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1 IEEE Itrtiol Cofr o Emrgig Thologis Sptmr - Islmd Khlid Mhmood Amir Mohmmd Ali Mud Asim Lo Lhor Uivrsity of Mgmt Sis Lhor Pist Emil: mmir lo}@lums.du.p Uivrsity of Mgmt d Thology Lhor Pist Emil: mmud@umt.du.p Astrt Th lssil Cooly-Tuy fst Fourir trsform (FFT) lgorithm hs th omputtiol ost of O( log ) whr is th lgth of th disrt sigl. Sptrum rsolutio is improvd through pddig zros t th til of th disrt sigl. If ( p ) zros r pddd (whr p is itgr) t th til of th dt squ th omputtiol ost through FFT oms O( p log p ). This ppr proposs ltrt ist of pddig zros to th dt squ tht rsults i omputtiol ost rdutio to O( p log ). It hs otd tht this modifitio usd to hiv o-uiform upsmplig tht would zoom-i or zoom-out prtiulr frquy d. I dditio it my usd for pruig th sptrum whih would rdu rsolutio of uimportt frquy d.. Itrodutio Suppos w hv disrt tim sigl [] for α < d whr α is itgr. Th disrt Fourir trsform (DFT) is usd to trsform th disrt tim sigl ito disrt frquy domi. Th lssil Cooly-Tuy fst Fourir trsform lgorithm []-[] for th omputtio of DFT hs th omputtiol ost of O( log ). My thiqus tht rdu its omplity li highr-rdi d splitrdi lgorithms [ 5] pruig [6]-[] lo trsforms [] d slid trsforms [5] mployd. Zro pddig t th til of th disrt sigl is rrid out to improv th frquy rsolutio. This pross is lld sptrl itrpoltio. Pddig ( p ) zros wh p > irss th omputtiol ost of th FFT lgorithm to ( p log p ) O. Th lssil FFT lgorithm s pplitio to pruig d zoomig r limitd s ouiform upsmplig is ot ovitly pplil. This ppr prsts ltrt zro pddig shm to th stdrd prti of pddig zros t th til of th dt squ usig rdutio i th omputtiol ost to p log ( ) of th zro pddd sigl. Th proposd zro pddig shm usd for o-uiform upsmplig i th frquy domi. This filitts zoomig-i of frquy ds of itrst. Similrly th sm lgorithm dptd for th purpos of pruig. This ppr is orgizd s follows. Th modifid zro pddig shm d its pplitio to th FFT lgorithm is prstd i Stio. Stio dsris wys i whih th proposd shm usd to hiv pruig zoomig d o-uiform upsmplig i th frquy domi.. Modifid FFT Algorithm. Proposd Modifitio Th disrt Fourir trsform for ompl dt squ of lgth is giv y: X [] [] () / for L Thr r my fst Fourir trsform lgorithms ut w will disuss oly th so lld rdi- dimtio i tim (DIT) vrit of th FFT. Cosidr sigl with ight dt poits with ight zros pddd. Figur shows lvls i FFT. --9-/5/$. 5 IEEE

2 IEEE Itrtiol Cofr o Emrgig Thologis Sptmr - Islmd 5 6 } 6} 5 } } } } } 6 5 l } } } 6 } } 5 } } } Figur-: Lvls i FFT of ight poit sigl. log lvls of th uttrfly lgorithm. Lt l itgr tht rprsts th stg umr of FFT lgorithm suh tht l < log ( ). W ll stg s th lowst lvl stg ( l ) wh th dimsio of th mtri Ω ( ) rprstig ostts d twiddl ftors rltig iput squ to output squ is miimum ( i rdi- DIT FFT) d th highst lvl ( l log [ ] ) wh th dimsios of th mtri Ω( log [ ] ) is mimum. Th lowst lvl stg mtri Ω for DIT FFT lgorithm is writt s: Thr r ( ) Ω( ) () Th proposl is to pd zros t th lowst lvl rthr th t th til of th ovrll dt squ to hiv upsmplig i th frquy domi. Suppos tht ftr zro pddig t th lowst lvl th umr of dt poits is p. This ft is lr from Figur. Equtio () oms: X p () [] [] [] L p Th lowst lvl Ω ( ) i this s is otid from qutio (). Th first olum of Ω ( ) is ll os ( ) d th sod olum of Ω ( ) is grtd y for < p ( ). Th rmiig dt poits for to p r ll pddd zros. So Ω ( ) writt s: Ω( ) () M M ( p) As spifi s if th lowst lvl squ is pddd with two zros ( Ω will : p ) th ( ) ( ) Ω (5) ( ) Ω is p mtri. Hr th lowst lvl hd two dt poits i th sust. If thr r q dt poits i th lowst lvl mtri th dimsios of ( ) Ω will qp q - th grliztio is sy d q Ω is grtd y for < p whr is itgr. Th grlizd mthmtil drivtio for th suggstd modifitio is show i Appdi A. strightforwrd. ( ) Empl-: Cosidr th futio () t s: ( t) os ( 6t ) os( t ) (6) Sigl is smpld t Hz. Th sigl ws 6. sods log. Figur- shows th powr sptrl dsity (PSD) fo th sigl otid with Ω ( ) of qutio (5) d omprd with th PSD of th sm sigl pddd with qul umr of zros t th til of th origil sigl.

