Part B: Transform Methods. Professor E. Ambikairajah UNSW, Australia

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1 Part B: Trasform Mthods

2 Chaptr 3: Discrt-Tim Fourir Trasform (DTFT) 3. Discrt Tim Fourir Trasform (DTFT) 3. Proprtis of DTFT 3.3 Discrt Fourir Trasform (DFT) 3.4 Paddig with Zros ad frqucy Rsolutio 3.5 Ivrs Discrt Tim Fourir Trasform (IDFT) 3.6 Computatioal Complxity of th DFT

3 3. Discrt Tim Fourir Trasform (DTFT) [] Fourir Trasform of a arbitrary discrt sigal (FTD) is also calld Discrt Tim Fourir Trasform (DTFT). For discrt-tim sigals, th Fourir trasform pair is Squc dfid by x FTD X ) ( X ( ) x x cotiuous variabl j X ( ) j d

4 Whr is th digital frqucy or rlativ frqucy (-) ad = T X() is priodic with priod. (i. X(+) = X ()) ot: Th Fourir trasform of a aalogu sigal is ot priodic. Exampl : x[ ] [ ] X ( ) j x() cotius f s -

5 Exampl : ) (si ) cos ( ) ( ) ( ) (, ] [ ja a X a a a X a a x j j j cos si ta ) ( ) ( cos ) ( a a X a a X

6 x() = a u() a a X

7 ot: X() ca b obtaid usig z-trasform as wll. x a X ( ) X u ( z) X z) az a ( z j j (ot: X() X( j ))

8 x Exrcis : Dtrmi X(), x() As : ( ) si X ( ) si

9 Magitud Discrt-tim Fourir Trasform ormalizd frqucy, corrspods to pi

10 % Exrcis. Discrt-tim Fourir Trasform = ; tim_ = :-; digital_frqucy = pi/4; sigal_x = cos(digital_frqucy*tim_); figur(), clf subplot(,,) stm(tim_, sigal_x) grid o samplig_frqucy =.; omga = (-:samplig_frqucy:)*pi; trasform_x = zros(,lgth(omga)); j = sqrt(-); for = :- trasform_x = trasform_x + sigal_x(+)*xp(-j**omga); d figur() subplot(,,) plot(omga/pi, abs(trasform_x)) xlabl('ormalizd frqucy, corrspods to pi') ylabl('magitud') titl ('Discrt-tim Fourir Trasform') grid o % Vrify that th digital frqucy dividd by pi is qual to th positio % of th spis. % That is, if th digital frqucy is pi/4, th spis should b locatd % at -/4 ad /4. Magitud Discrt-tim Fourir Trasform ormalizd frqucy, corrspods to pi

11 x x x 3. Proprtis of th Fourir Trasform of discrt sigal (FTD or DTFT) FTD FTD FTD X ( ) X ( ) X ( ) Frqucy shift Proof: : Liarity FTD : ax bx ax( ) bx ( ) FTD j X ( ) Tim shift : x x x FTD Z z X (z) -j X(), w dfi X( j ) X() j j( ) X ( ), X ( ) X ( )

12 Exampl: Th frqucy rspos of a idal low pass filtr, i th fudamtal itrval -is giv by H( ) (a) Fid th impuls rspos of th idal low pass filtr (b) Stch th impuls rspos for C. 4 C C

13 j d d H h c j j j j C C C C si 4 ) ( H() - - c c (a)

14 (b) si si 4 ( ) 4 c h h[] 3

15 Exrcis : Th frqucy rspos of a bad stop filtr is giv by Show that th impuls rspos h d [] of th bad stop filtr is giv by ) ( H d

16 h d 6 si 3 si H d ()

17 ad zro lswhr whr j j j j j j d d d d H h j j j j j j j j j j j j j d d ; si 3 si 6 si si ) ( /6 6 3 h d 6 si 3 si

18 3.3 Th Discrt Fourir Trasform (DFT) [] x x DTFT X ( ) DTFT DFT X DFT DFT mappig x[] to aothr squc. Th DFT dotd by X[], allows to valuat th Fourir Trasform X(). This complx valud squc X[] is obtaid by samplig th Fourir Trasform X() at a fiit umbr of frqucy poits. This samplig is covtioally prformd at qually spacd poits ovr th priod xtdig ovr -.

19 Th DFT allows us to dtrmi th frqucy cott of a sigal, that is, to prform spctral aalysis. Th DFT plays a ctral rol i th implmtatio of a varity of digital sigal procssig algorithms, as a rsult of th xistc of th fficit algorithm for th Fast Fourir Trasform (FFT).

