Hgh Dyac Rage 3-odul et wth Effcet Resdue to Bary Coverter Hgh Dyac Rage 3-odul et wth Effcet Reverse Coverter A. Harr, R. Rastegar, K. av Abstract-Resdue uber yste (R) s a valuable tool for fast ad parallel arthetc. It has a wde applcato dgtal sgal processg, fault tolerat systes, etc. I ths work, we troduce the 3-odul set {, -, +} ad propose ts resdue to bary coverter usg the Chese Reader Theore. We preset ts sple hardware pleetato that aly cludes oe Carry ave Adder (CA) ad a odular Adder (A). We copare the perforace ad area utlzato of our reverse coverter to the reverse coverters of the odul sets { -,, +, +} ad { -,, +, - (+)/ +, + (+)/ +} that have the sae dyac rage ad we deostrate that our archtecture s better ters of perforace ad area utlzato. Also, we show that our reverse coverter s faster tha the reverse coverter of { -,, +} for dyac rages lke 8-bt, 6-bt, 3-bt ad 64-bt however t requres ore area. Idex Ters Resdue arthetc, Resdue to bary coverter, Chese reader theore (CRT) I. ITRIDUCTIO Resdue uber yste (R) arthetc s a valuable tool for theoretcal studes of fast arthetc [5]. Wth ts carry-free operatos, parallels ad fault tolerace, R has bee used coputer arthetc sce 950s. These propertes have ade t very useful soe applcatos cludg dgtal sgal processg ad fault tolerat systes [4]. Dfferet odul sets have bee preseted for R that have dfferet propertes wth regards to reverse coverso (Resdue to Bary or R/B), Dyac Rage (DR) ad arthetc operatos. The odul of the fors, - ad + are very popular accordg to ther easy arthetc operatos. The ost faous odul set s { -,, +} ad several ethods have bee proposed for ts reverse coverso ad the best ethod has bee outled []. O the other had, there are soe other odul sets that have greater dyac rages coparso wth ths odul set. They clude; the odul sets { -,, +, + -} [] ad { -,, +, + +} [3] that have the dyac rages of ad + bts respectvely. I [], odul set { -,, +, A. Harr ad K. av are wth hahd Behesht Uversty, Tehra, Ira (eal: {harr, av}@eee.org). R. Rastegar s wth outher Illos Uversty, Carbodale, IL 690, UA (e-al: rrastegar@eee.org). +} has bee proposed that provdes the dyac rage of ( -). It has bee show that the reverse coverter of ths odul set has superor area-te coplexty coparso wth the reverse coverters of [] ad [3]. I [9] the odul set {, -, +, - (+)/ +, + (+)/ +} has bee focused o whch has the sae dyac rage of ( -) ad a ew reverse coverter has bee proposed that s ore effcet tha the prevous coverters cludg [8] ad [0]. I ths paper, we troduce the odul set {, -, +} that has the sae dyac rage as [] ad [9] but the reverse coverso ca be carred out faster ad t requres lower hardware area coparso wth [] ad [9]. Our reverse coverter s faster tha the reverse coverter of [] for dyac rages lke 8-bt, 6-bt, 3-bt ad 64-bt however t utlzes ore area tha the reverse coverter of []. I ecto II of ths paper we provde a short backgroud for R ad also troduce the odul set {, -, +}. I ecto III, we preset two leas ad cosder the reverse coverso schee for the proposed odul set usg the preseted leas ad the CRT. I ecto IV, we provde the hardware pleetato of the reverse coverter ad ecto V we evaluate ths coverter ad copare the results wth slar works. Fally, ecto VI we preset our coclusos. II. BACKGROUD R s defed by a set of tegers that are par-wse relatvely pre. That s {,,, } where gcd (, j } for, j ad j ad gcd eas the greatest coo dvsor. Every teger [0, -] ca be uquely represeted wth a -tuple where,, ( R, R,..., R ) ad R ( od ) ; for to The set ad the uber R are called the odul set ad the resdue of odulo respectvely. The arthetc operatos ca be carred out depedetly for each odulo, that s ( x, x,..., x ) ( y, y,..., y ) ( x y, x y,..., x y )
