Parallel Matched Filtering Algorithm with Low Complexity

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1 Cmputig, Prfrma a Cmmuiati systms (06) : 7- Clausius Sitifi Prss, Caaa Paralll ath Filtrig Algrithm with Lw Cmplxity Chg Xuhuag,a, Sha Gapig,ba Wag Yag, Cllg f Ifrmati Systm Egirig, Ifrmati Egirig Uivrsity, Chia a @qq.m, b @qq.m, wywy79@63.m Kywrs: vrlap-sav; lw mplxity; paralll math filtr; high sp mulati. Abstrat: Fr th prblm f ubl utig i th vrlap sav algrithm(osa), this papr prsts paralll math filtrig algrithm with lw mplxity. Th urrt ata is sgmt bas th filtr rr, th usig th quartr isrt furir trasfrm (QDFT) t ru th amut f alulati. Th alulati rsult f th prvius a urrt ata blk ar a t btai th blk filtr rsults. Aalysis a simulati rsults shw that th algrithm fftivly rus th mputatial mplxity. It is mr suitabl fr high-sp mulati whih has multipl paralll paths.. Itruti With th rapi vlpmt f ifrmati thlgy a mmuiati thlgy, th ata trasmissi rat f mmuiati systm a rah th rr f Gsps. High-sp ata a t b prss by traitial srial mulati mth at prst, s it s paralll prssig algrithm[]. ath filtr is high mplxity part f th ky algrithm, s It is sigifiat t stuy th paralll mathig filtr algrithm. T ru th mplxity f filtrig, i [] th mplxity f th filtr is ru by ptimizig th sig f stat ffiit multiplir. I [], th fast FIR algrithm (FFAs) is us t grat th lw mplxity filtrig algrithm by usig th haratristis f th filtr ffiit symmtry. I this papr, a lw mplxity paralll filtrig algrithm is prps th basis f vrlapp prsrvig filtrig algrithm. Th vrlap-prsrvig algrithm is mps it tw ipt prssig muls, a th lw-mplxity math filtr strutur is alulat.. Frquy - Dmai Paralll athig Filtrig Algrithm Bas QDFT. Lw mplxity f th frquy mai filtrig algrithm. Ovrlappig prsrvig mth, th sussiv iput sti vrlaps th ata part will brig th upliat mputati, limiat th upliati mputati part, a ru th algrithm th mplxity. Thrugh uti, th traitial vrlappig rsrvati mth a b mps it tw parts f th prvius ata blk a th urrt ata blk fr ipt prssig. Th prssig rsult f th prvius ata blk is rsrv by th last sgmt, a ly th urrt ata blk is prss. Th lgth f th ata that s t b ivlv i th alulati is ru by half.obsrvig th FFT algrithm by frquy xtrati (DIF), ah iput sgmt DFT rsult a 7

2 b mps it a shrtr tw-part lmt arig t th parity f its vtr ix. X D D xp X p X X x X p X () Whr th lmts X a X rprsts th v a vtr iis f X, K ar Kth rr isrt Furir trasfrm (ODFT) matris,a th (, k) lmts i th matrix ar ( k /) W.suprsript a Ɵ rsptivly rprst th ata fr DFT a ODFT trasfrm. K Divi G it th samg a G tw parts,si half f g is f a th rst is th -siz zr vtr 0.W hav G G D D f 0 F F Als frm () a th fiitis, th irular vluti a b rwritt as: y p D ( F X p) ( F X p) D ( F X ) ( F X ) y D ( F X p) ( F X p) D ( F X ) ( F X ) Th first half f th alulati th right si f quati (3) rlats ly t x p, whil th lattr half is ly rlat t x. Th lw mplxity f th frquy mai filtr algrithm is shw i Figur, whr J mul rprsts th ash part f th alulati, by th tw basi muls a mpts. Fr ah sussiv iput sgmt, th J-mul utput f th urrt ata blk is alulat a a t th J-mul rsult f th prvius ata, s that th rsult f th rrspig math filtr a b btai a th J-mul rsult f th urrt ata is sav. () (3) Fig. Shmati iagram f lw-mplxity frquy mai filtrig algrithm. Frquy mai mathig filtr algrithm bas th QDFT. Bas th aalysis, a w filtrig algrithm bas QDFT is prps. By usig th spial quarupl isrt Furir trasfrm (QDFT) i Graliz Disrt Furir Trasfrm (GDFT), th D ( G X ) a ( G X ) mputatis ar simplifi, a th mplxity f th math filtrig algrithm is furthr ru. pit iput ata QDFT rrspig iput a utput rlatiship is: 8

