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

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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

2 Copyright, S. K. Mitra DTFT Proprtis DTFT Proprtis Usig th diffrtiatio proprty of th DTFT giv i Tabl 3., w obsrv that th DTFT of is giv by xt usig th liarity proprty of th DTFT giv i Tabl 3. w arriv at x ω ω ω ω α α α ω ω d d d dx ω ω ω ω ω α α α α Y

3 DTFT Proprtis Exampl- Dtrmi th DTFT V of th squc v dfid by dv dv pδ pδ From Tabl 3., th DTFT of δ is Usig th tim-shiftig proprty of th DTFT giv i Tabl 3. w obsrv that ω th DTFT of δ is ad th DTFT ω ω of v is V ω 3 Copyright, S. K. Mitra

4 4 Copyright, S. K. Mitra DTFT Proprtis DTFT Proprtis Usig th liarity proprty of Tabl 3. w th obtai th frqucy-domai rprstatio of as Solvig th abov quatio w gt δ δ p p v d v d ω ω ω ω p p V d V d ω ω ω d d p p V

5 Ergy Dsity Spctrum 5 Th total rgy of a fiit-rgy squc g is giv by E g g From Parsval s rlatio giv i Tabl 3. w obsrv that π ω Eg g G dω π π Copyright, S. K. Mitra

6 Ergy Dsity Spctrum Th quatity S gg ω G ω is calld th rgy dsity spctrum Th ara udr this curv i th rag π ω π dividd by π is th rgy of th squc 6 Copyright, S. K. Mitra

7 Ergy Dsity Spctrum 7 Exampl- Comput th rgy of th squc si ω h c LP π, < < Hr π ω hlp H LP dω π π whr H ω LP,, ω c ω ω < c ω π Copyright, S. K. Mitra

8 Ergy Dsity Spctrum Thrfor h LP π ω c dω ω c ω π c < Hc, h LP is a fiit-rgy squc 8 Copyright, S. K. Mitra

9 DTFT Computatio Usig MATLAB 9 Th fuctio frqz ca b usd to comput th valus of th DTFT of a squc, dscribd as a ratioal fuctio i i th form of ω ωm ω p p... pm X ω ω d d... d at a prscribd st of discrt frqucy poits ω ω l Copyright, S. K. Mitra

10 DTFT Computatio Usig MATLAB For xampl, th statmt H frqzum,d,w rturs th frqucy rspos valus as a vctor H of a DTFT dfid i trms of th vctors um ad d cotaiig th cofficits { p i } ad { d i }, rspctivly at a prscribd st of frqucis btw ad π giv by th vctor w Copyright, S. K. Mitra

11 DTFT Computatio Usig MATLAB Thr ar svral othr forms of th fuctio frqz Th Program 3_ i th txt ca b usd to comput th valus of th DTFT of a ral squc It computs th ral ad imagiary parts, ad th magitud ad phas of th DTFT Copyright, S. K. Mitra

12 DTFT Computatio Usig MATLAB Exampl- Plots of th ral ad imagiary parts, ad th magitud ad phas of th DTFT ω ω X ω ω ω 3ω ω ω 4ω ar show o th xt slid Copyright, S. K. Mitra

13 DTFT Computatio Usig MATLAB Ral part Ral part.5.5 Amplitud Amplitud ω/π ω/π Magitud Spctrum 4 Phas Spctrum.8 Magitud.6.4 Phas, radias ω/π Copyright ω/π, S. K. Mitra

14 DTFT Computatio Usig MATLAB ot: Th phas spctrum displays a discotiuity of π at ω.7 This discotiuity ca b rmovd usig th fuctio uwrap as idicatd blow Phas, radias Uwrappd Phas Spctrum ω/π Copyright, S. K. Mitra

15 Liar Covolutio Usig DTFT 5 A importat proprty of th DTFT is giv by th covolutio thorm i Tabl 3. It stats that if y x * h, th th ω DTFT Y of y is giv by ω Y X H A implicatio of this rsult is that th liar covolutio y of th squcs x ad h ca b prformd as follows: ω ω Copyright, S. K. Mitra

16 Liar Covolutio Usig DTFT Comput th DTFTs X ad H of th squcs x ad h, rspctivly Form th DTFT ω ω ω Y X H ω 3 Comput th IDFT y of Y ω ω x h DTFT DTFT X H ω ω Y ω IDTFT y 6 Copyright, S. K. Mitra

