Journal of Emerging Trends in Computing and Information Sciences

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1 Joural of Eergg Treds Coputg ad Iforato Sceces CIS Joural. All rghts reserved. A Detaled Study o the Modul Nuber Effect o RNS Tg Perforace 1 Da Youes, 2 Pavel Steffa 1 Ph.D. Studet, Bro Uversty of Techology, Departet of Mcroelectrocs, Bro, Czech Republc 2 Assoc. Prof., Bro Uversty of Techology, Departet of Mcroelectrocs, Bro, Czech Republc E-al: 1 xyoue00@stud.feec.vutbr.cz, 2 steffa@feec.vutbr.cz ABSTRACT Ths paper studes the effect of the odul uber wth a odul set o the overall speed of the resdue uber syste (RNS). Choosg a proper odul set greatly affects the perforace of the whole syste. The wdely kow ssue s that as the uber of odul creases the speed of the resdue arthetc uts (RAUs) creases, whereas the resdue-to-bary coverters (RCs) becoe slower ad ore coplex. Thus, we carred out a detaled study o dfferet odul sets wth dfferet odul uber ad dfferet dyac rages (DRs) ad copared tg perforace of systes based o the order to detere the odul uber effect o the overall RNS tg perforace ad fd out the ost effcet set for each DR. Keywords: Resdue uber syste, odul set, dyac rage, reverse coverters, resdue arthetc uts 1. INTRODUCTION The carry-free, parallels, hgh-speed ad securty features of the resdue uber syste (RNS) have ade t very attractve to be used dgtal sgal applcatos (DSPs). The RNS dvdes the requred coputatos to a uber of parallel faster oes accordg to the uber of odul. These chaels are totally depedet ad perfor hgh-speed coputatos o saller resdues [1], [2]. Ths syste has bee tesvely used applcatos where addto, subtracto ad ultplcato are doat, such as, dgtal flters, dgtal coucatos, dscrete Fourer trasfor (DFT), age processg, ad vdeo codg [1], [2], [3]. However, the RNS could ot be wdely pleeted geeral-purpose processors, sce operatos as dvso, sg detecto, agtude coparso ad overflow detecto are probleatc, ad ca egatvely fluece the overall perforace of the desg. May solutos for these probleatc operatos were suggested. Most of the aly deped o the resdue-to-bary coverter (reverse coverter (RC)). Ths copoet represets the a over-head the whole RNS. O the other had, choosg a proper odul set s aother essetal ssue for buldg a effcet RNS wth a suffcet dyac rage (DR). The ost faous odul set s {2 1, 2, 2 + 1} [4]. Ths set has bee kow as a eas of splfyg the calculatos ecessary to pleet the RC. However, ths set has odulo (2 + 1) chael that represets the bottleeck of the syste. Its arthetc crcuts suffer fro the logest delay aog all three chaels. I geeral, arthetc crcuts odulo (2 k 1) are ore effcet tha those odulo (2 k + 1), therefore, t s better to reduce the uber of odul of the for (2 k + 1) [3]. Thus, order to splfy the coplexty caused by odulo (2 + 1) the set {2 1, 2, 2 + 1} [4], ew odul sets {2 1 1, 2 1, 2 } [5] ad {2 1, 2, } [6], that substtute ths odulo wth aother of the for (2 k 1), were suggested. These three sets have a 3-bt DR, whch s suffcet for applcatos that requre edu DRs (less tha 22-bts). However, ay DSP applcatos requre larger DRs, therefore, ew odul sets {2 1, 2, } [7] ad {2 1, 2 + 1, } [8] that provde 4-bt DR ad {2, 2 2 1, } [9] that provdes 5-bt DR, were suggested. Although the DR s larger, the delay of the resdue arthetc uts (RAUs) based o these sets has cosderably creased, due to utlzg odul wth greater agtudes. I order to elate ths drawback ad ata the large DR, sets of four ad fve odul have bee suggested, such as {2 1, 2, 2 + 1, } [10-I], {2 1, 2, 2 + 1, } [10-II], {2 /2 1, 2 /2 + 1, 2 + 1, } [20],{2 1, 2, 2 + 1, } [11], {2 1, 2, 2 + 1, } [12-I], {2 1, 2, 2 + 1, 2 2 (+1)/2 + 1, (+1)/2 + 1} [13], {2 1, 2 + 1, 2 2 2, } [14], {2 1, 2 2, 2 + 1, } [12-II], {2 1, 2, 2 + 1, 2 1 1, } [15], {2, 2 /2 1, 2 /2 + 1, 2 + 1, } [16], {2 + 1, 2 1, 2 2, } [21] ad {2 2+1, , 