3 X 3 ( k ) q 4 k xw 0 (4) 3/4 QDFT ultiply th tim sris by th rtati fatr W, a th alulat th DFT rsults, s th sam a b us FFT t quikly alulat.thrugh bsrvati a aalysis, i th tim mai D ( G X ) a b xprss as a yli vluti : (5) y x g x g k k k k k0 k A ( G X ) rrsps t th skw irular vluti as: (6) y x g x g k k k k k0 k Q G X ( ) a b xprss as a yli vluti frmula: y x g j x g k k k k k0 k QK ar Kth rr quartr isrt furir trasfrm (QDFT) matris,a th (, k) lmts i th ( 3/4) matrix ar W k,th suprsript q iiats that th ata urg QDFT K trasfrmati.cmparig (5) a (6) with (7), w a s that th ral part fq ( G X ) is quivalt t ( D ( G X ) ( G X ))/ a th imagiary part is quivalt t ( D ( G X ) ( G X ))/.I summary, (3) is quivalt t y ( ) ( ) p Q F X p Q F X y Q ( F X p) Q ( F X ) (7) (8).3 DQDFT paralll filtrig algrithm. I th high-sp mulati systm with mr paralll hals, th mputatial mplxity f th QDFT-bas frquy-mai math filtrig algrithm is still larg, a th J-mul is subivi it smallr muls t ru th mplxity f th math filtrig algrithm. Bas th abv aalysis, a paralll mathig filtr algrithm with lwr mplxity is prps. Th algrithm ivis th paralll iput ata it shrtr ata blks a iputs thm it svral J muls i paralll t btai th alulati rsults f th multipl J muls a arry ut th J mul rsults f th last ata blk whih is rtai by th prvius ata.th rrspig prati, t ahiv th purps f blk vluti.th imprv frquy mai paralll mathig filtr algrithm is quivalt t th fft f multipl vrlapp ata sgmts mathig filtrig at th sam tim, whih rus th mputatial mplxity a imprvs th ability f ral-tim prssig. I blk-bas vlutial math filtrig, th ata is ivi it smallr ata blks, a mr J muls ar us. W fi th smallst J mul as th uit J mul, th iput siz is th umbr f pwr, trmi by th filtr lgth. Fr xampl, wh th filtr lgth L is qual t 33, th iput t th uit J mul is 64. If th urrt ata paralll iput J mul fr QDFT mathig filtrig mth, w all QDFT mathig filtr algrithm. A J muls ar 9