Hgh Dyac Rage 3-odul et wth Effcet Resdue to Bary Coverter where deotes oe of the arthetc operatos of addto, subtracto, ad ultplcato. Here, we propose the ew odul set {, -, +} ad frst, we show that ths set eets the requreets of a R odul set. Theore : The set {, -, +} s a odul set for R. Proof: We should show that the odul are par-wse relatvely pre for ay atural uber. Obvously, the frst odulo s relatvely pre to the other odul therefore we oly show that the secod ad the thrd odul are relatvely pre. We assue that gcd ( -, +) d the we have d ( -) ad d ( +) therefore, d ( -+ +) so d ( ) or we have d ( + ) so d or d w (w ) but we kow that d w because - ad + are odd ubers so gcd ( -, +)d. o our proposed odul set ca be used R ad we ca cosder ts reverse coverter. III. REVERE COVERTER I ths secto, we preset the reverse coverter of the odul set {, -, +} but frst, we provde two leas whch are based o the propertes that have bee used calculatg the reverse coverters [][4][]. Lea : The resdue of a egatve resdue uber ( v) odulo ( -) s calculated by the oe s copleet operato where 0 v< -. Lea : The ultplcato of a resdue uber v by P odulo ( -) s carred out by P-bt crcular left shft where P s a atural uber. ow, to calculate the uber fro ts resdues, we ca apply the CRT. The CRT s forulated as; where ; ad R R ; Assug, - ad 3 + we have ; ˆ ( + ) ; ˆ ( ) () ˆ ( ) Theore : For the proposed odul set, we have ˆ 3 () (3) ˆ (4) ˆ (5) 3 3 3 Proof: For (3) we have: ( ) +. for (4) we have ( + ) ( ) ( ) +. ( ) ( ) ad for (5) we wrte ( ) ( + ) ( + ) ( + ) + Equato () ca be rewrtte as R R K where K s a teger uber ad depeds o the value of. By replacg ()-(5) (6) we have: ( ) ( ) R + ( ) R + + K ( ) R 3 By dvdg the both sde of (7) by ad calculatg the floor values odulo ( -) we have R + ( + ) R + ( ) or 3 ( ) ( ) R3 ( ) ( I ths case the uber ca be coputed by + R Equato (8) ca be wrtte as where 3 R ( ) + 3 ( ) + R 3 + ( ) R ( ) + + 3 ( ) 3 ( ) 3 ( ) ( ) (6) (7) (8) (9) (0) () R ()
Hgh Dyac Rage 3-odul et wth Effcet Resdue to Bary Coverter ( + ) R (3) 3 ( ) ( ) R (4) 3 3 3 ( ) ow, we cosder ()-(4) ad splfy the for pleetato a VLI syste. It s ecessary to ote that r,j eas the j-th bt of R. Evaluato of : The resdue R ca be represeted bts as follows; 3 Bts R 00 00r r r,( ),,0 by applyg Lea odulo ( -) we have 3 Bts 3 R r ( ),( ) r,( ) r,r,0 00 00 ad fally by applyg Lea we have 3 ( ),( ),,0 3 Bts R r r r wherer eas the copleet of r. Evaluato of : The resdue R ca be represeted bts as follows; Bts R 00 00r r r,,,0 we evaluate the two parts of separately usg Lea Bts 3 R r ( ), r,r,0 00 00r,( ) r,( + ) + Bts - Bts Bts R 00 00r ( ),( ) r,r,0 00 00 + Bts - Bts by addg (9) ad (0) we have the fal value of as R + R 3 ( 4 ) Bts,,,0,(),,0,( ),( + ) + Bts - Bts r r r r r r r r that s a -bt resdue uber. Evaluato of 3 : The resdue R 3 ca be represeted bts as follows; Bts R 00 00r r r 3 3, 3, 3,0 for the two parts of 3 we use Lea ad we wrte (5) (6) (7) (8) (9) (0) () () - Bts 3 R3 r ( ) 3, r3,r3,0 0 0r3, r 3,( + ) + Bts Bts + Bts R3 00 00r ( ) 3,( ) r3,r3,0 00 00 Bts - Bts for (4) we apply Lea ad we have + Bts R3 r ( ) 3,( ) r3,r3,0 Bts - Bts therefore, - Bts 3, 3, 