17 Discrt Fourir Trasform 7 Dfiitio - Th simplst rlatio btw a lgth- squc x, dfid for ω, ad its DTFT X is ω obtaid by uiformly samplig X o th ω-axis btw ω π at ω π/, From th dfiitio of th DTFT w thus hav X X ω ω π/ x π/ Copyright, S. K. Mitra,

18 Discrt Fourir Trasform 8 ot: X is also a lgth- squc i th frqucy domai Th squc X is calld th discrt Fourir trasform DFT of th squc x π / Usig th otatio W th DFT is usually xprssd as: X x W, Copyright, S. K. Mitra

19 Discrt Fourir Trasform Th ivrs discrt Fourir trasform IDFT is giv by x X W To vrify th abov xprssio w multiply both sids of th abov quatio by W l ad sum th rsult from to, 9 Copyright, S. K. Mitra

20 Copyright, S. K. Mitra Discrt Fourir Trasform Discrt Fourir Trasform rsultig i W W X W x l l W X l W X l

21 Discrt Fourir Trasform Maig us of th idtity l W, for l r,, othrwis w obsrv that th RHS of th last quatio is qual to X l Hc l x W X l r a itgr Copyright, S. K. Mitra

22 Discrt Fourir Trasform Exampl - Cosidr th lgth- squc x,, Its -poit DFT is giv by X x W x W Copyright, S. K. Mitra

23 Discrt Fourir Trasform 3 Exampl - Cosidr th lgth- squc y,, Its -poit DFT is giv by m, m m m Y y W y m W W m Copyright, S. K. Mitra

24 Discrt Fourir Trasform 4 Exampl- Cosidr th lgth- squc dfid for g cosπr/, r Usig a trigoomtric idtity w ca writ πr / πr / g r r W W Copyright, S. K. Mitra

25 5 Copyright, S. K. Mitra Discrt Fourir Trasform Discrt Fourir Trasform Th -poit DFT of g is thus giv by W g G, r r W W

26 Discrt Fourir Trasform 6 Maig us of th idtity l, for l r, r a itgr W, othrwis w gt /, for r G /, for r, othrwis Copyright, S. K. Mitra

27 Matrix Rlatios 7 Th DFT sampls dfid by X x W ca b xprssd i matrix form as X whr X x D x,... X X X T... x x x T Copyright, S. K. Mitra

28 8 Copyright, S. K. Mitra Matrix Rlatios Matrix Rlatios ad is th DFT matrix giv by D 4 W W W W W W W W W L M O M M M L L L D

29 Matrix Rlatios Liwis, th IDFT rlatio giv by x X W ca b xprssd i matrix form as x D X whr D is th IDFT matrix, 9 Copyright, S. K. Mitra

30 3 Copyright, S. K. Mitra Matrix Rlatios Matrix Rlatios whr ot: 4 W W W W W W W W W L M O M M M L L L D D* D

31 DFT Computatio Usig MATLAB 3 Th fuctios to comput th DFT ad th IDFT ar fft ad ifft Ths fuctios ma us of FFT algorithms which ar computatioally highly fficit compard to th dirct computatio Programs 3_ ad 3_4 illustrat th us of ths fuctios Copyright, S. K. Mitra

32 DFT Computatio Usig MATLAB Exampl- Program 3_4 ca b usd to comput th DFT ad th DTFT of th squc x cos6π/6, 5 as show blow Magitud idicats DFT sampls ormalizd agular frqucy Copyright, S. K. Mitra

33 DTFT from DFT by Itrpolatio 33 Th -poit DFT X of a lgth- squc x is simply th frqucy ω sampls of its DTFT X valuatd at uiformly spacd frqucy poits ω ω π/, Giv th -poit DFT X of a lgth- ω squc x, its DTFT X ca b uiquly dtrmid from X Copyright, S. K. Mitra

34 34 Copyright, S. K. Mitra DTFT from DFT by DTFT from DFT by Itrpolatio Itrpolatio Thus ω ω x X W X ω π ω / X S

35 DTFT from DFT by Itrpolatio 35 To dvlop a compact xprssio for th sum S, lt ωπ / r r r S r ThS r From th abov rs r r r r r S r Copyright, S. K. Mitra

36 DTFT from DFT by 36 Or, quivaltly, Itrpolatio S rs rs r Hc r S r ωπ si ω π si ω π ω π / ωπ/ / Copyright, S. K. Mitra

37 Thrfor DTFT from DFT by Itrpolatio X ω si X si ω π ω π ωπ/ / 37 Copyright, S. K. Mitra

38 Samplig th DTFT Cosidr a squc x with a DTFT X ω W sampl X at qually spacd poits ω π/, dvlopig th ω frqucy sampls { X } Ths frqucy sampls ca b cosidrd as a -poit DFT Y whos - poit IDFT is a lgth- squc y ω 38 Copyright, S. K. Mitra