2 + 1, 2 1} [22]. Each of these sets has ts ow advatages ad dsadvatages. Soe of the offer hgher DRs tha others, whle others have ore parallels. Soe ca result ore effcet RCs, whle others ore effcet RAUs. However, the a cocetrato of publcatos that troduce these sets s o the RCs. Moreover, each of the copares ts proposed set wth other two or three. Thus, we have observed a lack to a detaled coparatve study that farly copares these sets, ters of the DR, uber of odul, te ad hardware requreets for pleetg RCs ad RAUs based o the. Cosequetly, we have decded to do ths research, deeply study each of these sets, evaluate ad copare as far as possble ther RCs ad RAUs. The studed odul sets are classfed based o the DR they provde (3, 4, 5 ad 6). The effect of odul uber o the overall syste s tg perforace s a well-kow fact. More odul lead to ore parallels, whch theoretcally eas faster coputatos wth the depedet RAUs. However, ths advatage trasfors to a dsadvatage regardg desgg the RCs. Thus, the secod part of our research was dedcated for studyg the relato betwee the odul uber ad the overall syste tg perforace. Accordg to the RAUs ad RCs, te ad hardware requreets are coputed Sectos 4, 5, 6 ad 7, we estated the ost effcet ad effcet odul sets for three DRs (edu, large ad very large). For the sake of a far coparso, we have adopted the 85

2 Joural of Eergg Treds Coputg ad Iforato Sceces CIS Joural. All rghts reserved. ut gate odel [17]. Furtherore, we have used sae basc blocks aog all desgs (e.g. adders ad ultplers). The rest of ths paper s orgazed as follows; a bref overvew of RNS s preseted Secto 2. The basc blocks that were used durg our study ad coparsos, addto to ther te ad hardware requreets, are llustrated Secto 3. Sectos 4, 5, 6 ad 7 preset the te ad hardware requreets for pleetg the odular adders, odular ultplers ad RCs based o each of the studed odul sets for DR = 3, 4, 5 ad 6, respectvely. Secto 8 deostrates a coparso betwee all studed sets for three precse DRs, 12 bts (edu DR), 24 bts (large DR) ad 60 bts (very large DR). Ths secto copares the tg perforace of all copoets systes based o the odul sets uder study. Fally, the coclusos ad future work are draw Secto 9. a resdue uber (x 1, x 2,, x ) ca be coverted back to ts weghted equvalet X by, 1 X v v v v Where, v x,() v x v, v x v v v () The ew CRT-I s a odfcato of the orgal CRT, where the sze of the fal odular adder s reduced. By usg ths algorth, a resdue uber (x 1, x 2,, x ) ca be coverted back to ts weghted equvalet X by, 2 2 (4) 2. RNS OVERVIEW The RNS s defed by a set of postve parwse relatvely pre ubers { 1, 2,, } called odul. The dyac rage (DR) s defed as M = 1 2. I ths syste, ay teger X the rage [0, M 1] ca be uquely represeted by a ordered set of resdues (x 1, x 2,, x ). Each resdue x s represeted by, k ()() x x k x x k123 1() x x1 X x Where, k k k (5) x X od X ; 0 x (1) I ths syste, arthetc operatos (addto, subtracto ad ultplcato) are perfored totally parallel o those totally depedet resdues. ( x, x,,)( x, y, y,) y x, y x, y , x y ;(,,) 1 2 A resdue uber ( x 1, x 2,, x ) ca be coverted back to ts weghted equvalet, by usg oe of the resdueto-bary coverso algorths (reverse coverso algorths), such as, the Chese Reder Theore (CRT), the Mxed-Radx Coverso (MRC), the ew CRT-I, the ew CRT-II, etc. [1], [2]. The CRT ca be pleeted parallel. However, t eeds a large odular adder, whch ca be very dffcult for hardware pleetato. Accordg to the CRT, a weghted uber X ca be calculated fro ts resdues (x 1, x 2,, x ) by the followg equato, X x N M 1 M M 1 Where, M, N M, 1,2,, 1 M deotes the ultplcatve verse of M odulo. The MRC does ot eed ay odular adder, but t s a sequetal algorth, whch akes t ot sutable for systes wth ore tha four odul wth the set. By usg the MRC, (2) (3) As ca be otced equato 5, the fal odular adder s reduced by oe odulo. Ths ca brg