Digital Signal Processing, Fall E-Study -

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1 Digita Siga Procssig, Fa 5 - E-Study - Lctur : Fast Fourir Trasform Zhg-Hua Ta Dpartmt of Commuicatio Tchoogy aborg Uivrsity, Dmar zt@om.aau.d Digita Siga Procssig, X, Zhg-Hua Ta, 5 Cours at a gac MM Discrt-tim sigas ad systms Systm MM Fourir-domai rprstatio Sampig ad rcostructio Systm aaysis MM5 Systm structur MM6 MM4 Fitr dsig z-trasform DFTFFT MM7, MM8 MM3 MM9, MM Digita Siga Procssig, X, Zhg-Hua Ta, 5

2 Digita computatio of th DFT Th DFT of a fiit-gth squc of gth X x, Th ivrs DFT x X,,...,,,..., Du to th duaity, focus o th DFT oy. Us th umbr of arithmtic mutipicatios ad additios as a masur of computatioa compxity. Fast Fourir trasform FFT is a st of agorithms for th fficit ad digita computatio of th - poit DFT, rathr tha a w trasform., Digita Siga Procssig, X, Zhg-Hua Ta, 5 3 Part I: Dirct computatio of th DFT Dirct computatio of th DFT Dcimatio-i-tim FFT agorithms Dcimatio-i-frqucy FFT agorithms Fourir aaysis of sigas usig th DFT Digita Siga Procssig, X, Zhg-Hua Ta, 5 4

3 Dirct computatio of th DFT Th DFT of a fiit-gth squc of gth X x,,,..., Dirct computatio: compx mutipicatios ad - compx additios Comput ad stor oy ovr o priod π cosπ siπ,,,..., Comput th DFT usig stord ad iput x X x,,,..., ad x may b compx x X Digita Siga Procssig, X, Zhg-Hua Ta, 5 5 Dirct computatio of th DFT For ach X R{ x }R{ R{ x }Im{ } Im{ x }Im{ } Im{ x }R{ Thrfor, for ach vau of, th dirct computatio of X rquirs 4 ra mutipicatios ad 4- ra additios. Th dirct computatio of th DFT rquirs 4 ra mutipicatios ad 4 ra additios. Th fficicy ca b improvd by xpoitig th symmtry ad priodicity proprtis of }, },,..., Digita Siga Procssig, X, Zhg-Hua Ta, 5 6

4 Symmtry ad priodicity of compx xpotia Compx cougat symmtry Priodicity i ad For xamp R{ } Im{ * R{ x }R{ } R{ x }R{ R{ x } R{ x } R{ Th umbr of mutipicatios is rducd by a factor of. Digita Siga Procssig, X, Zhg-Hua Ta, 5 7 } } } Part II: Dcimatio-i-tim FFT agorithms Dirct computatio of th DFT Dcimatio-i-tim FFT agorithms Dcimatio-i-frqucy FFT agorithms Fourir aaysis of sigas usig th DFT Digita Siga Procssig, X, Zhg-Hua Ta, 5 8

5 FFT Cooy ad Tuy 965 pubishd a agorithm for th computatio of th DFT that is appicab wh is a composit umbr, i.., th product of two or mor itgrs. Latr, it rsutd i a umbr of highy fficit computatioa agorithms. Th tir st of such agorithms ar cad th fast Fourir trasform, FFT. FFT dcomposs th computatio of th DFT of a squc of gth ito succssivy smar DFTs. Digita Siga Procssig, X, Zhg-Hua Ta, 5 9 Dcimatio-i-tim FFT agorithms hr dcompositio is do by dcomposig th squc ito succssivy smar subsqucs, ad both th symmtry ad priodicity of compx π xpotia ar xpoitd. v Cosidr ad sparat x ito two - poit squcs X x, X x v,,..., odd x Digita Siga Procssig, X, Zhg-Hua Ta, 5