39 39 Copyright, S. K. Mitra Samplig th DTFT Samplig th DTFT ow Thus A IDFT of Y yilds ω ω l l l x X / X X Y π ω π l l l l l l W x x / W Y y

40 Samplig th DTFT 4 i.. y l xl W W l l x l W l Maig us of th idtity r W, for r m, othrwis Copyright, S. K. Mitra

41 Samplig th DTFT w arriv at th dsird rlatio 4 y x m, m Thus y is obtaid from x by addig a ifiit umbr of shiftd rplicas of x, with ach rplica shiftd by a itgr multipl of samplig istats, ad obsrvig th sum oly for th itrval Copyright, S. K. Mitra

42 To apply Samplig th DTFT y x m, m to fiit-lgth squcs, w assum that th sampls outsid th spcifid rag ar zros Thus if x is a lgth-m squc with M, th y x for 4 Copyright, S. K. Mitra

43 Samplig th DTFT If M >, thr is a tim-domai aliasig of sampls of x i gratig y, ad x caot b rcovrd from y Exampl- Lt { x } { 3 4 5} 43 By samplig its DTFT X at ω π/ 4, 3 ad th applyig a 4-poit IDFT to ths sampls, w arriv at th squc y giv by ω Copyright, S. K. Mitra

44 Samplig th DTFT y x x 4 x 4, 3 i.. { y } { 4 6 3} {x} caot b rcovrd from {y} 44 Copyright, S. K. Mitra

45 umrical Computatio of th DTFT Usig th DFT 45 A practical approach to th umrical computatio of th DTFT of a fiit-lgth squc ω Lt X b th DTFT of a lgth- squc x ω W wish to valuat X at a ds grid of frqucis ω π/ M, M, whr M >> : Copyright, S. K. Mitra

46 46 Copyright, S. K. Mitra umrical Computatio of th umrical Computatio of th DTFT Usig th DFT DTFT Usig th DFT Dfi a w squc Th π ω ω / M x x X,, M x x π ω / M M x X

47 umrical Computatio of th DTFT Usig th DFT 47 ω Thus X is sstially a M-poit DFT X of th lgth-m squc x Th DFT X ca b computd vry fficitly usig th FFT algorithm if M is a itgr powr of Th fuctio frqz mploys this approach to valuat th frqucy rspos at a prscribd st of frqucis of a DTFT ω xprssd as a ratioal fuctio i Copyright, S. K. Mitra

48 DFT Proprtis 48 Li th DTFT, th DFT also satisfis a umbr of proprtis that ar usful i sigal procssig applicatios Som of ths proprtis ar sstially idtical to thos of th DTFT, whil som othrs ar somwhat diffrt A summary of th DFT proprtis ar giv i tabls i th followig slids Copyright, S. K. Mitra

49 Tabl 3.5: Gral Proprtis of DFT 49 Copyright, S. K. Mitra

50 Tabl 3.6: DFT Proprtis: Symmtry Rlatios 5 x is a complx squc Copyright, S. K. Mitra

51 Tabl 3.7: DFT Proprtis: Symmtry Rlatios 5 x is a ral squc Copyright, S. K. Mitra

52 Circular Shift of a Squc This proprty is aalogous to th timshiftig proprty of th DTFT as giv i Tabl 3., but with a subtl diffrc Cosidr lgth- squcs dfid for Sampl valus of such squcs ar qual to zro for valus of < ad 5 Copyright, S. K. Mitra

53 Circular Shift of a Squc If x is such a squc, th for ay arbitrary itgr o, th shiftd squc x x o is o logr dfid for th rag W thus d to dfi aothr typ of a shift that will always p th shiftd squc i th rag 53 Copyright, S. K. Mitra

54 Circular Shift of a Squc Th dsird shift, calld th circular shift, is dfid usig a modulo opratio: x x c o For o > right circular shift, th abov quatio implis x c x, o for x o, for < o o 54 Copyright, S. K. Mitra

55 Circular Shift of a Squc Illustratio of th cocpt of a circular shift x x 6 x x 5 x 6 6 Copyright, S. K. Mitra

56 Circular Shift of a Squc As ca b s from th prvious figur, a right circular shift by o is quivalt to a lft circular shift by o sampl priods A circular shift by a itgr umbr o gratr tha is quivalt to a circular shift by o 56 Copyright, S. K. Mitra

