Radix-2/4 Streamlined Real Factor FFT Algorithms

Transcription

1 SHAIK QADEER et l: RADIX-/ STREAMLIED REAL FACTOR FFT ALGORITHMS Rdix-/ Stremlied Rel Ftor FFT Algorithms Shi Qdeer () Mohmmed Zfr Ali Kh () d Syed A. Sttr () () Deprtmet of Eletril d Eletrois egieerig MJ College of Egieerig Hyderbd Idi {emil:hqbei@gmil.om} () Deprtmet of Eletril egieerig Idi Istitute of Tehology Hyderbd Idi {emil: zfr@iith..i} () Deprtmet of Eletrois d ommuitio egieerig RITS Hyderbd Idi {emil: syedbdulsttr965@gmil.om } Abstrt I this pper three Rel Ftor FFT lgorithms re preseted. Two of them re bsed o Rdix- d oe o Rdix-. The omputtiol omplexity of Rdix- d Rdix- is show s order log d log respetively ulie ~ ~ their stdrd outerprts ~ 5log d ~ log. Moreover the proposed lgorithms lso require fewer multiplitios th their stdrd FFTs. e the show tht the fixed poit implemettio of 'rel ftor' FFT be modified with uique slig proedure so tht its oise to sigl rtio (SR) is lower th the SR of stdrd FFT. Filly implemettio issues re preseted whih verify the suitbility of the proposed 'rel ftor' FFT's. Keywords - Disrete Fourier Trsform () Deimtio i frequey (DIF) Fst Fourier Trsform (FFT) d rel-ftor FFT I. ITRODUCTIO The Disrete Fourier trsform () is mog the most fudmetl opertio i digitl sigl proessig [7 9]. he osiderig the lterte implemettios the FFT/IFFT lgorithm should be hose eepig i view the exeutio speed hrdwre omplexity flexibility d preisio [ 9]. Most of the bove metioed prmeters deped o the ext out of rithmeti opertios (rel dditios d multiplitios) herei lled flops (flotigpoit opertio) required for of give size whih remis itriguig usolved mthemtil questio. Amog the umerous developmets tht followed Cooley d Tuey's [] origil otributio re the iogrd Fourier trsform lgorithm (FTA) [] d the rel ftor lgorithms [ ] for redutio i the order of the multiplitive omplexity. However both FTA d 'relftor' FFT did ot meet expettios oe implemeted s the umber of dditios (d dt trsfers) lso mtter i the implemettio. I dditio Rder d Breer's 'rel ftor' FFT s re ill-oditioed i.e. "smll omputtiol errors led to lrge output errors"[] due to the lrge vlues tht the twiddle ftor te. I this pper the solutios for both the problems of Rder d Breer's 'rel ftor' FFT lgorithm is preseted. e first show tht the rithmeti omplexity (Multiplitios plus dditios lso ow s the 'Flop out') of the Rder d Breer's 'rel ftor' FFT be redued to bout ~ log whih is less th the rithmeti omplexity of Cooley-Tuey rdix- FFT the orrespodig Rdix- is lso disussed whih tes ~ log whih is gi lesser th its outerprt. It is the show tht the modified 'rel ftor' FFT is free from ill-oditios s the mgitude of ll the twiddle ftors is less th d hee hs lower SR th rdix- Cooley- Tuey FFT. As result the proposed FFT is more suitble s it requires oly hlf the umber of rel multiplitios. The rest of the pper is orgized s follows: I setio II modified DIT versio of Rder d Breer Rdix- FFT tht requires ~ 5log flops is first preseted i subsetio II-A. I subsetio II-B we derive the orrespodig rdix- DIF FFT. I subsetio II-C it is show tht the flop out be further redued to ~ log. otbly this is the best ow flop out tht be hieved by rdix- FFT. I subsetio II-D equivlet Rdix- FFT lgorithm is overed. e the preset i setio III implemettio issues of rel ftor FFT bsed o the rdix- FFTs developed i this pper d olusios re give i setio IV. II. E FFT I this setio we first modify Rder d Breer DIT FFT tht requires ~ 5log flops. The orrespodig DIF versio is derived d the lgorithm is further modified so tht the flop out is redued dow to ~ log. Filly similr rdix- DIF FFT lgorithm is lso derived. DOI.5/IJSSST...6. ISS: 7-x olie 7- prit

