Implementation of Digital Filters in Carry-Save Residue Number System

Transcription

1 Impementaton of Dgta Fters n Carry-Save Resdue Number System Andrea De Re, Aberto Nannare and Marco Re Department of Eectrca Engneerng Unversty of Rome "Tor Vergata" - Itay Abstract In ths work, we present the mpementaton of a Fnte Impuse Response (FIR)fter n the Resdue Number System (RNS), n whch we use a carry-save scheme n the bnary representaton of the resdues to speed-up moduar addtons. We compare the carry-save RNS mpementaton wth the mpementatons of the same fter n the tradtona bnary system and n pan RNS. Resuts show that the carry-save RNS fter s much faster and ts energy dsspaton per cyce comparabe. Furthermore, we show that a mutpe suppy votage approach for the pan RNSfter can ead to an addtona reducton n power dsspaton wthout performance degradaton. 1. Introducton The new generaton of teecommuncaton equpment requre the use of hgh order, hgh throughput Fnte Impuse Response (FIR) fters. On the other hand, ow power consumpton s desrabe for portabe devces and utra dense crcuts. In ths scenaro, computatonay ntensve sgna processng bocks can be effectvey mpemented by usng the Resdue Number System (RNS) arthmetc. The use of the Resdue Number System aows the decomposton of a gven dynamc range n sces of smaer range on whch the computaton can be mpemented n parae 1, 2, 3]. The drawback presented by the RNS s the overhead due to both nput and output conversons: bnary to RNS, and RNS to bnary. However, by usng effcent converson technques 4, 5], the overhead can be sgnfcanty reduced. In our work, we compare the performance, area and power of a FIR fter reazed n the tradtona bnary system, wth a RNS based one. Furthermore, we make the RNS fter faster than the tradtona one by ntroducng a redundant carry-save representaton, and on the other hand, we reduce the power dsspaton of the pan RNS fter wthout affectng ts deay by equazng the parae paths of the fter wth a mutpe suppy votage approach. Resuts show that the RNS fter can sustan the same throughput as the tradtona one, but dsspates ess power. The carry-save RNS mpementaton of the fter can be cocked amost at doube speed, wth respect to the tradtona one, wthout a sgnfcant ncrease n area and energy dsspated n a cyce. 2. RNS FIR Fter The startng pont of our desgn s a programmabe N taps FIR fter N v('~) - Z a~x(,~ - k) k=0 reazed n transposed form (Fgure 1). In RNS, the FIR fter s decomposed nto P fters workng n parae, as shown n Fgure 2, and both bnary to RNS and RNS to bnary converters are requred. In each tap, a moduar mutper s needed to compute the term (akx(n -- k))m,. Because of the compexty of moduar mutpcaton, we used the somorphsm technque 6] to mpement the product of resdues By usng somorphsms, the product of the two resdues s transformed nto the sum of ther ndces whch are obtaned by an somorphc transformaton. Accordng to 6], f m s prme there exsts a prmtve radx q such that ts powers moduo m cover the set 1, m - 1]: n = (qw')m wth n e 1,m-1] w~ e 0, m- 2]. Both transformatons n -+ w and w --+ n can be mpemented wth m - 1 entres tabes. Therefore, the product X/01/$ IEEE 1309

2 r x(t) I Z-1.--~ Z-1 Z-1 Fgure 1. FIR fter n transposed form. a 0 =.