Fast Fourier Transform
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1 Fst Fourer Trsform Alctos. Fst Fourer Trsform Perhs sgle lgorthmc dscover tht hs hd the gretest rctcl mct hstor. tcs coustcs qutum hscs telecommuctos sstems theor sgl rocessg seech recogto dt comresso. Progress these res lmted b lc of fst lgorthms. Hstor. Coole-Tue 5 revolutozed ll of these res. Delso-czos effcet lgorthm. Ruge-ög ld theoretcl groudwor. Guss 85 8 descrbes smlr lgorthm. Imortce ot relzed utl dvet of dgtl comuters. Je Btste Joseh Fourer 78-8 Polomls: Coeffcet Reresetto Degree oloml. q b b + b b Polomls: Pot-Vlue Reresetto Degree oloml. Uquel secfed b owg t dfferet vlues of. { } where - Addto: os q b b b Evluto: usg Horer s method ultlcto covoluto:. q b + b + b + + b + + b
2 Polomls: Pot-Vlue Reresetto Degree oloml. Addto:. { - } where { z z z } where z q - { z + z z } Best of Both Worlds C we get "fst" multlcto d evluto? Reresetto ultlcto Evluto coeffcet ot-vlue FFT log log! Yes! Covert bc d forth betwee two reresettos. ultlcto: but eed ots. { z z z } - - bb b- coeffcet multlcto c c c- Evluto: usg grge s formul. evluto FFT q q log ot-vlue multlcto r r r - - q - terolto verse FFT log 5 Covertg Betwee Reresettos: Nïve Soluto Evluto coeffcet to ot-vlue. Gve oloml choose dstct ots {... - } d comute for ech usg Horer s method.. Iterolto ot-vlue to coeffcet. Gve dstct ots {... - } d comute the coeffcets {... - } b solvg the followg ler sstem of equtos. Note Vdermode mtr s vertble ff re dstct.. Fst Iterolto: e Ide e de: choose {... - } to me comutto eser! Set? 7 8
3 Fst Iterolto: e Ide e de: choose {... - } to me comutto eser! Set? Use egtve umbers: set - so tht. set - / E eve E + / E - Fst Iterolto: e Ide e de: choose {... - } to me comutto eser! Set? Use egtve umbers: set - so tht. set - / + Use comle umbers: set where s th root of ut. - / + - / /8 + 8 Roots of Ut Roots of Ut: Proertes A th root of ut s comle umber z such tht z. e π / rcl th root of ut. : et be the rcl th root of ut. If > the / -. Proof: e π / / e π -. Euler s formul e t cos t + s t. -. There re ectl roots of ut: : et > be eve d let d ν be the rcl th d / th roots of ut. The ν. Proof: e π / e π / / ν. : et > be eve. The the squres of the comle th roots of ut re the / comle / th roots of ut Proof: If we squre ll of the th roots of ut the ech / th root s obted ectl twce sce: + / - thus + / both of these ν + / d hve the sme squre
4 Dvde-d-Coquer Gve degree oloml Assume s ower of d let be the rcl th root of ut. Defe eve d olomls: eve : / - : / - eve + FFT Algorthm FFT... - f // s ower of retur e π / e e e...e /- FFT/... - d d d...d /- FFT/ Reduces roblem of evlutg degree oloml t... - to evlutg two degree / olomls t: for to / - e + d +/ e - d retur... - comle multles f we re-comute. eve d ol evluted t / comle / th roots of ut. T T / + T log Recurso Tree Proof of Correctess Proof of correctess. Need to show for ech... - where s the rcl th root of ut. Bse cse.. Algorthm returs. Iducto ste. Assume lgorthm correct for /. let ν be the rcl / th root of ut e eve ν eve b emm d ν b emm recll eve e + d + eve + / e d eve eve + + eve + / + / + + / + "bt-reversed" order 5
5 7 Best of Both Worlds C we get "fst" multlcto d evluto?! Yes! Covert bc d forth betwee two reresettos. coeffcet Reresetto ultlcto Evluto ot-vlue FFT log log - - b b b - c c c q - - q q r - r r ot-vlue multlcto coeffcet multlcto log evluto FFT terolto verse FFT log 8 Forwrd FFT: gve {... - } comute {... - }. Iverse FFT: gve {... - } comute {... - }. Iverse FFT Gret ews: sme lgorthm s FFT ecet use - s "rcl" th root of ut d dvde b. F F Iverse FFT Iverse FFT: Proof of Correctess Summto lemm. et be rmtve th root of ut. The If s multle of the. Ech th root of ut s root of f we hve: Clm: F d F - re verses. otherwse mod otherwse f F F
6 Iverse FFT: Algorthm f // s ower of retur e -π/ e e e...e /- FFT/... - d d d...d /- FFT/ for to / - e + d / +/ e - d / retur... - IFFT... - Best of Both Worlds C we get "fst" multlcto d evluto? Reresetto ultlcto Evluto coeffcet ot-vlue FFT log log! Yes! Covert bc d forth betwee two reresettos. - bb b- coeffcet multlcto c c c- evluto FFT q q log ot-vlue multlcto r r r - - q - terolto verse FFT log Iteger Arthmetc ultl two -dgt tegers: -... d b b -... b b. Form two degree olomls. Note: b q. q b Comute roduct usg FFT log stes. Evlute r b. Problem: log comle rthmetc stes. r q Soluto. Strsse 8: crr out rthmetc to sutble recso. T Tlog T log log log +ε Schöhge-Strsse 7: use modulr rthmetc. T log log log
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