20 () Fourir trasform of a discrt sigal (DTFT or FTD) is () X ( X ( ) ) X ( z) j z X x j, (3) Discrt Fourir Trasform (DFT) x {Evaluat z-trasform o th uit circl} j i.. Samplig X() at qually spacd itrval, =,,,3, -. umbr of tim sampls = umbr of frqucy sampls () =

21 Th DFT corrspods to samplig th z-trasform of X(z) at -poits qually spacd i agl aroud th uit circl. = - = - = poits qually spacd o th uit circl

22 % Exrcis. Discrt Fourir Trasform % W actually us th Fast Fourir Trasform (FFT) algorithm, which dos th sam job % as DFT but at much fastr spd. = ; tim_ = :-; digital_frqucy = pi/4; sigal_x = cos(digital_frqucy*tim_); figur(), clf subplot(,,) stm(tim_, sigal_x) titl('sigal x') grid o trasform_x = fft(sigal_x); % trasform th sigal figur() subplot(,,) plot(abs(trasform_x)) titl('fourir Trasform of x') % ot that FFT always valuat th trasformatio for omga from to pi. % Ad th trasformd sigal has xactly th sam umbr of sampls as th sigal i tim domai. % Hc, th sampl at idx corrspods to th sampl at. % Similarly, th sampl at idx corrspod to th sampl at pi sigal x Fourir Trasform of x

23 ot: Discrt Cotiuous Discrt - - x[] tim sampls X() X[] frqucy Sampls DTFT or FTD X() x[] X() DFT (or usig FFT) x[]

24 Wh th DFT of a bloc of sampls is calculatd, th assumptio is that th origial sigal actually rpats itslf priodically, with priod. Clarly, for all ral sigals this will ot b tru. Ev for artificial sigals such as a pur siusoid this will oly b tru if is a multipl of th priod of th siusoid. This Widowig procss itroducs a slight distortio ito th frqucy rprstatio of th sigal big aalysd.

25 For xampl wh a 56 poit DFT is ta usig a rctagular widow (groupig sampls), o ca s that th wavform dos ot rpat ad as a rsult slight frqucy distortio will occur wh calculatig X[] x[] widow 56 sampls

26 Exampl: Lt f s = 8 Hz, umbr of sampl() = Frqucy rsolutio = X 999 x f 8 s 8 Hz f =, f = 8Hz, f = 6 Hz,. f 999 = 8Hz =,,,3, j

27 Exampl : A spch sigal is sampld at a rat of sampls/sc. A squc of lgth () 4 sampls is slctd ad th 4-poit DFT is computd. () What is th tim duratio of sgmt of spch? Duratio = o of sampls samplig priod. = 4 (/) = 5. ms () What is th frqucy rsolutio (spacig i Hz) btw th DFT valus. f s R solutio Hz

28 X X X X x x cos( ) j x DFT calculatio j,,, 3,... si( ) A B A B,,, 3,... f =(fs/) = f f f f s =samplig frqucy samplig This is a lgthly procdur, if for xampl, = 4 sampls

29 X[] =56 55 f f f f 55 Th frqucy rsolutio (f) ca b mad as small as dsird by icrasig th valu of (widow siz big aalysd)

30 Exampl : Cosidr th fiit lgth squc. x[] = [] +.[-] Fid th -poit DFT of x[] for = 5! Solutio: X(z) = +. z - X() = +. -j Th -poit DFT is obtaid by valuatig X() at poits qually spacd aroud th uit circl. Thrfor th DFT is giv by: X j. j. 5 =,,,. 49

31 Exrcis : Comput th -poit DFT H[] of th squc h[]. Show that wh = 8 th valu of H() = -. Exampl : h 3 3 othrwis Fid FTD ad DFT of th thr sampls avrag. h 3 othrwis

32 h[],, whr )] cos( [ 3 3 )) cos( ( 3 ) ( ) ( ) ( H H h H j j j

33 3.4 Paddig with Zros ad frqucy Rsolutio DFT : X j x,,,3,... To obtai mor poits i th DFT squc, w ca always icras th duratio of x[] by addig additioal zro-valud lmts. This procdur is calld paddig with zros. Ths zro-valud lmts cotribut othig to th sum i th abov quatio, but act to dcras th frqucy spacig (/).

34 Th zro paddig givs us a highdsity spctrum ad providd a bttr displayd vrsio for plottig. But it dos ot giv us a high rsolutio spctrum bcaus o w iformatio is addd to th sigal. Oly additioal zros ar addd i th data.

