Digital Sigal Procssig, Fall 6 Lctur 9: Th Discrt Fourir Trasfor Zhg-Hua Ta Dpartt of Elctroic Systs Aalborg Uivrsity, Dar zt@o.aau.d Digital Sigal Procssig, I, Zhg-Hua Ta, 6 Cours at a glac MM Discrt-ti sigals ad systs Syst MM Fourir-doai rprstatio Saplig ad rcostructio Syst aalysis MM5 Syst structur MM6 MM4 Filtr dsig z-trasfor DFTFFT MM7, MM8 MM MM9, MM Digital Sigal Procssig, I, Zhg-Hua Ta, 6
Th discrt-ti Fourir trasfor DTFT Th DTFT is usful for th thortical aalysis of sigals ad systs. But, accordig to its dfiitio coputatio of DTFT by coputr has svral probls: Th suatio ovr is ifiit Th idpdt variabl w is cotiuous Digital Sigal Procssig, I, Zhg-Hua Ta, 6 Th discrt Fourir trasfor DFT I ay cass, oly fiit duratio is of cocr Th sigal itslf is fiit duratio Oly a sgt is of itrst at a ti Sigal is priodic ad thus oly fiit uiqu valus For fiit duratio squcs, a altrativ Fourir rprstatio is DFT Th suatio ovr is fiit DFT itslf is a squc, rathr tha a fuctio of a cotiuous variabl Thrfor, DFT is coputabl ad iportat for th ipltatio of DSP systs DFT corrspods to sapls of th Fourir trasfor 4 Digital Sigal Procssig, I, Zhg-Hua Ta, 6
Part I: Th discrt Fourir sris Th discrt Fourir sris Th Fourir trasfor of priodic sigals Saplig th Fourir trasfor Th discrt Fourir trasfor Proprtis of th DFT Liar covolutio usig th DFT 5 Digital Sigal Procssig, I, Zhg-Hua Ta, 6 Th discrt Fourir sris A priodic squc with priod + r Priodic squc ca b rprstd by a Fourir sris, i.. a su of copl potial squcs with frqucis big itgr ultipls of th fudatal frqucy associatd with th Th frqucy of th priodic squc. Oly uiqu haroically rlatd copl potials sic 6 so + Digital Sigal Procssig, I, Zhg-Hua Ta, 6
4 Digital Sigal Procssig, I, Zhg-Hua Ta, 6 7 Th Fourir sris cofficits Th cofficits Th squc is priodic with priod For covic, dfi + + W Aalysisquatio Sythsis quatio W W Vry siilar quatios duality Digital Sigal Procssig, I, Zhg-Hua Ta, 6 8 DFS of a priodic ipuls trai Priodic ipuls trai Th discrt Fourir sris cofficits By usig sythsis quatio, a altrativ rprstatio of is r r δ W W W δ
5 Digital Sigal Procssig, I, Zhg-Hua Ta, 6 9 Part II: Th Fourir trasfor of priodic sigals Th discrt Fourir sris Th Fourir trasfor of priodic sigals Saplig th Fourir trasfor Th discrt Fourir trasfor Proprtis of th DFT Liar covolutio usig th DFT Digital Sigal Procssig, I, Zhg-Hua Ta, 6 Th Fourir trasfor of priodic sigals Fourir trasfor of copl potials Fourir trasfor of has th rquird priodicity with priod δ + < < r r a a, δ
Fourir trasfor of a priodic ipuls trai Priodic ipuls trai Th discrt Fourir sris cofficits P δ W Fourir trasfor P δ Fiit duratio sigal outsid of, Costruct * p * δ r r Its Fourir trasfor p δ r r r P δ Digital Sigal Procssig, I, Zhg-Hua Ta, 6 r Th Fourir trasfor of priodic sigals Copar P δ δ Coclud that First rprst it as Fourir sris ad th calculat Fourir trasfor i.. th DFS cofficits of ar sapls of th Fourir trasfor of th o priod of,, othrwis Digital Sigal Procssig, I, Zhg-Hua Ta, 6 6
