Topic 5:Discrete-Time Fourier Transform (DTFT)
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- Abraham Cooper
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1 ELEC64: Sigals Ad Systms Tpic 5:Discrt-Tim Furir Trasfrm DTFT Aishy Amr Ccrdia Uivrsity Elctrical ad Cmputr Egirig Itrducti DT Furir Trasfrm Sufficit cditi fr th DTFT DT Furir Trasfrm f Pridic Sigals DTFT ad LTI systms: Frqucy rsps Prprtis f DT Furir Trasfrm Summary Appdix: Trasiti frm DT Furir Sris t DT Furir Trasfrm Appdix: Rlatis amg Furir Mthds Figurs ad xampls i ths curs slids ar ta frm th fllwig surcs: A. Opphim, A.S. Willsy ad S.H. Nawab, Sigals ad Systms, d Editi, Prtic-Hall, 997 M.J. Rbrts, Sigals ad Systms, McGraw Hill, 004 J. McCllla, R. Schafr, M. Ydr, Sigal Prcssig First, Prtic Hall, 003
2 Furir rprstati A Furir fucti is uiqu, i.., tw sam sigals i tim giv th sam fucti i frqucy Th DT Furir Sris is a gd aalysis tl fr systms with pridic xcitati but cat rprst a apridic DT sigal fr all tim Th DT Furir Trasfrm ca rprst a apridic discrt-tim sigal fr all tim Its dvlpmt fllws xactly th sam as that f th Furir trasfrm fr ctiuus-tim apridic sigals
3 Ovrviw f Frqucy Aalysis Mthds 3
4 Ovrviw f Furir Aalysis Mthds Pridic i Tim Discrt i Frqucy Apridic i Tim Ctiuus i Frqucy Ctiuus i Tim Apridic i Frqucy CT Furir Sris : a x t T T 0 x t a 0 t 0 t dt CT - P CT IvrsFurir Sris : T DT DT CT - P T CT Furir Trasfrm: X x t x t X t dt t d CT IvrsCT Furir Trasfrm: CT CT CT Discrt i Tim Pridic i Frqucy DT Furir Sris X x N N 0 x IvrsDT Furir Sris N 0 0 X DT - P 0 N DT - P DT - P N N DT - P N DT Furir Trasfrm: X IvrsDT Furir Trasfrm: x x X d DT CT CT P P DT 4
5 Ovrviw f Furir symbls Variabl Prid Ctiuus Frqucy DT x N Discrt Frqucy / N CT xt t T / T 5 DT-FT: Discrt i tim; Apridic i tim; Ctius i Frqucy; Pridic i Frqucy DT-FS: Discrt i tim; Pridic i tim; Discrt i Frqucy; Pridic i Frqucy CT-FS: Ctiuus i tim; Pridic i tim; Discrt i Frqucy; Apridic i Frqucy CT-FT: Ctiuus i tim; Apridic i tim; Ctius i Frqucy; Apridic i Frqucy
6 Outli Itrducti DT Furir Trasfrm Sufficit cditi fr DTFT DT Furir Trasfrm f Pridic Sigals Prprtis f DT Furir Trasfrm DTFT & LTI systms: Frqucy rsps DTFT: Summary Appdix: Trasiti frm DT Furir Sris t DT Furir Trasfrm Appdix: Rlatis amg Furir Mthds 6
7 7 DT Furir Trasfrm DT Furir trasfrm ad th ivrs FT FT dscribs which frqucis ar prst i th rigial fucti Th rigial sigal ca b rcvrd frm wig th Furir trasfrm, ad vic vrsa Th fucti X ω is pridic i ω with prid π Th fucti ω is pridic with N=π d X x x X, f x f X x X : Frms
8 8 DT Furir Trasfrm DT sigal rprstatis: A sum f scald, dlayd impuls A liar cmbiati f wightd siusidal sigals x x d X x
9 DT Furir Trasfrm: Drivati Lt x b th apridic DT sigal W cstruct a pridic sigal x fr which x is prid x is cmprisd f ifiit umbr f rplicas f x Each rplica is ctrd at a itgr multipl f N N is th prid f x Csidr th fllwig figur which illustrats a xampl f x ad th cstructi f Clarly, x is dfid btw N ad N Csqutly, N has t b chs such that N > N + N + s that adact rplicas d t vrlap Clarly, as w lt as dsird 9