35 Thr Sampl Avragr h() = h() =6 h() =8 Paddig with zros H() H() H()

36 % Exrcis 3. Zro paddig % Somtims, th sigal i tim domai has oly a fw sampls such that w do't gt a smooth % curv for th trasformd sigal. % This ca b asily solvd by ``zro paddig". % Th ida hr is to add mor sampls with valu zro aftr th origial sampls i th sigal. = 8; tim_ = :-; digital_frqucy = pi/4; sigal_x = cos(digital_frqucy*tim_); figur(3), clf subplot(,,) stm(tim_, sigal_x) grid o trasform_x = fft(sigal_x); % trasform th sigal figur(3) subplot(,,3) plot(abs(trasform_x)) titl('fourir Trasform of x') % Zro paddig th origial sigal sampl = ; sigal_x_paddd = [sigal_x zros(,sampl-)]; figur(3) subplot(,,) stm(sigal_x_paddd) titl('zro-paddd sigal x') grid o trasform_x = fft(sigal_x_paddd); % trasform th sigal figur(3) subplot(,,4) plot(abs(trasform_x)) titl('fourir Trasform of zro-paddd sigal') % FFT also provids zro paddig mchaism. Th followig lis show how to do so. figur(4), clf plot(abs(fft(sigal_x,sampl))) % sampl is spcifid as a iput to th `fft' commad.

37 Fourir Trasform of x Zro-paddd sigal x - 5 Fourir Trasform of zro-paddd sigal

38 3.5 Ivrs Discrt Fourir Trasform (IDFT) [] Th ivrs DFT quatio coms dirctly from th Ivrs Fourir Trasform quatio : x[ ] Howvr, th sigal which it producs will b a priodic sigal rpatig vry sampls X ( ) j -

39 Us of DFT : Clarly, th DFT is usful i that it allows th spctral cott of a sigal to b dtrmid Additioally, oc i th frqucy domai th DFT of a sigal ca b procssd i ordr to filtr or altr th sigal is som dsird fashio Th IDFT ca th b usd to rgrat th procssd sigal If ay widow, othr tha rctagular, has b usd th DFT ad IDFT blocs must ovrlap by 5%, if prfct rcostructio is rquird

40 Computatios for valuatig th DFT : By cosidrig complx-valud squcs, w fid that both th trasform (DFT) ad its ivrs (IDFT) ca b valuatd with th sam algorithm. Ev though th tim squc may b ral-valud, th DFT squc is complx valud. Th DFT squc ca also b xprssd xplicitly i trms of its ral ad imagiary parts as h[]=h r []+jh i [], for - H[]=H r []+jh i [], for -

41 cos si si cos, i r i r j i r j h h j h h H jh h H h H DFT :

42 cos si ) si cos ] [, IDFT : i r i r j i r j H H j H H h jh H h H h

43 Summary (DFT/IDFT calculatios) Excpt for th diffrc i th sigs of th xpots ad th / scalig factor i th ivrs, th sam opratios ar prformd i both trasforms Both th DFT ad IDFT computatios rquir 4 ral-valud multiplicatios pr poit ( complx multiplicatios), ad 4 such opratios ar cssary to comput th - poit trasform

44 Fast Fourir Trasform : Th Fast Fourir Trasform (FFT) is simply a mathmatical tchiqu to acclrat th calculatio of th DFT. It was dvlopd by Cooly ad Tuy (965) rquirs to b a powr of. Typically, if th DFT is calculatd for a bloc of sampls.g. 5 or 4 sampls () it would ma th calculatio of th DFT quit dmadig. Th FFT simply uss rptitio ad rdudacy i th calculatio to spd it up. Th FFT is simply a TECHIQUE to calculat th DFT, OT a diffrt trasform.

45 Compariso of FFT/DFT complx multiplicatios As alrady statd, th FFT algorithm is most fficit wh th umbr of tim sampls to b trasformd is a powr of. If is ot a powr of th it may b paddd out with zro sampls to th arst powr of two bfor trasformatio.

46 A compariso of th umbr of complx multiplicatios rquird for dirct valuatio of th DFT ad th umbr dd for th Cooly-Tuy FFT is giv blow o. of Poits Complx Multiplicatios i DFT ( ) 56 65,536,48,576 FFT algorithm (/)log() Tims fastr tha dirct valuatio

47 FFT Aalysis usig Rctagular Spch Sigal Spctrum of Spch Sigal

48 Summary of Part B Chaptr 3 At th d of this chaptr, it is xpctd that you should ow: Th dfiitio of th discrt tim Fourir trasform ad its ivrs. Giv a filtr magitud rspos, fid th impuls rspos. Th proprtis of th DTFT ad its applicatios. Wh to us th discrt tim Fourir trasform (DTFT) ad wh to us th discrt Fourir trasform (DFT).

49 Wh ad how to us widowd discrt Fourir trasforms. Th fast Fourir trasform (FFT). Zro-paddig ad its rlatioship to frqucy rsolutio. Th rlativ complxity btw th DFT ad FFT i trms of multiplicatios.

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