Chapter 5 The Discrete-Time Fourier Transform

Transcription

1 ELG 30 Sigls d Systms Chptr 5 Chptr 5 Th Discrt-Tim ourir Trsform 5.0 Itroductio Thr r my similritis d strog prllls i lyzig cotiuous-tim d discrttim sigls. Thr r lso importt diffrcs. or xmpl, th ourir sris rprsttio of discrt-tim priodic sigl is fiit sris, s opposd to th ifiit sris rprsttio rquird for cotiuous-tim priod sigl. I this chptr, th lysis will b crrid out by tig dvtg of th similritis btw cotiuous-tim d discrt-tim ourir lysis. 5. Rprsttio of Apriodic Sigls: Th discrt-tim ourir Trsform 5.. Dvlopmt of th Discrt-Tim ourir Trsform Cosidr grl squc tht is fiit durtio. Tht is, for som itgrs d, x quls to zro outsid th rg, s show i th figur blow. W c costruct priodic squc ~ x usig th priodic squc x s o priod. As w choos th priod to b lrgr, ~ x is idticl to x ovr logr itrvl, s, ~ x x. Bsd o th ourir sris rprsttio of priodic sigl giv i Eqs d 3.8, w hv /5 Yo

2 ELG 30 Sigls d Systms Chptr 5 ~ π x / < >, 5. ~ π /. 5. x If th itrvl of summtio is slctd to iclud th itrvl, so ~ x c b rplcd by x i th summtio, x π / π /, 5.3 x Dfiig th fuctio x, 5. So c b writt s 0, 5.5 Th ~ x c b xprssd s x. 5.6 ~ 0 π / / 0 π 0 < > π < > As ~ x x, d th bov xprssio psss to itgrl, x d π, 5.7 π Th Discrt-tim ourir trsform pir: x d π π, 5.8 x. 5.9 /5 Yo

3 ELG 30 Sigls d Systms Chptr 5 3/5 Yo Eq. 5.8 is rfrrd to s sythsis qutio, d Eq. 5.9 is rfrrd to s lysis qutio d 0 is rfrrd to s th spctrum of x. 5.. Exmpls of Discrt-Tim ourir Trsforms Exmpl: Cosidr u x, <. 5.0 u x Th mgitud d phs for this xmpl r show i th figur blow, whr 0 > d 0 < r show i d b. Exmpl: x, < u Lt m i th first summtio, w obti 0 cos u m m m

4 ELG 30 Sigls d Systms Chptr 5 Exmpl: Cosidr th rctgulr puls, x, 5. 0, > / / si + si. 5.5 This fuctio is th discrt coutrprt of th sic fuctio, which pprs i th ourir trsform of th cotiuous-tim puls. Th diffrc btw ths two fuctios is tht th discrt o is priodic s figur with priod of π, whrs th sic fuctio is priodic Covrgc Th qutio x covrgs ithr if x is bsolutly summbl, tht is x <, 5.6 or if th squc hs fiit rgy, tht is x <. 5.7 /5 Yo

5 ELG 30 Sigls d Systms Chptr 5 Ad thr is o covrgc issus ssocitd with th sythsis qutio 5.8. If w pproximt pridic sigl x by itgrl of complx xpotils with frqucis t ovr th itrvl W, W xˆ d, 5.8 π W d x ˆ x for W π ourir trsform.. Thrfor, th Gibbs phomo dos ot xist i th discrt-tim Exmpl: th pproximtio of th impuls rspos with diffrt vlus of W. or W π /, 3π /8, π /, 3π /, 7π /8, π, th pproximtios r plottd i th figur blow. W c s tht wh W π, x x. 5/5 Yo

6 ELG 30 Sigls d Systms Chptr 5 5. ourir trsform of Priodic Sigls or priodic discrt-tim sigl, 0 x, 5.9 its ourir trsform of this sigl is priodic i with priod π, d is giv + l πδ πl ow cosidr priodic squc x with priod d with th ourir sris rprsttio x π /. 5. < > Th ourir trsform is π πδ Exmpl: Th ourir trsform of th priodic sigl x 0 0 cos0 +, with π 0, is giv s π π πδ + πδ +, 3 3 π < π. 5. 6/5 Yo

