Topic 5: Discrete-Time Fourier Transform (DTFT)

ELEC36: Signals And Systms Tpic 5: Discrt-Tim Furir Transfrm (DTFT) Dr. Aishy Amr Cncrdia Univrsity Elctrical and Cmputr Enginring DT Furir Transfrm Ovrviw f Furir mthds DT Furir Transfrm f Pridic Signals Prprtis f DT Furir Transfrm Rlatins amng Furir Mthds Summary Appndix: Transitin frm DT Furir Sris t DT Furir Transfrm Figurs and xampls in ths curs slids ar takn frm th fllwing surcs: A. Oppnhim, A.S. Willsky and S.H. Nawab, Signals and Systms, 2nd Editin, Prntic-Hall, 997 M.J. Rbrts, Signals and Systms, McGraw Hill, 2004 J. McClllan, R. Schafr, M. Ydr, Signal Prcssing First, Prntic Hall, 2003

DT Furir Transfrm (Nt: a Furir transfrm is uniqu, i.., n tw sam signals in tim giv th sam functin in frquncy) Th DT Furir Sris is a gd analysis tl fr systms with pridic xcitatin but cannt rprsnt an apridic DT signal fr all tim Th DT Furir Transfrm can rprsnt an apridic discrt-tim signal fr all tim Its dvlpmnt fllws xactly th sam as that f th Furir transfrm fr cntinuus-tim apridic signals 2

DT Furir Transfrm Lt x[n] b th apridic DT signal W cnstruct a pridic signal x[n] fr which x[n] is n prid x[n] is cmprisd f infinit numbr f rplicas f x[n] Each rplica is cntrd at an intgr multipl f N N is th prid f x[n] Cnsidr th fllwing figur which illustrats an xampl f x[n] and th cnstructin f Clarly, x[n] is dfind btwn N and N 2 Cnsquntly, N has t b chsn such that N > N + N 2 + s that adjacnt rplicas d nt vrlap Clarly, as w lt 3 as dsird

DT Furir Transfrm Lt us nw xamin th FS rprsntatin f Sinc x[n] is dfind btwn N and N 2 a k in th abv xprssin simplifis t 4 ω = 2π/N

DT Furir Transfrm Nw dfining th functin W can s that th cfficints a k ar rlatd t X( jω ) as whr ω 0 = 2π/N is th spacing f th sampls in th frquncy dmain Thrfr 5 As N incrass ω 0 dcrass, and as N th abv quatin bcms an intgral

DT Furir Transfrm On imprtant bsrvatin hr is that th functin X( jω ) is pridic in ω with prid 2π Thrfr, as N, (Nt: th functin jω is pridic with N=2π) This lads us t th DT-FT pair f quatins 6

7 DT Furir Transfrm: Frms = + = + = π ω ω π ω ω π ω π 2 2 2 ) ( 2 ] [ DT P CT Invrs DT Furir Transfrm : ] [ ) ( P CT DT DT Furir Transfrm : d X n x n x X n j j n n j j

DT Furir Transfrm: Exampls 8

DT Furir Transfrm: Exampl 9

Outlin DT Furir Transfrm Ovrviw f Furir mthds DT Furir Transfrm f Pridic Signals Prprtis f DT Furir Transfrm Rlatins amng Furir Mthds DTFT: Summary Appndix: Transitin frm DT Furir Sris t DT Furir Transfrm 0

Ovrviw f Furir Analysis Mthds: Typs f signals

Ovrviw f Furir Analysis Mthds: Typs f signals 2

Ovrviw f Furir Analysis Mthds: Cntinuus-Valu and Cntinuus-Tim Signals All cntinuus signals ar CT but nt all CT signals ar cntinuus 3

Ovrviw f Furir Analysis Mthds 4 Cntinuus in Tim Apridic in Frquncy Discrt in Tim Pridic in Frquncy Pridic in Tim Discrt in Frquncy CT Furir Sris : jkω0t ak = x( t) dt T 0 CT Invrs Furir Sris : x( t) = X[ k] = T a k k = N n= 0 k = 0 jkω t DT Furir Sris x[ n] 0 jω kn CT - P Invrs DT Furir Sris N jω0kn x[ n] = X[ k] N 0 DT - P N T DT DT CT - P DT - P DT - P N N T DT - P N Apridic in Tim Cntinuus in Frquncy CT Furir Transfrm : X ( jω) = x( t) = DT Furir Transfrm : X ( jω x[ n] = ) = n= Invrs DT Furir Transfrm : 2 π 2 π x[ n] X ( jω jωt Invrs CT Furir Transfrm : 2π x( t) X ( jω) jωn ) dt jωt jωn dω dω CT CT DT CT + P CT CT CT + P 2π 2π DT