10 DT Furir Trasfrm: Drivati Lt us w xami th FS rprstati f Sic x is dfid btw N ad N a i th abv xprssi simplifis t 0 ω = π/n
11 DT Furir Trasfrm: Drivati Nw dfiig th fucti W ca s that th cfficits a ar rlatd t X ω as whr ω 0 = π/n is th spacig f th sampls i th frqucy dmai Thrfr As N icrass ω 0 dcrass, ad as N th abv quati bcms a itgral
12 DT Furir Trasfrm: Drivati O imprtat bsrvati hr is that th fucti X ω is pridic i ω with prid π Thrfr, as N, Nt: th fucti ω is pridic with N=π This lads us t th DT-FT pair f quatis
13 DT Furir Trasfrm: Exampls x X r r Th pridic impuls trai Lt x a u a X a 3
14 DT Furir Trasfrm: Exampls Apridic Pridic 4
15 5 Furir trasfrm pairs
16 Outli Itrducti DT Furir Trasfrm Sufficit cditi fr DTFT DT Furir Trasfrm f Pridic Sigals Prprtis f DT Furir Trasfrm DTFT & LTI systms: Frqucy rsps DTFT: Summary Appdix: Trasiti frm DT Furir Sris t DT Furir Trasfrm Appdix: Rlatis amg Furir Mthds 6
17 Sufficit cditi fr DTFT Cditi fr th cvrgc f th ifiit sum X x x x If x is abslutly summabl, its FT xists sufficit cditi 7
18 8 Exampl: Exptial squc DTFTds t xist : : : a X a a X a u a x
19 Outli Itrducti DT Furir Trasfrm Sufficit cditi fr DTFT DT Furir Trasfrm f Pridic Sigals Prprtis f DT Furir Trasfrm DTFT & LTI systms: Frqucy rsps DTFT: Summary Appdix: Trasiti frm DT Furir Sris t DT Furir Trasfrm Appdix: Rlatis amg Furir Mthds 9
20 FT f Pridic DT Sigals Csidr th ctiuus tim sigal This sigal is pridic Furthrmr, th Furir sris f this sigal is ust a impuls f wight ctrd at ω= ω 0 Nw csidr this sigal It is als pridic ad thr is impuls pr prid Hwvr, th sparati btw adact impulss is π I particular, th DT Furir Trasfrm fr this sigal is 0 DTFT f a pridic sigal with prid N X X ; N
21 DTFT: Pridic sigal Th sigal ca b xprssd as W ca immdiatly writ Equivaltly prid π
22 DT FT f pridic sigals FS vs. FT
23 Outli Itrducti DT Furir Trasfrm Sufficit cditi fr DTFT DT Furir Trasfrm f Pridic Sigals Prprtis f DT Furir Trasfrm DTFT & LTI systms: Frqucy rsps DTFT: Summary Appdix: Trasiti frm DT Furir Sris t DT Furir Trasfrm Appdix: Rlatis amg Furir Mthds 3
24 Cmplx umbrs *Cartsia rprstati : z x y Magitud f It is is * Plar rprstati : Cmplx Cugat : z Phasargumtf ca chagby ay multipl radias t dgrsar big usd z * * * 4 z x y ; z z ad zz ar ral r th distacf th agl t thral psitiv z a z z x f y pit z frm th rigi z z axis ad still cs ta y x giv z si thsamagl
25 Prprtis f th DTFT Th fucti ω is pridic with N=π 5
26 6 Prprtis f th DTFT
27 7 Prprtis f th DTFT
28 8 Exampl: Tim shift Dtrmiig th DTFT f Sluti F a a X u a x a x a X X u a x x a X u a x 5 i.. 5 i u a x
29 9 Prprtis f th DTFT
30 30 Prprtis f th DTFT
31 3
32 Symmtry prprtis f th DTFT
33 Symmtry prprtis f th DTFT Duality prprty
34 34 Prprtis f th DTFT
35 Prprtis f th DT FT Accumulati : m m - - xm xm - - ω whr thimpuls trai thright - had sid rflcts th avrag valur dc cmptthat may rsult frm thsummati f X ω Xf πx 0 m X 0 cmb f δω πm 35 Trai f impulss cmb
36 36 Prprtis f th DT FT
37 37 Prprtis f th DT FT
38 38 Prprtis f th DT FT
39 39 Prprtis f th DT FT