~x x(t) Bnary --- FIR fter mod rn 11--~ -~ FIRftermodm21-~ -~FIR fter mod m pi--~ RNS to to RNS Bnary ---~ FRftermodm-~ Converter Converter y(t) of a and a2 moduo ra can be obtaned as: (a. a2)m -- (qw)m Fgure 2. RNS FIR fter. where w- (W 2c-W2)m--1 wth a -- (qw~)m a2 - (q~2),~ n ~r It Isomorph ~ Mutper A Because of the transposed form of the FIR fter, the nput x s the mutpcand of a the mutpcatons (see Fgure 1). For ths reason ony one drect somorphc transformaton, ncorporated n the bnary to RNS converson, s necessary for a the taps. On the other hand, because the coeffcents of the fter (mutpcators) are constant terms oaded once at start-up, t s convenent to oad drecty the somorphc representaton moduo m - 1. As a resut, n each tap, we reduce the moduar mutpcaton to a moduar addton, mpemented as expaned beow, foowed by an access to tabe (nverse somorphsm). The tabe s mpemented as syntheszed ogc. The moduar addton (b + b2)m, conssts of two bnary addtons. If the resut of b~ + b2 exceeds the moduo (t s arger than m - 1), we have to subtract the moduo m. In order to speed-up the operaton we can execute n parae the two operatons: (b + b2) and (b + b2 - m). If the sgn of the three-term addton s negatve t means than the sum (b + b2)< m and the moduar sum s b + b2, otherwse the moduar addton s the resut of the three-term addton. The resutng mpementaton of the tap for one moduo s depcted n Fgure 3. Due to the transposed form of the fter (Fgure 1), x must be buffered to avod ong deays. The crtca path of the RNS fter, gven by the sowest moduo, s: trns -- tbufer "Jr" tmodmult -]- tmodadd -1- treg (1) Because we chose as mnmum cock perod the deay (crtca path) n the sowest tap, the converters had to be ppened to mantan the fter throughput. Ths resuted n Yk-1 t Moduar Adder (9 o~ n- cock cyce ~ "YE Fgure 3. Structure of one moduo RNS tap. a atency of Nov requred for the conversons. The expresson for the area s AreaRNs -- AreaconvN + AreaTAP. N + AreaconvOUT where AreaTAP s the sum of the area of the P parae taps" AreaTAP -- ~P=o AreaTAP~. As for the power dsspaton, we get a smar expresson PRNS -- PconvN Jr- PTAP " N + P~o~,IN wth the mtaton that the term PTAP strongy depends on the swtchng actvty and consequenty on the vaue of the coeffcents. 3. Carry-Save RNS Fter In order to speed-up the operatons by makng the cock perod shorter, we can resort to a carry-save representaton for the bnary representaton of resdues (ys~, yck) and avod to compute the moduar addton n every tap. The operands to be added n a tap are three: the product Pk = (akx(n - k))m~ and the carry-save representaton of Yk-1 y8 k -- sum(ysk_, YCk-I,Pk), YCk = carry(ysk-1, YCk_I,Pk ). 1310

3 t n CSA YSk- " ~ I n I H YCk_ -.-, Isomorph n,~,--- A E Mutper n 3:2+ HA HA I o & cock cyce L_ L n,d Ys "-~ o~ I~ k ~ Yc m I 1 k Fgure 4. Tap structure for RNS carry-save. x ~ o 1 utperoxo -~---- A1 k t Psk /pc 1 ~20 ~ ~ Y YSk~ -I 1 20 " -# ] I~ ~ L. ~1 S k 2o CSA 4:2.-~ Yc k YCkTI -', / 7 re cock cyce " Fgure 6. Tap structure for the tradtona fter mpementaton.,,d X.. Regster X I Ys k-~ Yc k " ~ E 11 CPA & rood H Regster n--~ Y k cock cyce Fgure 5. Reay staton. The structure of the carry-save RNS tap s shown n Fgure 4. Wth the new representaton, we reduce the term tmodadd of Expr. (1) to the deay of a haf-adder and the crtca path s now: tcs-rns : tbuffer + tmodmult + txor + treg (2) The crtca path s essentay determned by the mutper atency, beng tbu//er and treg unavodabe, and beng txor the mnmum deay attanabe for a haf-adder. However, the CS-representaton mpes the doubng of the regsters, and, as the number of taps ncreases, a ogarthmc ncrease n the bt-wdth of CSAs and regsters. For ths reason, t mght be convenent to nsert some reay statons (RS), whch assmate the carry-save representaton of Yk and extract ts moduo m, to prevent the bt-wdth from growng too much. Reay statons are depcted n Fgure 5 and ther deay s trs -- tcpa&mod + treg (3) They ntroduce an extra cyce of atency, but they can aso be used to better dmenson the bufferng of x (.e. reduce The spacng of reay statons (.e. the number n of taps between two reay statons) can be determned by combn- w..