Part III: Saplig th Fourir trasfor Th discrt Fourir sris Th Fourir trasfor of priodic sigals Saplig th Fourir trasfor Th discrt Fourir trasfor Proprtis of th DFT Liar covolutio usig th DFT Digital Sigal Procssig, I, Zhg-Hua Ta, 6 Saplig th Fourir trasfor A apriodic squc ad its Fourir trasfor Saplig th Fourir trasfor grats a priodic squc i with priod sic th Fourir trasfor is priodic i with priod d 4 Digital Sigal Procssig, I, Zhg-Hua Ta, 6 7
Saplig th Fourir trasfor ow w wat to s if th saplig squc is th squc of DFS cofficits of a squc this ca b do by usig th sythsis quatio W 5 r r W W Digital Sigal Procssig, I, Zhg-Hua Ta, 6 p A priodic squc rsultig fro apriodic covolutio Eapls Cas Fig 8.8 I this cas, th Fourir sris cofficits for a priodic squc ar sapls of th Fourir trasfor of o priod 6 Digital Sigal Procssig, I, Zhg-Hua Ta, 6 8
Eapls Cas Fig 8.9 I this cas, still th Fourir sris cofficits for ar sapls of th Fourir trasfor of. But, o priod of is o logr idtical to This is ust saplig i th frqucy doai as copard i th ti doai discussd bfor. 7 Digital Sigal Procssig, I, Zhg-Hua Ta, 6 Saplig i th frqucy doai Th rlatioship btw ad o priod of i th udrsapld cas is cosidrd a for of ti doai aliasig. Ti doai aliasig ca b avoidd oly if has fiit lgth, ust as frqucy doai aliasig ca b avoidd oly for sigals big badliitd. If has fiit lgth ad w ta a sufficit ubr of qually spacd sapls of its Fourir trasfor spcifically, a ubr gratr tha or qual to th lgth of, th th Fourir trasfor is rcovrabl fro ths sapls, quivaltly is rcovrabl fro. 8 Digital Sigal Procssig, I, Zhg-Hua Ta, 6 9
Digital Sigal Procssig, I, Zhg-Hua Ta, 6 9 Saplig i th frqucy doai Rcovrig i.. rcovrig dos ot rquir to ow its Fourir trasfor at all frqucis Applicatio: rprst fiit lgth squc by usig Fourir sris cofficits DFT othrwis,,, DFS Digital Sigal Procssig, I, Zhg-Hua Ta, 6 Saplig th Fourir trasfor Fourir trasfor Discrt-ti Fourir trasfor Discrt Fourir trasfor d Ω Ω Ω Ω Ω d t dt t t t
Part IV: Th DFT Th discrt Fourir sris Th Fourir trasfor of priodic sigals Saplig th Fourir trasfor Th discrt Fourir trasfor Proprtis of th DFT Liar covolutio usig th DFT Digital Sigal Procssig, I, Zhg-Hua Ta, 6 Th discrt Fourir trasfor Cosidr a fiit lgth squc of lgth sapls if sallr tha, appdig zros Costruct a priodic squc r r Assuig o ovrlap btw r odulo Rcovr th fiit lgth squc,, othrwis To aitai a duality btw th ti ad frqucy doais, choos o priod of as th DFT,, othrwis Digital Sigal Procssig, I, Zhg-Hua Ta, 6
Th DFT Priodic squc ad DFS cofficits W W Sic suatios ar calculatd btw ad - W, Grally, othrwis W, W, othrwis W Digital Sigal Procssig, I, Zhg-Hua Ta, 6 Th DFT A fiit or priodic squc has oly uiqu valus, for << Spctru is copltly dfid by distict frqucy sapls DFT: uifor saplig of DTFT spctru 4 Digital Sigal Procssig, I, Zhg-Hua Ta, 6