3,( ) 3, 3,0 00 00 3, 3,( ) 3,( + ) + Bts Bts r r r r r r r + Bts 3, 3,( ) 3,() 3, 3,0 Bts r r r r - Bts so, 3 cludes two -bt ubers that are 3, ad 3,. (3) (4) (5) (6) (7) IV. HARDWARE IPLEETAIO To pleet the reverse coverter, four -bt ubers should be sued up odulo ( -). Ths requres a - level Carry ave Adder (CA) tree that cludes two -bt CAs. evertheless by cosderg (7) ad (7), t s clear that the 3 rghtost bts of ad also the leftost bts of 3, are oes. o, we replace the 3 rghtost bts of 3, wth the sae bts of. Based o ths apulato, the ew ubers have bee show (8) ad (9). Cosequetly, ow 3, cotas oes ad we kow that t s equvalet to zero odulo ( -). ow, we have 3 ubers ad therefore, the requred -level CA ca be replaced by oly oe CA. + Bts,( ),0 3, 3,0 r r r r Bts 3 Bts 3, Bts (8) (9) Fg. shows the hardware archtecture of the reverse coverter. The Operad Preparato (OP) copoet cludes soe wres ad verters ad prepares the -bt ubers for the ult Operad odular Adder (OA). The CA tree cludes oly oe -bt CA wth Ed-Aroud Carry (EAC) [6]. The last copoet OA s a odular Adder (A) ad ca be pleeted usg the ethods of [6], [7] or [5]. The output of ths adder s equvalet to ad cosequetly, ca be coputed by usg (9).
Hgh Dyac Rage 3-odul et wth Effcet Resdue to Bary Coverter R 3 R Fg. Hardware archtecture of the proposed reverse coverter V. EALUATIO AD COPARIO odul sets of [] ad [9] provde the sae dyac rage as our odul set. o, ths secto we copare two propertes of our odul set to the odul sets of [] ad [9]; ) Te ad area coplextes of the reverse coverso ad ) Te coplexty of the arthetc operatos ther odul. Fally, we copare our reverse coverter to the reverse coverter of a 3 odul set proposed []. ow, we copute the hardware utlzato of our reverse coverter ters of adders ad basc gates. As outled the prevous secto, we should su up three -bt ubers, ad 3,. For ths purpose, oe CA whch cludes Full Adders (FAs) s suffcet. But by cosderg the operads, t s clear that soe of these FAs could be splfed further. For the (- ) rghtost bts, we eed (-) pars of OR/OR gates stead of (-) FAs, sce oe of the puts of each FA s. larly, for the ddle (-) bts, we replace the (-) FAs wth (-) pars of OR/AD gates, sce oe of the puts of each FA s 0. For the rest of the bts, we use (+) FAs. Besdes ths OA, the operad preparato cludes soe wres ad verters. Igorg the wres, t cludes (3+) verters. The total aout of the used hardware s show Table I. TABLE I HARDWARE UTILIZATIO OF THE REVERE COVERTER R/B Coverter Our work [] [9] [] DR ( -) ( -) ( -) ( -) Iverters (OP) 3+ 5+3 + FAs + 7+6 5 OR/AD Pars - - 7 - OR/OR Pars - - Other - -3 verter - OR+HA U - - Oe 4 Two A -bt -bt -bt -bt It s clear fro Table I that our proposed reverse coverter requres very low hardware area coparso wth the reverse coverter of [] ad also our reverse coverter s superor to the reverse coverter of [9] whch s the ost R Operad Preparato (OP) 3, -bt CA -bt s Copleet Adder OA effcet coverter for the odul set {, -, +, - (+)/ +, + (+)/ +}. I [9], oe 4 ultplexer s requred for geeratg oe of the -bt operads of the CA tree. o ths operad ca have four possble values ad they would oly cota fxed oes ad zeros. To cosder ts assocated CA, we have assued that the uber of oes s approxately equal to the uber of zeros ad ths assupto does ot affect the coparso. The total delay of our reverse coverter s the su of the delays of three copoets: the operad preparato, CA ad A. The delay of operad preparato s