- 1 ng Expr. (2), n whch the term tbu/fer ncreases wth n, and Expr. (3), where tcpa~mod depends on n. However, by rewrtng tbu//er -- tnt + ho~d " n, the sum tnt + tmodmult + txor + treg becomes domnant over Expr. (3) and, as a consequence, we can choose n, whch satsfes our tmng constrants, by Expr. (2) ony: tnt + ttoad " n + tmodmult + txor + treg < Tcock 4. Tradtona FIR Fter In order to evauate the performance of these RNS fters, we have mpemented a N-tap error-free FIR fter (20 bts dynamc range) n the tradtona two's compement system (T2S) 7], n RNS 7] and n RNS usng the carrysave scheme (CS-RNS). For the tradtona T2S fter we opted for a carry-save representaton n the taps to keep the cyce tme as short as possbe (Fgure 6). The product Pk = akx(n -- k) s reazed wth a Booth mutper 8] and the resutng parta products are accumuated n a Waace tree structure whch produces a carry-save representaton of the product. Because the sum k--1 at the (k - 1)-th tap Yk-1 -- ~=o ax(n- ) s stored n carry-save representaton, an array of 4:2 compressors 9] s requred to reduce the CS representaton of Pk and the CS representaton of Yk-1 to the CS representaton of Yk = Yk-x + Pk n the k-th tap. The carry-save representaton s fnay converted nto the two's compement representaton by a carry-propagate adder (reazed wth a carry-ook-ahead scheme) n the ast stage of the fter. 1311

4 Fter T2S RNS CS-RNS MV-RNS Cyce Area Ins] (NAND2 equv.) IN* N N N MHz N* N N N 6 (est.) N 3 Tabe 1. Summary of resuts for mpemented fters. The crtca path s" tt2s -- tbuffer -}- tmult ~- tcsa-4:2 -~- treg An extra cock cyce s needed to compute y from ts CSrepresentaton (y - y8 + yc). The expresson for the area as a functon of the number of taps s: AreaT2s -- AreaTAP N + AreacPA where AreacpA s the area of the fna carry-propagate adder. For the power dsspaton the expresson s smar to that of the area: PT2S -- PTAP " N + PCPA. 5. Resuts and Comparsons For the RNS and CS-RNS fters, n order to have a dynamc range of 20 bts, as n the case of the tradtona mpementaton, we chose the foowng set of modu: m -- {3, 5, 7, 11,17, 64} such that og2 ( ) > 20. The mutpcaton was mpemented wth somorphsm n a the modu but 3 (mutpcaton can be easy computed n tabuar way) and 64 (moduar product corresponds to the 6 east-sgnfcant bts of the conventona product). Ppene regsters are added n the converters to mantan the cock cyce, determned by the crtca path n the taps, to ts mnmum. As for the mpementaton of carry-save RNS, n our desgn, the carry-save representaton s not used for moduo 3 because ts tap deay s suffcenty sma. However, we have to pace reay statons Oust two 2-bt regsters and, eventuay, buffers) to synchronze (X)3 wth the other (X)r,,. For moduo 64, the use of CS-representaton does not mpy an ncrease of bt-wdth n the CSAs and regsters. For a the modu, n our carry-save RNS (CS-RNS) fter we paced reay statons spaced by 8 taps, because ths confguraton gves us the best deay-area tradeoff. The fters were mpemented n a 0.35#m brary of Standard Ces. Deay, area and power dsspaton have been determned wth Synopsys toos. The resuts are shown n Tabe 1 aong wth an evauaton of a possbe mpementaton wth mutpe suppy votages (MV-RNS), descrbed n the next secton. In the tabe, area, reported as number of NAND2 equvaent gates, and power, computed at 100 MHz, are expressed as a functon of the number of taps (N). Coumn N* ndcates the N for whch the RNS and the CS-RNS fter occupy ess area and dsspate ess power than the correspondng T2S fter. Athough nterconnectons are not taken nto account, oca and goba routng for the RNS s expected to be no worse than for the tradtona fter. 6. Mut-Votage RNS Fter In addton to the above presented mpementatons, we add a possbe approach to reduce the power dsspaton n a RNS fter wthout penazng ts speed. The power dsspated n a ce depends on the square of the suppy votage (VDD) SO that a sgnfcant amount of energy can be saved by reducng ths votage 10]. However, by owerng the votage