Th DFT of a rctagular puls Eapl 8.7 pp.56 5 Digital Sigal Procssig, I, Zhg-Hua Ta, 6 Th DFT of a rctagular puls 6 Digital Sigal Procssig, I, Zhg-Hua Ta, 6
Part V: Proprtis of th DFT Th discrt Fourir sris Th Fourir trasfor of priodic sigals Saplig th Fourir trasfor Th discrt Fourir trasfor Proprtis of th DFT Liar covolutio usig th DFT 7 Digital Sigal Procssig, I, Zhg-Hua Ta, 6 Proprtis of th DFT liarity Liarity DFT a + b a + b Th lgths of squcs ad thir DFTs ar all qual to th aiu of th lgths of ad 8 Digital Sigal Procssig, I, Zhg-Hua Ta, 6 4
Circular shift of a squc Giv DFT DFT Th,, othrwis 9 Digital Sigal Procssig, I, Zhg-Hua Ta, 6 Circular shift of a squc a apl Digital Sigal Procssig, I, Zhg-Hua Ta, 6 5
6 Digital Sigal Procssig, I, Zhg-Hua Ta, 6 Duality, DFT DFT Digital Sigal Procssig, I, Zhg-Hua Ta, 6 Circular covolutio I liar covolutio, o squc is ultiplid by a ti rvrsd ad liarly shiftd vrsio of th othr. For covolutio hr, th scod squc is circularly ti rvrsd ad circularly shiftd. So it is calld a -poit circular covolutio,,,
Circular covolutio with a dlayd ipuls Th dlayd ipuls squc δ W W Digital Sigal Procssig, I, Zhg-Hua Ta, 6 Suary of proprtis of th DFT 4 Digital Sigal Procssig, I, Zhg-Hua Ta, 6 7
Part VI: Liar covolutio of th DFT Th discrt Fourir sris Th Fourir trasfor of priodic sigals Saplig th Fourir trasfor Th discrt Fourir trasfor Proprtis of th DFT Liar covolutio usig th DFT 5 Digital Sigal Procssig, I, Zhg-Hua Ta, 6 Liar covolutio usig th DFT Procdur Coput th -poit DFTs ad of two squcs ad, rspctivly Coput th product of for Coput th squc as th ivrs DFT of As w ow, th ultiplicatio of DFTs corrspods to a circular covolutio of th squcs. To obtai a liar covolutio, w ust sur that circular covolutio has th ffct of liar covolutio. 6 Digital Sigal Procssig, I, Zhg-Hua Ta, 6 8
9 Digital Sigal Procssig, I, Zhg-Hua Ta, 6 7 Liar covolutio of two fiit-lgth squcs Digital Sigal Procssig, I, Zhg-Hua Ta, 6 8 Th circular covolutio corrspodig to is idtical to th liar covolutio corrspodig to if th lgth of DFTs satisfis Circular covolutio as liar covolutio with alaisig othrwis,, : th ivrs DFT of So,, Also, DFT : Dfi a : Fourir trasfor of r r p p + P L
Circular covolutio as liar covolutio with alaisig 9 Digital Sigal Procssig, I, Zhg-Hua Ta, 6 Suary Th discrt Fourir sris Th Fourir trasfor of priodic sigals Saplig th Fourir trasfor Th discrt Fourir trasfor Proprtis of th DFT Liar covolutio usig th DFT 4 Digital Sigal Procssig, I, Zhg-Hua Ta, 6
Cours at a glac MM Discrt-ti sigals ad systs Syst MM Fourir-doai rprstatio Saplig ad rcostructio Syst aalysis MM5 Syst structur MM6 MM4 Filtr dsig z-trasfor DFTFFT MM7, MM8 MM MM9, MM 4 Digital Sigal Procssig, I, Zhg-Hua Ta, 6