equal to the delay of a OT gate. For the CA, the delay s the delay of a FA. For the A, dfferet ethods ca be appled that have dfferet delays [6][7][5]. Here we have used the odular adder of [5]. Adoptg the ut gate delay [][3]5], we assue t v t ad, t ux, t FA, t xor ad cosequetly usg the othod of [5], t A() log ()+3. Table II shows the delays of the reverse coverters. It ca be cocluded for Table II that we have elated the delay of two FAs coparso wth [] ad the delay of three FAs coparso wth [9]. I addto to ths delay proveet, we have utlzed uch lower hardware tha [] ad [9]. TABLE II DELAY OF THE REVERE COVERTER R/B Delay Ut Gate Delay [] t CLA() +t OT + 3t FA [9] t CLA() +t OT + 4t FA [ ] t CLA() +t OT + t U + t FA log( ) + 7+ 7 log( ) + 7+ 9 Ours t CLA() +t OT + t FA log( ) + 7+ 3 log ( ) + 7 + 5; f log ( ) log ( ) () log ( ) + 5 + 5; f log ( ) log ( ) + () o far, we have show that our coverter has better area ad te coplextes tha those of [] ad [9], but we have left oe questo uaswered. For a equal dyac rage, s a 4 or 5-odul set always faster tha a 3-odul set? It s the agtude of the largest odulo that dctates the speed of arthetc operatos; however, speed ad cost do ot just deped o the wdth of the resdues but also deped o the odul chose [5]. Cosequetly, for the odul set of [], odulo + deteres the overall speed of the R. The sae s true for our proposed odul set. Therefore our odul set ad the odul set of [], are both restrcted to the te perforace of odulo +. The odul set of [9] cludes two odul of ( - (+)/ +) ad ( + (+)/ +). Here, we copute the delay of addto odulo ( + (+)/ +) by usg the ethod of [] ad we copare t to delay of addto odulo ( +) that has bee coputed by usg the ethod of [3]. Table III shows that addto odulo ( +) s uch faster tha addto odulo ( + (+)/ +). o, we ca coclude that although [9] has fve odul, t s ot faster tha our proposed odul set. Therefore our odul set
Hgh Dyac Rage 3-odul et wth Effcet Resdue to Bary Coverter outperfors both odul sets of [] ad [9]. TABLE III DELAY OF ADDITIO I TWO ODULI Addto odulo ( + (+)/ +) Addto odulo ( +) 4 log ()+7 log ()+6 log ()+8 I addto to coparg [] ad [9], we would lke to copare our reverse coverter to the reverse coverters of 3- odul sets. I [4], t has bee show that odul set { -,, +} has the fastest ad the ost area effcet reverse coverter aog the other 3-odul sets for the dyac rages of 8-bt, 6-bt, 3-bt ad 64-bt. o, we copare our reverse coverter to the reverse coverter of [] whch s the ost effcet reverse coverter for { -,, +}. For the sake of a far coparso, we cosder the odul set { -,, +} where s chose a way that provdes slar dyac rages to our odul set ad ore or less ca be the floor or celg value of 5/3. By usg ths approxato, the hardware utlzato of the reverse coverter of [] has bee derved ad cluded Table I. I Table II, we have copared our reverse coverter to the reverse coverter of [] cosderg two cases. I case () our reverse coverter s faster tha the reverse coverter of [] ad t s worthwhle to eto that for exaple, for [, 50], ths case covers 73% of dyac rages cludg 8-bt, 6-bt, 3-bt ad 64- bt. I case () whch covers 6% of dyac rages, our reverse coverter ad the reverse coverter of [] have the sae delay but [] requres less hardware area. Table IV shows the area ad delay coparso of the proposed reverse coverter ad that of the [] usg the ut-gate odel where the hardware area utlzato of the gates are A OT A AD A OR ad A OR. The hardware area utlzato of the odular adder has bee coputed usg the adder of [5]. TABLE IV COPARIO OF REVERE COVERIO I TWO 3-ODULI ET Extra peed - DR A ours A [] t Area% ours t [] up % 8-bt 3 5 36.0 4 4. 