7 ELG 30 Sigls d Systms Chptr 5 Exmpl: Th priodic impuls tri + x δ. 5.5 Th ourir sris cofficits for this sigl c b clcultd < > x π /. 5.6 Choosig th itrvl of summtio s 0, w hv. 5.7 Th ourir trsform is π π δ /5 Yo

8 ELG 30 Sigls d Systms Chptr Proprtis of th Discrt-Tim ourir Trsform ottios to b usd { x }, { } x, x Priodicity of th Discrt-Tim ourir Trsform Th discrt-tim ourir trsform is lwys priodic i with priod π, i.., + π 5.3. Lirity. 5.9 If x, d x, th x + bx + b Tim Shiftig d rqucy Shiftig If x, th 0 x 5.3 d x 5.3 8/5 Yo

9 ELG 30 Sigls d Systms Chptr Cougtio d Cougt Symmtry If x, th x* * 5.33 If x is rl vlud, its trsform is cougt symmtric. Tht is * 5.3 rom this, it follows tht R{ } is v fuctio of d Im{ } is odd fuctio of. Similrly, th mgitud of is v fuctio d th phs gl is odd fuctio. urthrmor, { x } R{ } Ev, 5.35 d { x } Im{ } Od Diffrcig d Accumultio If x, th x x or sigl y x m, 5.38 m its ourir trsform is giv s 9/5 Yo

10 ELG 30 Sigls d Systms Chptr 5 m x m + π 0 + m δ π Th impuls tri o th right-hd sid rflcts th dc or vrg vlu tht c rsult from summtio. or xmpl, th ourir trsform of th uit stp x u c b obtid by usig th ccumultio proprty. W ow g δ G, so x m g m Tim Rvrsl + 0 G + πg δ π + π + δ π. 5.0 If x, th x Tim Expsio or cotiuous-tim sigl, w hv x t. 5. or discrt-tim sigls, howvr, should b itgr. Lt us dfi sigl with positiv itgr, x /, if is multipl of x , if is ot multipl of x is obtid from x by plcig zros btw succssiv vlus of th origil sigl. Th ourir trsform of x is giv by 0/5 Yo

11 ELG 30 Sigls d Systms Chptr 5 Tht is, + + r x x r x r r + r r. 5. x. 5.5 or >, th sigl is sprd out d slowd dow i tim, whil its ourir trsform is comprssd. Exmpl: Cosidr th squc x displyd i th figur blow. This squc c b rltd to th simplr squc y s show i b. x y + y, whr y /, y 0, if if is v is odd Th sigls y d y r dpictd i c d d. As c b s from th figur blow, y is rctgulr puls with, its ourir trsform is giv by Y si5 /. si / Usig th tim-xpsio proprty, w th obti /5 Yo

12 ELG 30 Sigls d Systms Chptr 5 y si5 si y 5 si5 si Combiig th two, w hv si5 +. si Diffrtitio i rqucy If x, Diffrtit both sids of th lysis qutio d d + x x. 5.6 Th right-hd sid of th Eq. 5.6 is th ourir trsform of x. Thrfor, multiplyig both sids by, w s tht d x d Prsvl s Rltio If x, th w hv + x d π π 5.8 /5 Yo

