 5.: DFT Proprtis: Symmtry Rlatios x is a complx squc opyrit 5 S. K. itra x is a ral squc opyrit 5 S. K. itra Tabl 5.: ral Proprtis of DFT ircular Sift of a Squc Tis proprty is aaloous to t timsifti proprty of t DTFT as iv i Tabl. but wit a subtl diffrc osidr lt- squcs dfid for Sampl valus of suc squcs ar qual to zro for valus of < ad 5 opyrit 5 S. K. itra opyrit 5 S. K. itra

7 ircular Sift of a Squc ircular Sift of a Squc If x is suc a squc t for ay arbitrary itr o t siftd squc x x o is o lor dfid for t ra tus d to dfi aotr typ of a sift tat will always p t siftd squc i t ra T dsird sift calld t circular sift is dfid usi a modulo opratio: x c x o For o > (rit circular sift) t abov quatio implis x x o for o c x o for < o 7 opyrit 5 S. K. itra 8 opyrit 5 S. K. itra ircular Sift of a Squc ircular Sift of a Squc Illustratio of t cocpt of a circular sift x x x As ca b s from t prvious fiur a rit circular sift by o is quivalt to a lft circular sift by o sampl priods A circular sift by a itr umbr o ratr ta is quivalt to a circular sift by o 9 x 5 x opyrit 5 S. K. itra opyrit 5 S. K. itra Tis opratio is aaloous to liar covolutio but wit a subtl diffrc osidr two lt- squcs ad rspctivly Tir liar covolutio rsults i a lt- ( ) squc y iv by y m m m opyrit 5 S. K. itra I computi y w av assumd tat bot lt- squcs av b zropaddd to xtd tir lts to T lor form of y rsults from t tim-rvrsal of t squc ad its liar sift to t rit T first ozro valu of y is y ad t last ozro valu is y opyrit 5 S. K. itra 7

8 To dvlop a covolutio-li opratio rsulti i a lt- squc y w d to dfi a circular tim-rvrsal ad t apply a circular tim-sift Rsulti opratio calld a circular covolutio is dfid by y m m m opyrit 5 S. K. itra Sic t opratio dfid ivolvs two lt- squcs it is oft rfrrd to as a -poit circular covolutio dotd as y T circular covolutio is commutativ i.. opyrit 5 S. K. itra 5 Exampl- Dtrmi t -poit circular covolutio of t two lt- squcs: { } { } { } { } as stcd blow opyrit 5 S. K. itra T rsult is a lt- squc y iv by y From t abov w obsrv m m m m y m m ( ) ( ) ( ) ( ) opyrit 5 S. K. itra 7 iwis y m m m ( ) ( ) ( ) ( ) 7 y m m m ( ) ( ) ( ) ( ) opyrit 5 S. K. itra 8 ad y m m m ( ) ( ) ( ) ( ) 5 y T circular covolutio ca also b computd usi a DFT-basd approac as idicatd i Tabl 5. opyrit 5 S. K. itra 8

9 9 9 opyrit 5 S. K. itra Exampl- osidr t two lt- squcs rpatd blow for covic: T-poit DFT of is iv by 5 opyrit 5 S. K. itra Trfor iwis 5 opyrit 5 S. K. itra c T two-poit DFTs ca also b computd usi t matrix rlatio iv arlir 5 opyrit 5 S. K. itra D D is t -poit DFT matrix D 5 opyrit 5 S. K. itra If dots t -poit DFT of t from Tabl.5 w obsrv Tus y 5 opyrit 5 S. K. itra A-poit IDFT of yilds * y y y y D 5 7

10 55 Exampl-ow lt us xtdd t two lt- squcs to lt 7 by appdi ac wit tr zro-valud sampls i.. opyrit 5 S. K. itra 5 xt dtrmi t 7-poit circular covolutio of ad : y m From t abov m m 7 y 5 opyrit 5 S. K. itra 57 otiui t procss w arriv at y ( ) ( ) y ( ) ( ) ( ) 5 y ( ) ( ) ( ) ( ) 5 y ( ) ( ) ( ) 58 opyrit 5 S. K. itra y 5 ( ) ( ) y ( ) As ca b s from t abov tat y is prcisly t squc y obtaid by a liar covolutio of ad y opyrit 5 S. K. itra 59 T-poit circular covolutio ca b writt i matrix form as y y y O ot: T lmts of ac diaoal of t matrix ar qual Suc a matrix is calld a circulat matrix opyrit 5 S. K. itra Tabular tod illustrat t mtod by a xampl osidr t valuatio of y O* wr {} ad {} ar lt- squcs First t sampls of t two squcs ar multiplid usi t covtioal multiplicatio mtod as sow o t xt slid opyrit 5 S. K. itra

11 opyrit 5 S. K. itra : : 5 T partial products ratd i t d rd ad t rows ar circularly siftd to t lft as idicatd abov opyrit 5 S. K. itra T modifid tabl aftr circular sifti is sow blow T sampls of t squc ar obtaid by addi t partial products i t colum abov of ac sampl : : : : yc yc yc yc yc } { y c opyrit 5 S. K. itra Tus y c y c y c y c

DTFT Properties. Example - Determine the DTFT Y ( e ) of n. Let. We can therefore write. From Table 3.1, the DTFT of x[n] is given by 1

DTFT Properties. Example - Determine the DTFT Y ( e ) of n. Let. We can therefore write. From Table 3.1, the DTFT of x[n] is given by 1 DTFT Proprtis Exampl - Dtrmi th DTFT Y of y α µ, α < Lt x α µ, α < W ca thrfor writ y x x From Tabl 3., th DTFT of x is giv by ω X ω α ω Copyright, S. K. Mitra Copyright, S. K. Mitra DTFT Proprtis DTFT