DSP First, 2/e. Sample the DTFT DFT. This Lecture: Lecture 17 DFT: Discrete Fourier Transform. Chapter 8, Sections 8-1, 8-2 and 8-4

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1 DSP First READIG ASSIGMETS This ctur: Chaptr 8 Sctios 8-8- ad 8-4 ctur 7 DFT: Discrt Fourir Trasfor Aug 6-6 JH McCllla & RW Schafr ECTURE OBJECTIVES Discrt Fourir Trasfor DFT fro DTFT by frqucy saplig DFT coputatio FFT DFT pairs ad proprtis Priodicity i DFT ti & frqucy Sapl th DTFT DFT Wat coputabl Fourir trasfor Fiit sigal lgth Fiit ubr of frqucis Priodic : is th frqucy id Aug 6-6 JH McCllla & RW Schafr 4 Aug 6-6 JH McCllla & RW Schafr 5

2 Wat a Coputabl IVERSE Fourir Trasfor IVERSE Fourir Trasfor Writ th ivrs DTFT as a fiit Ria su: li ot that Propos: whr This is th ivrs Discrt Fourir Trasfor IDFT Aug JH McCllla & RW Schafr DFT b forward will Ivrs DFT wh = proof proof Copl potials ar ORTHOGOA Copl potials ar ORTHOGOA Aug 6-6 JH McCllla & RW Schafr 7 Orthogoality of Copl Epotials Epotials Th squc st: for Th squc st: for othrwis b bcaus ad l li Aug 6-6 JH McCllla & RW Schafr 8 ad l l li 4-pt DFT: urical Eapl p p Ta th 4 pt DFT of th followig sigal Ta th 4-pt DFT of th followig sigal } { 4 9 Aug 6-6 JH McCllla & RW Schafr 9 4

3 -pt DFT: urical Eapl 4-pt idft: urical Eapl Ta th -pt DFT of th ipuls { } { } Aug 6-6 JH McCllla & RW Schafr Aug 6-6 JH McCllla & RW Schafr Matri For for -pt DFT I MATAB DFT atri is dftt Obtai DFT by = dftt* Or or fficitly by = fft Fast Fourir trasfor FFT algorith latr DFT atri Sigal vctor Aug 6-6 JH McCllla & RW Schafr FFT: Fast Fourir Trasfor FFT is a algorith for coputig th DFT log vrsus opratios g Cout ultiplicatios ad additios For apl wh = 4 = ops vs. opratios tis fastr What about =56 how uch fastr? Aug 6-6 JH McCllla & RW Schafr

4 Zro-Paddig givs dsr FREQUEC SAMPIG FREQUEC SAMPIG W t l f DTFT Wat ay sapls of DTFT WH? to a a sooth plot Fiit sigal lgth Fiit ubr of frqucis Thus w d Aug 6-6 JH McCllla & RW Schafr 4 Zro-Paddig with th FFT Zro-Paddig with th FFT Gt ay sapls of DTFT Gt ay sapls of DTFT Fiit sigal lgth Fi it b f f i Fiit ubr of frqucis Thus w d I MATAB Us = fft With =lgth lss tha Dfi padto = zros-; Ta th pt DFT of padto Aug 6-6 JH McCllla & RW Schafr 5 Ta th -pt DFT of padto DFT priodic i frqucy doai frqucy doai Sic DTFT is priodic i frqucy th DFT Sic DTFT is priodic i frqucy th DFT ust also b priodic i What about gativ idics ad Cougat What about gativ idics ad Cougat Sytry? Aug 6-6 JH McCllla & RW Schafr 6 9 DFT Priodicity i Frqucy Id Frqucy Id Aug 6-6 JH McCllla & RW Schafr 7.g.

5 DFT pairs & proprtis Rcall DTFT pairs bcaus DFT is sapld DTFT S t two slids DFT acts o a fiit-lgth sigal so w ca us DTFT pairs & proprtis for fiit sigals Wat DFT proprtis rlatd to coputatio Ad w will coctrat t o o or pair: DTFT ad DFT of fiit siusoid or cp gth- sigal -pt DFT Aug 6-6 JH McCllla & RW Schafr 8 Ths sigals hav ifiit it lgth Aug 6-6 JH McCllla & RW Schafr 9 Suary of DTFT Pairs Ths proprtis ivolv circular idig Aug 6-6 JH McCllla & RW Schafr Aug 6-6 JH McCllla & RW Schafr

6 Covolutio Proprty ot th sa Alost tru for DFT: Covolutio aps to ultiplicatio of trasfors d a diffrt id of covolutio CIRCUAR COVOUTIO ATER i a advacd DSP cours iwis is for Ti-Shiftig Has to b circular Bcaus th doai is also priodic Aug 6-6 JH McCllla & RW Schafr Aug 6-6 JH McCllla & RW Schafr Dlay Proprty of DFT Rcall DTFT proprty for ti shiftig: y d Epctd DFT proprty via frqucy saplig y Idics such as odulo- bcaus d d d ust b valuatd d d d Aug 6-6 JH McCllla & RW Schafr 4 DTFT of a gth- Puls Kow DTFT of fiit rctagular puls Dirichlt for ad a liar phas tr si si othrwis Us frqucy-saplig to gt DFT si D si si si D Aug 6-6 JH McCllla & RW Schafr 5

7 DTFT of a Fiit gth Copl Epotial Copl Epotial Kow DTFT of fiit rctagular puls Kow DTFT of fiit rctagular puls Dirichlt for ad a liar phas tr si si si othrwis Us frqucy shift proprty si si D Us frqucy-shift proprty i si th i y Aug 6-6 JH McCllla & RW Schafr 6 si othrwis DTFT of a Fiit gth Copl Epotial Copl Epotial Kow DTFT so w ca sapl i frqucy Kow DTFT so w ca sapl i frqucy si si othrwis y Thus th -poit DFT is othrwis y Dirichlt Fuctio si at si si D Aug 6-6 JH McCllla & RW Schafr 7 si -pt DFT of Copl Epotial Epotial.5 si si si othrwis Aug 6-6 JH McCllla & RW Schafr 8 for so th outli is a shiftd Dirichlt si si D -pt DFT of Copl Ep: o th grid th grid si si si othrwis Aug 6-6 JH McCllla & RW Schafr 9 for so th outli is a shiftd Dirichlt si si D

8 5-pt DFT of Siusoid: zro paddig 5.4 RECA: BadPass Filtr BPF Frqucy shiftig up ad dow is do by cosi ultiplicatio i th ti doai Acos othrwis 4 4 AD AD Aug 6-6 JH McCllla & RW Schafr 5-pt DFT of Siusoid: zro paddig 5.4? BPF is frqucy shiftd vrsio of PF blow h cos si diff BP id DTFT H co BP co co co Aug 6-6 JH McCllla & RW Schafr? Zro-crossigs of Dirichlt? Width of Dirichlt? Dsity of frqucy sapls? Thus w hav a sipl BPF Aug 6-6 JH McCllla & RW Schafr