6-bt 4 6 34 98 4.4 4 6.5 3-bt 7 674 604.58 6 8. 64-bt 3 400 330 5.6 8 0 0 It ca be cocluded that the coparso of our work ad [] s purely dctated by the chose dyac rage. However, for the dscussed dyac rages, our reverse coverter s faster tha the reverse coverter of [] whle [] requres less area. sae dyac rages. We also showed that for ajorty of the slar dyac rages, our reverse coverter s faster tha the reverse coverter of { -,, +} but the reverse coverter of { -,, +} has less area. ACKOWLEDGET The authors wsh to ackowledge the valuable help of Dr. T. Vergos wth the odular adders. REFRECE [] B. Cao, C. Chag ad T. rkatha, A effcet reverse coverter for the 4-odul et { -,, +, +} based o the ew Chese reader theore, IEEE Trasacto o Crcuts ad ystes I, Vol. 50 Issue 0, Oct. 003 Page(s): 96 303. [] A. P. Vod ad A. B Prekuar, A eoryless reverse coverter for the 4-odul superset { -,, +, + -}, Joural o Crcuts, yst., Coput., Vol 0, 0, o.&, Page(s).85 99,000. [3] Bhardwaj, T. rkatha ad C. T. Clarke, A reverse coverter for the 4-odul superset { -,, +, + +}, I the Proceedg of the 4 th IEEE yposu o Coputer Arthetc, Adelade, Australa, 4-6 Aprl 999 Page(s): 68 75. [4]. zabo ad R. I. Taaka, Resdue uber syste ad ts applcato to coputer techology, cgraw Hll ew York 967. [5] B. Parha, Coputer arthetc, Oxford Uversty Press, 000. [6]. J. Pestrak, A Hgh peed Realzato of a Resdue to Bary Coverter, IEEE Trasacto o Crcuts ad ystes II, Volue 4, Issue 0, Oct. 995 Page(s): 66 663. [7]. Bhardwaj, A. B. Prekuar ad T. rkatha, Breakg the bt carry propagato barrer resdue to bary coverso for the { -,, +} odul set, IEEE Trasacto o Crcuts ad ystes I, Volue 45, Issue 9, ept. 998 Page(s): 998 00. [8] A. kavatzos, A effcet resdue to weghted coverter for a ew resdue uber syste, Proceedgs of the 8th Great Lakes yposu o VLI, Feb. 998 Page(s): 85 9. [9] A. A. Hasat, VLI pleetato of ew arthetc resdue to bary decoders, IEEE Trasacto o Very Large cale Itegrato (VLI) ystes, Volue 3, Issue, Ja. 005 Page(s): 53 58. [0] Y. Wag, Resdue-to-bary coverters based o ew Chese reader theores, IEEE Trasacto o Crcuts ad ystes II, Volue 47, Issue 3, arch 000 Page(s): 97-05. [] Y. Wag,. og,. Aboulhad ad H. he, Adder based resdue to bary uber coverters for ( -,, +), IEEE Trasactos o gal Processg, Volue 50, Issue 7, July 00 Page(s):77 779. [] A.A. Hasat, Hgh-speed ad reduced-area odular adder structures for R, IEEE Trasactos o Coputers, Volue 5, Issue, Ja. 00 Page(s):84 89. [3] C. Efstathou, H.T. Vergos ad D. kolos, Fast parallel-prefx odulo + adder, IEEE Trasactos o Coputers, Volue 53, Issue 9, ept. 004 Page(s): -6. [4] W. Wag,.. way,.o. Ahad ad Y. Wag, A study of the resdue-to-bary coverters for the three-odul sets, IEEE Trasactos o Crcuts ad ystes I: Fudaetal Theory ad Applcatos, Volue 50, Issue, Feb. 003 Page(s):35-43 [5] L. Kalapoukas, D. kolos, C. Efstathou, H.T. Vergos ad J. Kalaataos, Hgh-speed parallel-prefx odule - adders IEEE Trasactos o Coputers, Volue 49, Issue 7, July 000 Page(s):673-680. VI. COCLUIO I ths paper we proposed the odul set {, -, +} ad ts reverse coverter. Ths odul set provdes the dyac rage of ( -) ad the pleetato results have show that ts reverse coverter has better area ad te coplextes coparso wth the odul sets wth the