13 ELG 30 Sigls d Systms Chptr 5 Exmpl: Cosidr th squc x whos ourir trsform is dpictd for π π i th figur blow. Dtrmi whthr or ot, i th tim domi, x is priodic, rl, v, d /or of fiit rgy. Th priodicity i tim domi implis tht th ourir trsform hs oly impulss loctd t vrious itgr multipls of th fudmtl frqucy. This is ot tru for. W coclud tht x is ot priodic. Sic rl-vlud squc should hv ourir trsform of v mgitud d phs fuctio tht is odd. This is tru for d. W coclud tht x is rl. If x is rl d v, th its ourir trsform should b rl d v. Howvr, sic, is ot rl, so w coclud tht x is ot v. Bsd o th Prsvl s rltio, itgrtig qutity. W coclud tht x hs fiit rgy. from π to π will yild fiit 5. Th covolutio Proprty If x, h d y r th iput, impuls rspos, d output, rspctivly, of LTI systm, so tht y x h, 5.9 th, Y H, /5 Yo

14 ELG 30 Sigls d Systms Chptr 5 whr, H d Y r th ourir trsforms of x, h d y, rspctivly. Exmpl: Cosidr th discrt-tim idl lowpss filtr with frqucy rspos H illustrtd i th figur blow. Usig π π s th itrvl of itgrtio i th sythsis qutio, w hv h π π π H d π π π d sic π Th frqucy rspos of th discrt-tim idl lowpss filtr is show i th right figur. Exmpl: Cosidr LTI systm with impuls rspos h α u, α <, d suppos tht th iput to th systm is x β u, β <. Th ourir trsforms for h d x r H α, d, β so tht Y H. α β /5 Yo

15 ELG 30 Sigls d Systms Chptr 5 If α β, th prtil frctio xpsio of Y is giv by Y α β A B α β α β + +, α β α β W c obti th ivrs trsform by ispctio: + α u u y α β α u β u α β α β α β ββ or α β, +. Y α, which c b xprssd s d Y. α α d Usig th frqucy diffrtitio proprty, w hv d α u, d α To ccout for th fctor, w us th tim-shiftig proprty to obti + d + α u + d α, illy, ccoutig for th fctor y + α u +. / α, w hv Sic th fctor + is zro t, so y c b xprssd s y + α u. Exmpl: Cosidr th systm show i th figur blow. Th LTI systms with frqucy rspos H r idl lowpss filtrs with cutoff frqucy π / d uity gi i th pssbd. lp 5/5 Yo

16 ELG 30 Sigls d Systms Chptr 5 π w x x π W. π W H. lp π w w w 3 π π π W W H. 3 lp π π W W H Discrt-ourir trsforms r lwys 3 priodic with priod of π. W H. lp π Y W + W H + H 3 Th ovrll systm hs frqucy rspos π H H H + lp which is show i figur b. lp Th filtr is rfrrd to s bdstop filtr, whr th stop bd is th rgio π / < < 3π /. lp lp lp, It is importt to ot tht ot vry discrt-tim LTI systm hs frqucy rspos. If LTI systm is stbl, th its impuls rspos is bsolutly summbl; tht is, + h <, 5.5 lp. 5.5 Th multiplictio Proprty Cosidr y qul to th product of x d x, with Y,, d dotig th corrspodig ourir trsforms. Th 6/5 Yo

17 ELG 30 Sigls d Systms Chptr 5 y x dθ θ x π π 5.5 Eq. 5.5 corrspods to priodic covolutio of d, d th itgrl i this qutio c b vlutd ovr y itrvl of lgth π. Exmpl: Cosidr th ourir trsform of sigl x which th product of two sigls; tht is x x x whr si3π / x, d π si π / x. π Bsd o Eq. 5.5, w my writ th ourir trsform of x π π π θ dθ Eq rsmbls priodic covolutio, xcpt for th fct tht th itgrtio is limitd to th itrvl of π < θ < π. Th qutio c b covrtd to ordiry covolutio with itgrtio itrvl < θ < by dfiig for π < < π 0 othrwis ˆ Th rplcig i Eq by ˆ, d usig th fct tht ˆ is zro for π < < π, w s tht π θ π π π θ θ dθ ˆ d. Thus, is / π tims th priodic covolutio of th rctgulr puls ˆ d th priodic squr wv. Th rsult of thus covolutio is th ourir trsform, s show i th figur blow. 7/5 Yo