3 IEEE Itrtiol Cofr o Emrgig Thologis Sptmr - Islmd 6} 5 } } 6 } } } 5 } 6 } 5 } } Figur-: Modifid suggstd FFT lvls. Figur-: Compriso tw ovtiol d proposd shm. Ω ( ) is s i (5).. Computtiol Cost Th Cooly-Tuy fst Fourir trsform lgorithm hs omplity/ost of O( log ). I ft thr r log stgs of th FFT lgorithm h with O ( ) omputtios. Cosidr disrt tim sigl α [] whr < d α is positiv itgr. For improvig th sptrl rsolutio zros r ppdd to th sigl. Suppos w pd ( p ) zros whr p is positiv itgr grtr th o. Th umr of stgs i th FFT lgorithm is ow log ( p). Thrfor th omputtiol ost of th lgorithm is p log ( p ) h with O ( p ) omputtios. If th modifitio suggstd i Stio. is implmtd whr h sust hs lst possil umr of smpls th th umr of stgs log stgs with would rmi uhgd i.. ( ) ( p ) O omputtios du to zro pddig t this stg. Cosqutly zro pddig t th lowst stg rdus O p log p. th omputtiol ost y ( [ ]). Colusios Wh w r prformig frquy-domi lysis zoom FFT is usful for zoomig i o rrow frquy d. W us zoom FFT to fous o rrowd hl y prformig oly th lultios dd to oti th FFT dt i th frquy rg of itrst. For lrg dt siz w my us pruig to rdu th omputtiol lod. Ω for Ω t Ω i rquird prtiulr fshio will yild o-uiform As show i Stio w pd ( ) ttr sptrl stimt. W pd y ( l) y stg of th lgorithm. Epdig ( l)

4 IEEE Itrtiol Cofr o Emrgig Thologis Sptmr - Islmd upsmplig. This is do wh w r itrstd i sptrum of th sigl i prtiulr frquy d. Ω () l is squzd wh w r ot itrstd i som frquy d. Suppos som of th twiddl ftors hd vry smll mgitud th th orrspodig rhs of th uttrfly oprtios ould droppd (prud) to rdu omplity. Thus w squz Ω ( l) for pruig. Vrious pruig thiqus still pplid ftr this modifitio. It is ovious tht omputtiol ost hs rdud y ftor of O( p log [ p] ). Morovr th proposd mthod usd for pruig d o-uiform upsmplig i th frquy domi. Aowldgmts This rsrh wor ws fudd y Highr Edutio Commissio (HEC) Islmd Pist. Thir support is grtfully owldgd. Rfrs [] Z. J. Mou P Duhml I-pl Buttrfly- Styl FFT of D Rl Squs IEEE Trstios o Aoustis Sph d Sigl Prossig vol. 6 o. Ot. 9. [] A. Frtr Computtiolly ffiit mthods for lysis d sythsis of rl sigls usig FFT d IFFT IEEE Trs. Sigl Prossig vol. pp. 6-6 April 999. [] J. G. Prois d D. G. Molis Digitl Sigl Prossig Priipls Algorithms d Applitios rd Editio Prti-Hll I [] P. Duhml Implmttio of Split-rdi FFT lgorithms for ompl rl d rlsymmtri dt IEEE Trs. o ASSP vol. pp April 96. [5] H. V. Sors M. T. Hidm C. S. Burrus O Computig th Split rdi FFT ICASSP-5 Prodigs Mrh 95. [6] H. V. Sorso d C. S. Burrus Effiit omputtio of th DFT with oly sust of iput or output poits IEEE Trs. Sigl Prossig vol. o. pp. 99 Mr. 99. [] D. P. Sir Pruig th dimtio-itim FFT lgorithm IEEE Trs. Aoust. Sph Sigl Prossig vol. ASSP- o. pp. 9 9 Apr. 96. [] S. B. ry d K. M. M. Prhu Fst Hrtly trsform pruig IEEE Trs. Sigl Prossig vol. 9 o. pp. J. 99. [9] S. Brsh Y. Ritov Logrithmi pruig of FFT frquis IEEE Trs. Sigl Pross. vol. o [] S.R. Rgr S. Sriivs Grlizd Mthod for Pruig FFT Typ of Trsform VISP() o. pp. 9-9 August 99. [] H. Hug Y. L P. Lo A ovl lgorithm for omputig th D split-vtor-rdi FFT Sigl Prossig vol. o. Mrh. [] S. Frz S. K. Mitr Grhrd Doligr Frquy stimtio usig wrpd disrt Fourir trsform Sigl Prossig vol. o. August. [] S. Brsh d Y. Ritov Pruig FFT frquis IEEE trstios o Sigl Prossig vol. pp [] G. P. M. Eglmrs d P. C. W. Somm Rursiv omputtio of lo-fft s Pro. ProRISC/IEEE Symp. Ciruits Systms Sigl Prossig thrlds pp Mr [5] T. Sprigr Slidig FFT omputs frquy sptr i rl tim ED pp. 6 Sp. 9. Appdi A Cosidr L }

5 Lt d (A-) ow w suppos tht two zros r pddd. So hgig th summds with four poit DFT d lso hgig th twiddl ftors i qutio (A-) (A-) 6 This is DFT of ight dt poit DFT with ight zros pddd t th d. W grliz th rsults of qutio (A-) s: / / odd v Whr } L. This rltioship is usd rursivly. IEEE Itrtiol Cofr o Emrgig Thologis Sptmr - Islmd 5