2 SHAIK QADEER et l: RADIX-/ STREAMLIED REAL FACTOR FFT ALGORITHMS A. DIT versio of Rder d Breer FFT: The essee of Rder d Breer's DIT FFT is s follows: Let { } deotes -poit of the sequee of = M i.e. { }= / { }. The rdix- DIT FFT is give by A B.. D () where =exp (-/) d the /-poit sequees re defied s b = = d {B }= / {b } d {D }= / {d }. ote tht for -poit sequee if the idex is greter th we ssume mod. Sie i geerl D d re both omplex i ture the bsi butterfly opertio idited i () requires '' rel multiplitios d '' rel dditios. Rder d Breer itrodued the followig sequee [] d d q.. () where q d. It follows tht () be rewritte s m m A B C A / B C d A B s & / where { }= / { }. d s(x) is the oset of x. If we hge the sig to plus i the defiitio of the the oeffiiet of i () is /se(/) where se(x) is set of x. I either se () requires rel multiplitios d rel dditios. But this is t the expese of more dditios to fid q d from b. Therefore totl rel dditios (A()) will be A()=++=5. The rel multiplitios (M()) re give by M()= d the totl flop out (T()) is give by T()=~ 6log. To further redue the rithmeti omplexity here we modify s C d d.. It follows tht () is still vlid with the exeptio tht is defied s C 5 Deote P( ) s the the DIT omputtio for the ew lgorithm is shemtilly represeted for =6 is show i Figure. The rithmeti omplexity of this DIT FFT will beome T( ) 5 log due to the redutio i the umber of dditio to ompute usig () isted of (). ote tht the differee betwee Rder d Breer's FFT d our proposed FFT is i the omputtio of C. I Rder d Breer's FFT C q is dded to ll whih requires '(-)' rel dditios while i our versio C is diretly omputed usig (5) whih requires '(-) rel dditios. B. Derivtio of DIF versio of ew FFT Aordig to the priiple of DIF FFT the eve d odd oeffiiets re give by A.. (6) A.. (7) Fig. The deompositio of poit ito two DIT Rdix- FFT lgorithm for =6. - poit s of ew Fig. The deompositio of poit ito two - poit s for ew DIF Rdix- FFT lgorithm for =6. DOI.5/IJSSST...6. ISS: 7-x olie 7- prit

3 SHAIK QADEER et l: RADIX-/ STREAMLIED REAL FACTOR FFT ALGORITHMS To elimite the omplex multiplitio ftor from (7) we defie the sequee { } s ( ).. () where δ() is uit impulse futio. Equtio () be simplified s (9) s..( ) () Applyig / poit o both sides of () we get.. where (). The Rdix- DIF omputtio for the ew lgorithm is shemtilly represeted for =6 is show i Figure. The umber of multiplitios be lulted from (). For eh vlue of '' '' rel multiplitios re eeded d the totl umber of rel multiplitios with give two / -poits s is (-). Therefore the totl umber of rel multiplitios for -poit is give by M( ) if M( ) if Solvig () by repeted substitutios we hve M ( ) log The umber of rel dditios give two /-poits s is lulted s follows (/) rel dditios for evlutig (6) for =../-. (/+)rel dditios for evlutig () for =../-. (/)rel dditios for evlutig () for =../-. It follows tht T( ) A( ) if A( ) if Solvig () by repeted substitutios we hve A( ) log () () () (5) The totl flop out is the give by 5 log. Importtly the DIF versio of the FFT preseted i sub setio II-A d II-B requires the sme umber of omputtios (rel dds + rel multiplies) s orml Rdix- DIT or DIF FFT while requirig lesser multiplies. Also s ompred to DIT versio of Rder d Breer's FFT the modified DIT show i subsetio II-A d DIF i II-B requires fewer rel dditios. C. Modified DIF versio of ew FFT I this sub setio we show tht the omputtiol omplexity be redued dow further. If we dd to () the for = (/) s( / ) (6) Fig. the deompositio of poit ito two - poit s for ew modified DIF Rdix- FFT lgorithm for =6. Ad rest of the sequee { } = /- / +.. /- is give by (9)-(). Equtio () for this se modifies to C C for evevlues of A (7) C C otherwise The DIF omputtio for the ew lgorithm is shemtilly represeted for =6 re show i Figure DOI.5/IJSSST...6. ISS: 7-x olie 7- prit

4 SHAIK QADEER et l: RADIX-/ STREAMLIED REAL FACTOR FFT ALGORITHMS DOI.5/IJSSST...6. ISS: 7-x olie 7- prit d Figure. ith these substitutios it is obvious tht the umbers of rel multiplitios re sme. However s ompred to the sub setio II-A d II-B the umber of rel dditios re redued by /-. Aordigly the umber of rel dditios is give by 7 if if ) A( ) A( () Solvig () by repeted substitutios we hve log ) ( A 7 (9) The totl flop out is give by log ) ( T. Importtly flop out of this Rdix- FFT is less th y other Rdix- FFTs (iludig stdrd DIT d DIF Rdix- FFTs). D. ew Rdix- DIF FFT: Aordig to the priiple of Rdix- DIF FFT the eve d odd oeffiiets re give by for.. To elimite the omplex multiplitio ftor from () we similrly defie the sequee { } s.. ( ) where δ() is uit impulse futio. Equtio () be simplified s Fig.. The omplete i ple omputtio digrm of ew DIF Rdix- FFT lgorithm for = / s Applyig / poit o both sides of () we get 6.. A where d As i subsetio II-C if we dd to (5) the for =/ s Ad for the rest of the sequee { } = /- / + /- is give by (5). Equtio (6) for this se modifies to