40 40 Prprtis f th DT FT
41 4 Prprtis f th DT FT * impuls rspsh: a LTI systmwith fr It fllws: Multiplicati & Cvluti duality : X H Y x h y Y X y x Y X y x
42 4 Prprtis f th DT FT
43 43 Prprtis f th DT FT
44 Prprtis f th DT FT: Diffrc quati DT LTI Systms ar charactrizd by Liar Cstat-Cfficit Diffrc Equatis A gral liar cstat-cfficit diffrc quati fr a LTI systm with iput x ad utput y is f th frm Nw applyig th FT t bth sids f th abv quati, w hav But w w that th iput ad th utput ar rlatd t ach thr thrugh th impuls rsps f th systm, dtd by h, i.., 44
45 Prprtis f th DT FT : Diffrc quati Applyig th cvluti prprty if is giv a diffrc quati crrspdig t sm systm, th FT f th impuls rsps f th systm ca fud dirctly frm th diffrc quati by applyig th Furir trasfrm FT f th impuls rsps = Frqucy rsps Ivrs FT f th frqucy rsps = Impuls rsps 45
46 Prprtis f th DT FT: Exampl With a <, csidr th causal LTI systm that is charactrizd by th diffrc quati Th frqucy rsps f th systm is Frm tabls r by applyig ivrs FT, w gt 46
47 47 Tabl.
48 Outli Itrducti DT Furir Trasfrm Sufficit cditi fr DTFT DT Furir Trasfrm f Pridic Sigals Prprtis f DT Furir Trasfrm DTFT & LTI systms: Frqucy rsps DTFT: Summary Appdix: Trasiti frm DT Furir Sris t DT Furir Trasfrm Appdix: Rlatis amg Furir Mthds 48
49 49 Frqucy rsps f LTI systms If iput is cmplx xptials Dfi h & H: Frqucy ad impuls rspss ar a FT pair h h T y x } { H y h H d H X y d X x if
50 50 Frqucy rsps Th frqucy rsps f discrt-tim LTI systms is always a pridic fucti f th frqucy variabl w with prid Oly spcify vr th itrval Th lw frqucis ar cls t 0 Th high frqucis ar cls t Frqucy rsps is grally cmplx H h h H H I R H H H H dscribs chags t x i magitud ad phas
51 5 Frqucy rsps: Exampl Frqucy rsps f th idal dlay systm d d I d R d d d H H H H H h x y d, si, cs Idaldlay :
52 5 Frqucy rsps FT pair & impuls rspssar a Frqucy rsps thfrqucy with ; Rsps: iput If Cvluti thrm: thimpuls rsps with ; systms: LTI Rspsf H H X X h x y x x h h y x
53 53 Frqucy rsps: Exampl
54 Idal frqucy-slctiv LTIsystms r filtrs Idal frqucy-slctiv filtr hav uity frqucy rsps vr a crtai rag f frqucis, ad is zr at th rmaiig frqucis Exampl: Idal lw-pass filtr: passs ly lw frqucis ad rcts high frqucis f a iput sigal x 54
55 Exampl : idal lwpass filtr Frqucy rsps H lp, 0, c c h lp c c d si c, h is t abslutly summabl Filtr causal 55
56 Outli Itrducti DT Furir Trasfrm Sufficit cditi fr DTFT DT Furir Trasfrm f Pridic Sigals Prprtis f DT Furir Trasfrm DTFT & LTI systms: Frqucy rsps DTFT: Summary Appdix: Trasiti frm DT Furir Sris t DT Furir Trasfrm Appdix: Rlatis amg Furir Mthds 56
57 DTFT: Summary DT Furir Trasfrm rprsts a discrt tim apridic sigal as a sum f ifiitly may cmplx xptials, with th frqucy varyig ctiuusly i -π, π DTFT is pridic ly d t dtrmi it fr 57
58 58 Summary: Sigal & Systm rprstatis Sigal: A sum f scald, dlayd impuls Sigal: A liar cmbiati f wightd siusidal sigals LTI systm: Cvluti LTI systm: Diffrc quati: x x d X x * y H X Y h x h x
59 59 Ral-wrld applicati: Imag cmprssi Ergy Distributi f trasfrm DCT Cfficits i Typical Imags