18 ELG 30 Sigls d Systms Chptr Tbls of ourir Trsform Proprtis d Bsic ourir Trsform Pris 8/5 Yo

19 ELG 30 Sigls d Systms Chptr 5 9/5 Yo

20 ELG 30 Sigls d Systms Chptr Dulity or cotiuous-tim ourir trsform, w obsrvd symmtry or dulity btw th lysis d sythsis qutios. or discrt-tim ourir trsform, such dulity dos ot xist. Howvr, thr is dulity i th discrt-tim sris qutios. I dditio, thr is dulity rltioship btw th discrt-tim ourir trsform d th cotiuous-tim ourir sris Dulity i th discrt-tim ourir Sris Cosidr th priodic squcs with priod, rltd through th summtio f m If w lt r< > g r r π / m m d r, Eq. 5.5 bcoms. 5.5 f < > g r r π / Compr with th two qutios blow, x π /, 3.80 π / x. 3.8 w fod tht g r corrspods to th squc of ourir sris cofficits of f. Tht is S f g This dulity implis tht vry proprty of th discrt-tim ourir sris hs dul. or xmpl, x 0 m π / S S π / m /5 Yo

21 ELG 30 Sigls d Systms Chptr 5 r dul. Exmpl: Cosidr th followig priodic sigl with priod of 9. si5π /9, 9 si π/9 x 5, 9 multipl of 9 multipl of W ow tht rctgulr squr wv hs ourir cofficits i form much s i Eq Dulity suggsts tht th cofficits of x must b i th form of rctgulr squr wv. Lt g b rctgulr squr wv with priod 9,, g, , < Th ourir sris cofficits b for g c b giv rfr to xmpl o pg 7/3 b si5π /9, 9 si π/9 5, 9 multipl of multipl of Th ourir lysis qutio for g c b writt b 9 π / Itrchgig th ms of th vribl d d otig tht x b, w fid tht x 9 π / 9. Lt ' i th sum o th right sid, w obti x 9 + π '/ 9. /5 Yo

22 ELG 30 Sigls d Systms Chptr 5 illy, movig th fctor / 9 isid th summtio, w s tht th right sid of th qutio hs th form of th sythsis qutio for x. Thus, w coclud tht th ourir cofficits for x r giv by / 9,, 0, < with priod of Systm Chrctriztio by Lir Costt-Cofficit Diffrc Equtios A grl lir costt-cofficit diffrc qutio for LTI systm with iput x d output x is of th form 0 M y b x, which is usully rfrrd to s th-ordr diffrc qutio. Thr r two wys to dtrmi H : Th first wy is to pply iput form H x to th systm, d th output must b of th. Substitutig ths xprssios ito th Eq. 5.63, d prformig som lgbr llows us to solv for H. Th scod pproch is to us discrt-tim ourir trsform proprtis to solv for H. Bsd o th covolutio proprty, Eq c b writt s Y H. 5.6 Applyig th ourir trsform to both sids d usig th lirity d tim-shiftig proprtis, w obti th xprssio 0 Y M 0 b /5 Yo

23 ELG 30 Sigls d Systms Chptr 5 3/5 Yo or quivltly M b Y H Exmpl: Cosidr th cusl LTI systm tht is chrctrizd by th diffrc qutio, x y y, <. Th frqucy rspos of this systm is Y H. Th impuls rspos is giv by u h. Exmpl: Cosidr cusl LTI systm tht is chrctrizd by th diffrc qutio 8 3 x y y y +.. Wht is th impuls rspos?. If th iput to this systm is u x, wht is th systm rspos to this iput sigl? Th frqucy rspos is 3 8 H +. Aftr prtil frctio xpsio, w hv H, Th ivrs ourir trsform of ch trm c b rcogizd by ispctio, u u h.

24 ELG 30 Sigls d Systms Chptr 5 /5 Yo Usig Eq. 5.6 w hv 3 3 H Y. Aftr prtil-frctio xpsio, w obti H Y + 8 Th ivrs ourir trsform is 8 u y + +.