6 Dcimatio-i-tim FFT agorithms X r r r v xr xr xr G x r H,,,..., oy comput for,,..., r r odd r x xr r r xr xr r r r du to th priodicity Digita Siga Procssig, X, Zhg-Hua Ta, 5 Fow graph of th dcimatio-i-tim Priodicity is appid,.g. G7G3 Digita Siga Procssig, X, Zhg-Hua Ta, 5

7 Digita Siga Procssig, X, Zhg-Hua Ta, 5 3 Dcimatio-i-tim FFT Furthr bra dow r r h h H g g g g r g G Digita Siga Procssig, X, Zhg-Hua Ta, 5 4 Combiatio of Fig. 9.3 ad 9.4

8 -poit DFT Digita Siga Procssig, X, Zhg-Hua Ta, 5 5 Fow graph og stags ad ach stag has compx mutipicatios ad compx additios. I tota, og compx mutipicatios ad additios..g. 4,48,576 og,4 rductio of ordrs! Digita Siga Procssig, X, Zhg-Hua Ta, 5 6

9 Fow graph of buttrfy computatio Digita Siga Procssig, X, Zhg-Hua Ta, 5 7 Fow graph Th umbr of compx mutipicatios ar rducd by a factor of ovr th umbr i Fig Digita Siga Procssig, X, Zhg-Hua Ta, 5 8

10 Part III: Dcimatio-i-frqucy FFT agorithms Dirct computatio of th DFT Dcimatio-i-tim FFT agorithms Dcimatio-i-frqucy FFT agorithms Fourir aaysis of sigas usig th DFT Digita Siga Procssig, X, Zhg-Hua Ta, 5 9 Dcimatio-i-frqucy FFT agorithms Fig. 9. Digita Siga Procssig, X, Zhg-Hua Ta, 5

11 Part IV: Fourir aaysis of sigas usig th DFT Dirct computatio of th DFT Dcimatio-i-tim FFT agorithms Dcimatio-i-frqucy FFT agorithms Fourir aaysis of sigas usig th DFT Digita Siga Procssig, X, Zhg-Hua Ta, 5 Fourir aaysis of sigas usig th DFT Fiit-duratio rquirmt of DFT widowig Digita Siga Procssig, X, Zhg-Hua Ta, 5

12 Digita Siga Procssig, X, Zhg-Hua Ta, 5 3 Fourir aaysis of sigas usig th DFT Fig.. Taprs off but is ot bad-imitd. ot ida. Low-pass fitrd ad modifid. r c T r T X T X π Digita Siga Procssig, X, Zhg-Hua Ta, 5 4 Fourir aaysis of sigas usig th DFT π π π d V X V idowig. V v V,,...,, π π

13 Digita Siga Procssig, X, Zhg-Hua Ta, 5 5 Effct of idowig o Fourir aaysis Fig..3 Effct of idowig o Fourir aaysis rctaguar widow of gth 64. cos cos cos cos cos cos Ω Ω c V w w v x t t t s Digita Siga Procssig, X, Zhg-Hua Ta, 5 6 Effct of idowig o Fourir aaysis Fig..3 Th DTFT of a siusoida siga is a pair of impuss. idowig broads th impuss ad rducs th distictio of sigas that ar cos i frqucy

14 Summary Dirct computatio of th DFT Dcimatio-i-tim FFT agorithms Dcimatio-i-frqucy FFT agorithms Fourir aaysis of sigas usig th DFT Digita Siga Procssig, X, Zhg-Hua Ta, 5 7 Cours at a gac MM Discrt-tim sigas ad systms Systm MM Fourir-domai rprstatio Sampig ad rcostructio Systm aaysis MM5 Systm structur MM6 MM4 Fitr dsig z-trasform DFTFFT MM7, MM8 MM3 MM9, MM Digita Siga Procssig, X, Zhg-Hua Ta, 5 8

15 Th d. Thas for your atttio! Digita Siga Procssig, X, Zhg-Hua Ta, 5 9