5 SHAIK QADEER et l: RADIX-/ STREAMLIED REAL FACTOR FFT ALGORITHMS C A C C for evevlues of C otherwise Methods M() T() Stdrd log 5log Rdix- RB-FFT log 6log Proposed log Rdix- log Stdrd log Rdix- log Proposed Rdix- log log (7) oly Rdix- DIF d other be similrly obtied [ d 5]). The oise to sigl rtio (SR) of Rder d Breer's lgorithm is ow to be very poor beuse of the lrge vlues of s []. Aordigly we first modify the rel ftor FFT for fixed poit implemettio i setio III-A to elimite the problem of ill omputtio; the we disussed the remiig implemettio issues i III-B d expeted exeutio times is give i setio III-C. A. Modifitio to rel ftor FFT for fixed poit implemettio To elimite the problem of ill-oditioed we proposed slig method to (9)-() s Tble I. Computtiol Complexity of vrious FFT's Deote P( ) s the the DIF omputtio for the ew lgorithm is shemtilly represeted for =6 is show i Figure. The totl dditio for eh butterfly will be A( ) d totl multiplitio will be M ( ). Hee the totl flop outs T( ) 6 per butterfly. The ost of the ew Rdix- FFT lgorithm ow be represeted by the followig reurree igorig otrivil multiplitios: T( ) T 6 6 if if Solvig () by repeted substitutios we hve () T( ) log (.) E. Compriso of omputtiol Complexity of vrious FFT lgorithms: The omputtiol omplexity ompriso for Rdix- / FFTs is give i Tble. I (ote tht RF-FFT mes Rel Ftor while RB-FFT mes Rder d Breer's FFT). Observe tht the ew DIF versio of modified rel ftor FFT preseted i setio II requires ot oly lesser multiplitios but lso the totl flop outs (T ()). III. IMPLEMETATIO ISSUES OF PROPOSED FFT I this setio we osider the implemettio issues of 'rel ftor' FFTs preseted i setio II (ote tht we over here ĉ l s ĉ l l.. where l is hose suh tht s(/) l+. It follows tht s(/)/ l+ for ll. This modifitio esures tht the rel multiplitio ftors re less th. Applyig o both sides of () with this modifitio we get l Ĉ Ĉ ĉ A.. () where Ĉ ĉ d ĉ ( ) ĉ. The purely imgiry twiddle ftor is ow give by P( ) s. A similr lgorithm with purely l rel twiddle ftor be similrly developed. ote tht i both the ses P(). The dvtge of the preseted fixed poit lgorithm be observed from () wherei the effet of lrge vlues of s(θ) hs bee ptured i l d the effetive preisio t the output is l+b bits [6 ]. A. I-ple omputtio of proposed FFT lgorithm: Figure 5 shows the i-ple omputtio digrm of the proposed rel ftor FFT whih implemets (6) through (7) for =. It is observed tht Prt-I of lgorithm is similr to Cooley-Tuey's rdix- DIF FFT lgorithms [] exept the twiddle ftor 'P( )' whih is defied erlier (but ot show i lgorithm ). Oe butterfly opertio is performed o pir of omplex umber ( b) to produe (A B) there is o eed to sve the iput pir ( b) exept the first vlue of vetor '' s show i Figure 5. Hee we store the results (A B) i the sme lotios s ( b). 9 DOI.5/IJSSST ISS: 7-x olie 7- prit