Outlin DT Furir Transfrm Ovrviw f Furir mthds DT Furir Transfrm f Pridic Signals Prprtis f DT Furir Transfrm Rlatins amng Furir Mthds DTFT: Summary Appndix: Transitin frm DT Furir Sris t DT Furir Transfrm 5

Furir Transfrm f Pridic DT Signals Cnsidr th cntinuus tim signal This signal is pridic Furthrmr, th Furir sris rprsntatin f this signal is just an impuls f wight n cntrd at ω= ω 0 Nw cnsidr this signal It is als pridic and thr is n impuls pr prid Hwvr, th sparatin btwn adjacnt impulss is 2π, which agrs with th prprtis f DT Furir Transfrm In particular, th DT Furir Transfrm fr this signal is 6

Furir Transfrm f Pridic DT Signals: Exampl Th signal can b xprssd as W can immdiatly writ Or quivalntly whr X( jω ) is pridic in ω with prid 2π 7

Outlin DT Furir Transfrm Ovrviw f Furir mthds DT Furir Transfrm f Pridic Signals Prprtis f DT Furir Transfrm Rlatins amng Furir Mthds DTFT: Summary Appndix: Transitin frm DT Furir Sris t DT Furir Transfrm 8

Prprtis f th DT Furir Transfrm Nt: th functin jω is pridic with N=2π 9

Prprtis f th DT Furir Transfrm 20

Prprtis f th DT Furir Transfrm 2

Prprtis f th DT Furir Transfrm 22

Prprtis f th DT Furir Transfrm 23

Prprtis f th DT Furir Transfrm 24

Prprtis f th DT Furir Transfrm 25

Prprtis f th DT Furir Transfrm 26

Prprtis f th DT Furir Transfrm 27

Prprtis f th DT Furir Transfrm 28

Prprtis f th DT Furir Transfrm 29

Prprtis f th DT Furir Transfrm 30

Prprtis f th DT Furir Transfrm 3

Prprtis f th DT Furir Transfrm 32

Prprtis f th DT Furir Transfrm: Exampl 33

Prprtis f th DT Furir Transfrm 34

Prprtis f th DT Furir Transfrm: Diffrnc quatin DT LTI Systms ar charactrizd by Linar Cnstant- Cfficint Diffrnc Equatins A gnral linar cnstant-cfficint diffrnc quatin fr an LTI systm with input x[n] and utput y[n] is f th frm Nw applying th Furir transfrm t bth sids f th abv quatin, w hav 35 But w knw that th input and th utput ar rlatd t ach thr thrugh th impuls rspns f th systm, dntd by h[n], i..,

Prprtis f th DT Furir Transfrm : Diffrnc quatin Applying th cnvlutin prprty if n is givn a diffrnc quatin crrspnding t sm systm, th Furir transfrm f th impuls rspns f th systm can fund dirctly frm th diffrnc quatin by applying th Furir transfrm Furir transfrm f th impuls rspns = Frquncy rspns Invrs Furir transfrm f th frquncy rspns = Impuls rspns 36

Prprtis f th DT Furir Transfrm: Exampl With a <, cnsidr th causal LTI systm that us charactrizd by th diffrnc quatin Frm th discussin, it is asy t s that th frquncy rspns f th systm is Frm tabls (r by applying invrs Furir transfrm), n can asily find that 37

Outlin DT Furir Transfrm Ovrviw f Furir mthds DT Furir Transfrm f Pridic Signals Prprtis f DT Furir Transfrm Rlatins amng Furir Mthds DTFT: Summary Appndix: Transitin frm DT Furir Sris t DT Furir Transfrm 38

Rlatins Amng Furir Mthds 39 Cntinuus in Tim Apridic in Frquncy Discrt in Tim Pridic in Frquncy Pridic in Tim Discrt in Frquncy CT Furir Sris : jkω0t ak = x( t) dt T 0 CT Invrs Furir Sris : x( t) = X[ k] = T a k k = N n= 0 k = 0 jkω t DT Furir Sris x[ n] 0 jω kn CT - P Invrs DT Furir Sris N jω0kn x[ n] = X[ k] N 0 DT - P N T DT DT CT - P DT - P DT - P N N T DT - P N Apridic in Tim Cntinuus in Frquncy CT Furir Transfrm : X ( jω) = x( t) = DT Furir Transfrm : X ( jω x[ n] = ) = n= Invrs DT Furir Transfrm : 2 π 2 π x[ n] X ( jω jωt Invrs CT Furir Transfrm : 2π x( t) X ( jω) jωn ) dt jωt jωn dω dω CT CT DT CT + P CT CT CT + P 2π 2π DT