60 Ral-wrld applicatis: Imag cmprssi Imags Apprximatd by Diffrt Numbr f trasfrm DCT Cfficits Origial With 6/64 Cfficits With 8/64 Cfficits With 4/64 Cfficits 60 Wavfrm-basd vid cdig
61 DTFT: Summary Kw hw t calculat th DTFT f simpl fuctis Kw th gmtric sum: Kw Furir trasfrms f spcial fuctis,.g. δ, xptial Kw hw t calculat th ivrs trasfrm f ratial fuctis usig partial fracti xpasi Prprtis f DT Furir trasfrm Liarity, Tim-shift, Frqucy-shift, 6
62 6 DT-FT Summary: a quiz A discrt-tim LTI systm has impuls rsps Fid th utput y du t iput Sluti : Us th cvluti prprty: u h 7 u x * X H Y x h y, a a M u a m X H 7 ad
63 63 DT-FT Summary: a quiz ct. Usig partial fracti xpasi mthd f fidig ivrs FT givs: Thrfr, sic a FT is uiqu, i.. tw sam sigals i tim giv th sam fucti i frqucy ad sic It ca b s that a FT f th typ shuld crrspd t a sigal. Thrfr, th ivrs FT f is th ivrs FT f is Thus th cmplt utput 7 Y Y 7 / 5 7/5 a M u a m a a u 7 /5 7 5 u 7/5 5 7 u u u y
64 Outli Itrducti DT Furir Trasfrm Sufficit cditi fr DTFT DT Furir Trasfrm f Pridic Sigals Prprtis f DT Furir Trasfrm DTFT & LTI systms: Frqucy rsps DTFT: Summary Appdix: Trasiti frm DT Furir Sris t DT Furir Trasfrm Appdix: Rlatis amg Furir Mthds 64
65 Trasiti: DT Furir Sris t DT Furir Trasfrm DT Puls Trai Sigal 65 This DT pridic rctagular-wav sigal is aalgus t th CT pridic rctagularwav sigal usd t illustrat th trasiti frm th CT Furir Sris t th CT Furir Trasfrm
66 Trasiti: DT Furir Sris t DT Furir Trasfrm DTFS f DT Puls Trai As th prid f th rctagular wav icrass, th prid f th DT Furir Sris icrass ad th amplitud f th DT Furir Sris dcrass 66
67 Trasiti: DT Furir Sris t DT Furir Trasfrm Nrmalizd DT Furir Sris f DT Puls Trai By multiplyig th DT Furir Sris by its prid ad plttig vrsus istad f, th amplitud f th DT Furir Sris stays th sam as th prid icrass ad th prid f th rmalizd DT Furir Sris stays at 67
68 Trasiti: DT Furir Sris t DT Furir Trasfrm Th rmalizd DT Furir Sris apprachs this limit as th DT prid apprachs ifiity 68
69 Outli Itrducti DT Furir Trasfrm Sufficit cditi fr DTFT DT Furir Trasfrm f Pridic Sigals Prprtis f DT Furir Trasfrm DTFT & LTI systms: Frqucy rsps DTFT: Summary Appdix: Trasiti frm DT Furir Sris t DT Furir Trasfrm Appdix: Rlatis amg Furir Mthds 69
70 Rlatis Amg Furir Mthds Pridic i Tim Discrt i Frqucy Apridic i Tim Ctiuus i Frqucy Ctiuus i Tim Apridic i Frqucy CT Furir Sris : a x t T T 0 x t a 0 t 0 t dt CT - P CT IvrsFurir Sris : T DT DT CT - P T CT Furir Trasfrm: X x t x t X t dt t d CT IvrsCT Furir Trasfrm: CT CT CT Discrt i Tim Pridic i Frqucy DT Furir Sris X x N N 0 x IvrsDT Furir Sris N 0 0 X DT - P 0 N DT - P DT - P N N DT - P N DT Furir Trasfrm: X IvrsDT Furir Trasfrm: x x X d DT CT CT P P DT 70
71 Rlatis Amg Furir Mthds 7
72 7 CT Furir Trasfrm - CT Furir Sris
73 73 CT Furir Trasfrm - CT Furir Sris
74 74 CT Furir Trasfrm - DT Furir Trasfrm
75 75 CT Furir Trasfrm - DT Furir Trasfrm
76 76 DT Furir Sris - DT Furir Trasfrm
77 77 DT Furir Sris - DT Furir Trasfrm
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