4 uit J mul wh w all DQDFT mathig filtrig algrithm. 3. Cmplxity aalysis a simulati xprimt I this papr, i rr t failitat th mplxity aalysis,, m usig th pwr f 4 tims th umbr f harwar us i th wily us bas-4 FFT alulatis. Th mputatial mplxity f th bas-4 FFT a b summariz as fllws: 3 3 lg ; lg ; 8 lg ; lg 8 whr a ma th umbr f multipliatis a aitis fr th -pit trasfrm rsptivly, a th suprsript a ma th mplx a ral ata rsptivly. Th ttal umbr f multipliatis fr th traitial vrlap-prsrvig filtrig algrithm is, a th umbr f aiti is. Th QDFT algrithm multiplis th (3/4) iput ata by th rtati fatr (multipliati f W N t th -th iput) a uss th staar FFT t mplt th alulati. Thrfr, th th ttal umbr f multipliatis fr QDFT filtrig algrithm is 3, a th umbr f aiti is.aftr th J mul agai aliqut f th QDFT mathig filtr mth rquirs 4 / 3 mplx multipliati tims, a th umbr f aiti is 4 /. It a b s that, i th blk vlutial mathig filtr, th prvius ata is ivi it smallr ata blks agai, a th mr J muls ar us, th lwr th mplxity f th algrithm is. Th siz f th miimum ata blk ps th siz f th ll J mul a is trmi by th filtr lgth. Tabl mpars th mputatial mplxity f DFT-bas vrlap-prsrvig math filtr algrithm a QDFT-bas math filtr algrithm a QDFT math filtr algrithm. Tabl Cmparis f algrithm mplxity algrithm multipliati aiti DFT 3 lg/4+5+ lg+ QDFT 3 lg/4+3 lg+ Tabl mpars th mputatial mplxity (th sum f multipliati a aiti) f th thr algrithms fr svral lassial sgmt lgths. As shw i Tabl, th urrt iput ata blk lgth is 5, th QDFT-bas math filtr algrithm savs.6% f th mputati tim, a th DQDFT math filtr algrithm savs 38.5%. Th QDFT-bas math filtr algrithm savs 0.0% f th mputatial mplxity whil th DQDFT math filtr algrithm savs 5.% wh th iput ata blk lgth is Thrfr, th DQDFT algrithm prps i this papr a ru th mputatial mplxity gratly. I prati, it a ru th ruig tim a harwar rsur sumpti t imprv th mulati sp. Tabl Th mputatial mplxity f th thr algrithms ur typial lgth Data lgth DFT QDFT sav(%) DQDFT sav(%)

5 4. Clusis I this papr, th vrlap-prsrvig mth is mps. By stuyig th rlatiship btw QDFT a vrlap-prsrvig mth, a lw-mplxity frquy-mai paralll mathig filtr algrithm is prps. Oly t alulat th urrt ata blk, tha th traitial vrlap filtr algrithm t prsrv th fr shrtr FFT lgth. Wh th umbr f paralll paths is larg, th algrithm prps savs mr mputatial mplxity tha th DFT algrithm. Wh th sgmt lgth rahs 4096, th savig fft rahs mr tha half. Thrfr, th DQDFT math filtrig algrithm a b appli i th high-sp mulati systm a s. Rfrs [] Ramai, Siar E, Byr A, t al. Th Eltrmagti Cmpatibility f Itgrat Ciruits Past, Prst, a Futur[J]. IEEE Trasatis Eltrmagti Cmpatibility, 009, 5(): [] SHIHONG D, YAU H, SAWAN. A high ata rat QPSK mulatr fr iutivly pwr ltris implats[c]//ieee Itratial Sympsium Ciruits a Systms Isla f Ks. Gr: IEEE, 006: [3] Chghua X, H C, Shua Z. Dsig a Implmtati f a High-Sp Prgrammabl Plyphas FIR Filtr[C]// IEEE 5th ASIC, 003, : [4] Gustafss O, Dmpstr A G. O th Us f ultipl Cstat ultipliati i Plyphas FIR Filtrs a Filtr Baks[C]// IEEE NORSIG04, 004: [5] Sriiasa,Ch C-C, Gray A. A All-Digital, High Data-Rat Paralll Rivr[R].Jt Prpulsi Lab TDA Prgrss Rprt, 997, vl [6] Liu Cli, A Jiapig, Wag Cuilia, a s. Lw Cmplxity Frquy Dmai Paralll Dmulati Arhittur fr Jit Symbl Syhrizati [J]. Spa Eltri Thlgy, 03,0 (): 7-9. [7] Park S Y, hr P K. Lw-pwr, high-thrughput, a lw-ara aaptiv FIR filtr bas istribut arithmti[j]. IEEE Trasatis Ciruits a Systms II: Exprss Brifs, 03, 60(6):