the deay ncreases, so that to mantan the performance ths technque s apped ony to ces not n the crtca path. Tabe 2 reports detas on the mpementaton of the pan RNS fter for the dfferent paths correspondng to the modu m. Deay s normazed to the crtca path (.e. cock cyce), whe area and power dsspaton are normazed to ther per tap totas. Because the snge tap deay of modu 3, 5, 7 and 64 s ess than the crtca path, we can use the avaabe tme sack and reduce the suppy votage for these modu wthout affectng the overa performance. In Tabe 3 we report the possbe suppy votage whch can be used n the moduo m fters wthout ncreasng the crtca path. The brary of standard ces we used normay operates at VDD V. Tabe 3 aso reports the power savngs obtaned wth the sted suppy votage. These vaues have been computed assumng that the swtchng actvty does not change when scang the votage. Ths assump- 1312

5 Moduus Deay Area Power tota Tabe 2. Deay, area and power dsspaton per tap n RNS FIR. snge VDD mutpe votage m Deay VDD Power Deay VDD Power V V V V V V V V V V V V 0.16 tot Tabe 3. Power dsspaton for mutpe suppy votage n tap. ton seems to be reasonabe because an ncreased shortcrcut energy, due to onger transton tmes, s compensated by a suppresson of some gtches, due to a onger gate deay. The tabe shows a reducton of about 15% n the power dsspaton per tap. The use of a mutpe suppy votage requres eveshftng crcutry when gong from the ower votage to the hgher one 11]. In our case, votage eve shfters are ony requred for a few bts before the output converson stage. 7. Concusons In ths work, we compared the performance, area and power of FIR fters reazed wth the tradtona bnary arthmetc, the RNS, and the RNS wth a carry-save representaton of resdues. Tabe 1 shows that the RNS fter can sustan the same throughput as the tradtona one, but dsspates ess power for fters wth more than 4 taps. The CS-RNS mpementaton of the fter can be cocked amost at doube speed, wth respect to the tradtona one, wthout a sgnfcant ncrease n area and energy dsspated n a cyce (the power ncreases neary wth frequency). Fnay, we showed a possbe mpementaton of the RNS fter wth mutpe suppy votage, whch can ead to an addtona reducton of the power dsspated, wthout affectng the performance of the fter. Acknowedgments Ths work was partay supported by MURST project: Codesgn methods for ow power ntegrated crcuts. References 1] N.S. Szabo and R.I. Tanaka. Resdue Arthmetc and ts Appcatons n Computer Technoogy. New York: McGraw-H, ] M.A. Sodestrand, W.K. Jenkns, G. A. Juen, and F. J. Tayor. Resdue Number System Arthmetc: Modern Appcatons n Dgta Sgna Processng. New York: IEEE Press, ] M.A. Soderstrand and K.A1 Marayat. VLSI mpementaton of very hgh-order FIR fters. IEEE Internatona Symposum on Crcuts and Systems (IS- CAS'95), 2: , ] S.Pestrak. A hgh-speed reazaton of a resdue to bnary number system converter. IEEE Trans. Crcuts Systems-H Anaog and Dgta Sgna Processng, 42: , Oct ] M. Re, A. Nannare, GC. Cardar, and R. Lojacono. FPGA Impementaton of RNS to Bnary Sgned Converson Archtecture. Proc. of IEEE Internatona Symposum on Crcuts and Systems, IV: , May ] I.M. Vnogradov. An Introducton to the Theory of Numbers. New York: Pergamon Press, ] A. Nannare, M. Re, and GC. Cardar. Tradeoffs between Resdue Number System and Tradtona FIR Fters. Proc. of IEEE Internatona Symposum on Crcuts and Systems, II: , May ] Israe Koren. Computer Arthmetc Agorthms. Prentce-Ha, Inc., ] M.D. Ercegovac and T. Lang. Dvson and Square Root: Dgt-Recurrence Agorthms and Impementatons. Kuwer Academc Pubsher, ] A. P. Chandrakasan and R. W. Brodersen. Low Power Dgta CMOS Desgn. Kuwer Academc Pubshers, ] K. Usam and M. Horowtz. Custered votage scang technque for ow-power desgn. Proc. of Internatona Symposum on Low Power Desgn, pages 3-8, Apr