57 Circular Covolutio 57 This opratio is aalogous to liar covolutio, but with a subtl diffrc Cosidr two lgth- squcs, g ad h, rspctivly Thir liar covolutio rsults i a lgth- squc y L giv by y L g m h m, m Copyright, S. K. Mitra

58 Circular Covolutio 58 I computig y L w hav assumd that both lgth- squcs hav b zropaddd to xtd thir lgths to Th logr form of y L rsults from th tim-rvrsal of th squc h ad its liar shift to th right Th first ozro valu of y L is y L g h, ad th last ozro valu is g h y L Copyright, S. K. Mitra

59 Circular Covolutio 59 To dvlop a covolutio-li opratio rsultig i a lgth- squc y C, w d to dfi a circular tim-rvrsal, ad th apply a circular tim-shift Rsultig opratio, calld a circular covolutio, is dfid by y C m g m h m, Copyright, S. K. Mitra

60 Circular Covolutio Sic th opratio dfid ivolvs two lgth- squcs, it is oft rfrrd to as a -poit circular covolutio, dotd as y g h Th circular covolutio is commutativ, i.. g h h g 6 Copyright, S. K. Mitra

61 Circular Covolutio 6 Exampl - Dtrmi th 4-poit circular covolutio of th two lgth-4 squcs: { g } { }, as stchd blow g 3 { h } { } h 3 Copyright, S. K. Mitra

62 6 Circular Covolutio Th rsult is a lgth-4 squc giv by C g y 4 h From th abov w obsrv y C 3 3 y C g m h m 4, m g m h m 4 m 3 g h g h3 g h g3 h 6 Copyright, S. K. Mitra

63 63 Copyright, S. K. Mitra Circular Covolutio Circular Covolutio Liwis 3 4 m C m h m g y 3 3 h g h g h g h g m C m h m g y 3 3 h g h g h g h g 6

64 64 Copyright, S. K. Mitra Circular Covolutio Circular Covolutio ad Th circular covolutio ca also b computd usig a DFT-basd approach as idicatd i Tabl h g h g h g h g m C m h m g y y C

65 65 Copyright, S. K. Mitra Circular Covolutio Circular Covolutio Exampl - Cosidr th two lgth-4 squcs rpatd blow for covic: Th 4-poit DFT G of g is giv by 3 g 3 h 4 / g g G π / / g g π π 3 3, / / π π

66 66 Copyright, S. K. Mitra Circular Covolutio Circular Covolutio Thrfor Liwis,, G, G G 3, 4 G 4 / h h H π / / h h π π 3 3, / / π π π

67 Circular Covolutio Hc, H 6, H, H, H 3 Th two 4-poit DFTs ca also b computd usig th matrix rlatio giv arlir 67 Copyright, S. K. Mitra

68 68 Copyright, S. K. Mitra Circular Covolutio Circular Covolutio g g g g G G G G D h h h h H H H H D is th 4-poit DFT matrix D 4

69 69 Copyright, S. K. Mitra Circular Covolutio Circular Covolutio If dots th 4-poit DFT of th from Tabl 3.5 w obsrv Thus 3 H G Y C, Y C y C H G H G H G H G Y Y Y Y C C C C

70 7 Copyright, S. K. Mitra Circular Covolutio Circular Covolutio A 4-poit IDFT of yilds Y C * C C C C C C C C Y Y Y Y y y y y D

71 Circular Covolutio 7 Exampl - ow lt us xtdd th two lgth-4 squcs to lgth 7 by appdig ach with thr zro-valud sampls, i.. g, g, h, h, Copyright, S. K. Mitra

72 Circular Covolutio 7 W xt dtrmi th 7-poit circular covolutio of g ad h : y 6 From th abov g m h m m, 7 y g h g h 6 6 g h 4 g 4 h 3 g 5 h g 6 h 3 g h Copyright, S. K. Mitra

73 73 Copyright, S. K. Mitra Circular Covolutio Circular Covolutio Cotiuig th procss w arriv at, 6 h g h g y h g h g h g y, h g h g h g h g y, h g h g h g y, 4

74 Circular Covolutio y 5 g h 3 g 3 h, y 6 g 3 h 3 As ca b s from th abov that y is prcisly th squc y L obtaid by a liar covolutio of g ad h y L 74 Copyright, S. K. Mitra

75 Circular Covolutio Th -poit circular covolutio ca b writt i matrix form as yc yc M y C h h h M h h h h M h h h h M h 3 L L L O L h g h g h3 g M M h g 75 ot: Th lmts of ach diagoal of th matrix ar qual Such a matrix is calld a circulat matrix Copyright, S. K. Mitra