Rlatins Amng Furir Mthds 40

4 CT Furir Transfrm - CT Furir Sris

42 CT Furir Transfrm - CT Furir Sris

43 CT Furir Transfrm - DT Furir Transfrm

44 CT Furir Transfrm - DT Furir Transfrm

45 DT Furir Sris - DT Furir Transfrm

46 DT Furir Sris - DT Furir Transfrm

Outlin DT Furir Transfrm Ovrviw f Furir mthds DT Furir Transfrm f Pridic Signals Prprtis f DT Furir Transfrm Rlatins amng Furir Mthds DTFT: Summary Appndix: Transitin frm DT Furir Sris t DT Furir Transfrm 47

DTFT: Summary DT Furir Transfrm rprsnts a discrt tim apridic signal as a sum f infinitly many cmplx xpnntials, with th frquncy varying cntinuusly in (-π, π) DTFT is pridic nly nd t dtrmin it fr 48

DTFT: Summary Knw hw t calculat th DTFT f simpl functins Knw th gmtric sum: Knw Furir transfrms f spcial functins,.g. δ[n], xpnntial Knw hw t calculat th invrs transfrm f ratinal functins using partial fractin xpansin Prprtis f DT Furir transfrm Linarity, Tim-shift, Frquncy-shift, 49

50 DT-FT Summary: a quiz A discrt-tim LTI systm has impuls rspns Find th utput y[n] du t input (Suggstin: wrk with and using th cnvlutin prprty) Slutin This can b slvd using cnvlutin f h[n] and x[n]. Hwvr, th pint was t us th cnvlutin in tim multiplicatin in frquncy prprty. Thrfr, It can b radily shwn that Thrfr, ] [ 2 ] [ n u n h n = ] [ 7 ] [ n u n x n = ) ( ) ( ) ( ] [ ]* [ ] [ ω ω ω j j j X H Y n x n h n y = = ( ), ) ( ] [ ] [ < = = a a M n u a n m j j n ω ω ω ω ω ω j j j j X H = = 7 ) ( and 2 ) (

5 DT-FT Summary: a quiz Expliting th cnvlutin in tim multiplicatin in frquncy prprty givs: Using partial fractin xpansin mthd f finding invrs Furir transfrm givs: Thrfr, sinc a Furir transfrm is uniqu, (i.. n tw sam signals in tim giv th sam functin in frquncy) and sinc It can b sn that a Furir transfrm f th typ shuld crrspnd t a signal. Thrfr, th invrs Furir transfrm f is th invrs transfrm f is Thus th cmplt utput ) 2 )( 7 ( ) ( ω ω ω j j j Y = + = ω ω j j Y 7 5 2 / ) ( jω 2 7 /5 ( ) ω ω j j n a M n u a n m = = ) ( ] [ ] [ a jω a n u[n] jω 7 2/5 ] [ 7 5 2 n u n jω 2 7 /5 ] [ 2 5 7 n u n + = ] [ 7 5 2 ] [ n u n y n ] [ 2 5 7 n u n

Outlin DT Furir Transfrm Ovrviw f Furir mthds DT Furir Transfrm f Pridic Signals Prprtis f DT Furir Transfrm Rlatins amng Furir Mthds DTFT: Summary Appndix: Transitin frm DT Furir Sris t DT Furir Transfrm 52

Transitin: DT Furir Sris t DT Furir Transfrm DT Puls Train Signal 53 This DT pridic rctangular-wav signal is analgus t th CT pridic rctangularwav signal usd t illustrat th transitin frm th CT Furir Sris t th CT Furir Transfrm

Transitin: DT Furir Sris t DT Furir Transfrm DTFS f DT Puls Train As th prid f th rctangular wav incrass, th prid f th DT Furir Sris incrass and th amplitud f th DT Furir Sris dcrass 54

Transitin: DT Furir Sris t DT Furir Transfrm Nrmalizd DT Furir Sris f DT Puls Train By multiplying th DT Furir Sris by its prid and pltting vrsus instad f k, th amplitud f th DT Furir Sris stays th sam as th prid incrass and th prid f th nrmalizd DT Furir Sris stays at n 55

Transitin: DT Furir Sris t DT Furir Transfrm Th nrmalizd DT Furir Sris apprachs this limit as th DT prid apprachs infinity 56