6 SHAIK QADEER et l: RADIX-/ STREAMLIED REAL FACTOR FFT ALGORITHMS Cosequetly we require fixed umber of storge registers i order to store the results (ote tht is omplex umber) of the omputtios t eh stge. Sie storge lotios re used throughout the omputtio of the -Poit we sy tht the omputtios re doe i-ple. A seod observtio is eed of dt shufflig (DS) i Prt-II of proposed lgorithm. TABLE II. Expeted Exeutio time of FFT vrits for Rdix- = υ S.o Problem Legth C-T i m se Proposed RDIF i m se Proposed RDIF i m se log log log.5.5*.5* *ote tht our lgorithms wors for >. IV. COCLUSIO I this pper we hve show tht the rithmeti omplexity of rel ftor FFTs is sigifitly less th tht of stdrd FFT. e hve show tht these FFTs hve sigifit dvtge ompred to orrespodig stdrd FFTs. he it omes to fixed poit implemettio s it hs lower umber of oise soures (rel multiplitios) d hee lower SR. This lgorithm is of prtil importe s Rdix-/ FFTs re geerlly preferred i ASIC implemettio d for softwre defied rdio (ogitive rdio) pplitios due to ese of re-ofigurbility. Over ll these ftors re i fvor of rel ftor FFTs. However the disdvtge is dditiol memory storge of C required for eh stge. But s the ost of memory is deresig storge my ot be sigifit ftor. A prt of this pper ws preseted t EUSIPCO []. There is eed of dt deimtio fter eh dditiol dditio s show i Figure 6 s t the ed of Prt-I the order of dt of our lgorithm is bit reversed. e defie this deimtio of dt s DS. The DS eeded for = is show i Figure 6. Observe tht '' DS opertios re required for =. I geerl if the DS opertios re required to ompute -poit FFT. ote tht the umber of times dt deimtios/shufflig re eeded i the omputtio of usig C-T FFT is. Hee the omputtio requiremet for both DS d Bit reversed opertios re equl. B. Expeted Exeutio Time of Vrious FFT lgorithms: To help thigs i perspetive it is istrutive to see the differees i opertio outs i terms of expeted exeutio times []. The bove expeted times re lulted by osiderig the se of TMSC5 DSP proessor whose speed of opertio is 5s (MIPS) for proposed Rdix- FFT. Assumig tht both multiplitio d dditio tes sme time we get the results s show i bove Tble II. Fig. 5 I-Ple omputtio digrm for = of the Proposed Rel Ftor FFT lgorithm. () () () (6) () (5) () (7) Dt shufflig Bit Reversed Order () () () (6) () () (5) (7) Dt shufflig A () A () A () A () A () A (5) A (6) A(7) turl Order Fig.6 Shufflig of Dt d bit reversl DOI.5/IJSSST ISS: 7-x olie 7- prit

7 SHAIK QADEER et l: RADIX-/ STREAMLIED REAL FACTOR FFT ALGORITHMS REFERECES [] J.. Cooley d J.. Tuey A lgorithm for the mhiecomputtio of the omplex Fourier series Mth. Computtio vol. 9 pp. 97 Apr [] S. iogrd O omputig the disrete Fourier trsform Pro. t. Ad. Si. USA Vol. 7 April 976 pp [] C. M. Rder d. M. Breer A ew priiple for fst Fourier trsformtio IEEE Trs. Aoust. Speeh Sigl Proessig vol. pp [] K.M. Cho d G.C. Temes Rel-ftor FFT lgorithms Pro.ICASSP 7 Tuls UK April 97 pp [5] A. V. Oppeheim d C. J. eistei Effets of fiite registerlegth i digitl filterig d the fst Fourier trsform Pro. of IEEE. 97 pp [6] T. Tr B. Liu Fixed-poit fst Fourier trsform error lysis IEEE Trs. o ASSP 976 vol.(6) pp [7]. M. Getlem d G. Sde Fst Fourier trsform for fu d profit i 966 Fll Joit Corput. Cof. AFIPS Pro.v ol. 9. ew Yor: Sprt pp [] de Lowdermil d Fred Hrris Fiite Arithmeti Cosidertios for the FFT Implemeted i FPGA-Bsed EmbeddedProessors i Sytheti Istrumets I IEEE Istrumettio d Mesuremet Mgzie August 7. [9].-H. Chg d T. guye O the Fixed-Poit Aury Alysis of FFT Algorithms IEEE Trstios O Sig Proessig Vol. 56 o. pp Otober. [] P. D. elh A fixed poit fst Fourier trsform error lysis IEEE Trs. Audio Eletrooust. vol. AU-7 pp Jue 969. [] A. V. Oppeheim d R. Shfer Digitl Sigl Proessig.Perso Edutio. [] Sit Mitr Digitl Sigl Proessig: A Computer Approh rd editio Chpter MGrw- Hill 6. [] Eleor Chu d Al George Iside the FFT Bl Box: Seril d Prllel FFT lgorithms Pge o. Appedix A CRC Press. [] M.Z.A. Kh d Shi Qdeer Stremlied Rel Ftor EUSIPCO AlborgDemrAugust-7. [5] J.G. Prois d D.G. Molis"Digitl Sigl ProessigPriiplesAlgortihmsd pplitios"perso rd Editio. DOI.5/IJSSST ISS: 7-x olie 7- prit