a great beeft whe the frst odulo s of the 2 k for, ad the ultplcato of the rest odul s of the (2 k 1) for. The ew CRT-II eve further reduces the sze of the fal odular adder. A resdue uber ( x 1, x 2,, x ) ca be coverted back to ts weghted equvalet X by the ew CRT- II by, X Z k () Y Z Z x k () x x Y x k () x x Where, k 1, k 1, k CONSIDERATIONS AND BASIS OF THE COMPARISONS The coparso betwee odul sets has bee doe based o te ad hardware requreets for pleetg the RCs ad RAUs. For the sake of a far coparso, the ut gate odel has bee used [17]. Before begg, the cosderatos that have bee take to accout ust be lsted. I order to ake the coparso process uversal, cost effectve ad hgh-speed desgs were ot cosdered. That s the reaso why soe te ad area values are dfferet fro those the orgal publcatos. The te ad area requreets (referred to as T ad A ) for each of the followg copoets are as follows, gates ad ultplexers: the te ad area requreets for a verter (NOT gate) were gored. Each 2- put ootoc gate (AND, OR, NAND, NOR): T AND = 1, A AND = 1. Each 2-put XOR, XNOR gate: T XOR = 2, A XOR = 2. A 2:1 ultplexer: T MUX = 2, A MUX = 3. (6) 86

3 Joural of Eergg Treds Coputg ad Iforato Sceces CIS Joural. All rghts reserved. Fg. 1: Structures of the utlzed odular adders (a) Geeral odular adder. (b) Modulo (2 1) adder. (c) Modulo (2 + 1) adder. Bary adders: A half adder (HA): T HA = 2, A HA = 3. A full adder (FA): T FA = 4, A FA = 7. Ay bary adder was cosdered as a carry propagate adder for -bt (CPA). A CPA wth ed-aroud carry (CPA-EAC) has a delay twce that of a regular CPA, ad the sae area. The structures of the used odular adders are show Fg. 1. The structure of the utlzed geeral odular adder cossts of two ( -bt) CPAs, a OR gate ad a 2:1 ultplexer of -bt [3]. The (2 1) odular adder (1 st copleet adder) s cosdered as a CPA- EAC [2]. The used (2 + 1) odular adder s the oe based o the seres ethod reported [18], whch cossts of two bary adders of (+1)-bt, a OR gate ad a 2:1 ultplexer of (+1) bt. The structure of ths adder s very slar to that of the geeral odular adder. However, we have used t, due to ts wde utlzato ad splcty. Bary ultplers: The used bary ultpler for - bt s a array ultpler [2]. The structures of the utlzed odular ultplers are show Fg. 2. The utlzed geeral odular ultpler s the parttoed-operad odulo- ultpler, whch s costructed based o the productparttog ethod preseted [2]. The utlzed odular ultplers are based o those reported [19]. Therefore, odulo (2 1) ultpler cossts of a bary ultpler of - bt ad odulo (2 1) adder. The utlzed odulo (2 + 1) ultpler cossts of a bary ultpler of ( +1) bt ad odulo (2 + 1) subtractor. The delay ad coplexty of crcular shftg or bts rearrageet was gored, sce t s just a rearrageet of wres, whch does ot vrtually preset ay addtoal delay or hardware cost. The last ssue to be observed s that the coputed delay the followg tables represets the delay of the crtcal chael (odulo), whereas the hardware requreets (coplexty) are those for all chaels together. The values of the least delays ad hardware requreets are bold ad uderled order to hghlght the. 4. MODULI SETS WITH 3N DYNAMIC RANGE The frst category of the studed odul sets s the oe that provdes a DR of 3-bts. The three sets that have bee vestgated are, {2 1, 2, 2 + 1} [4], {2 1 1, 2 1, 2 } [5] ad {2 1, 2, } [6]. They provde the followg DRs, (3 ), (3 1), ad (3 + 1), respectvely. However, we have cosdered that these sets belog to the 3- DR category. The ost faous ad wdely used odul set s the oe troduced [4]. Nevertheless, as aforeetoed before, odulo (2 + 1) represets a bottleeck to the whole syste. I order to get rd of the coplextes resulted fro usg ths odulo, the authors of [5] ad [6] substtuted t wth aother odulo of (2 k 1) for. Fg. 2: Structures of the utlzed odular ultplers (a) Geeral odular ultpler. (b) Modulo (2 1) ultpler. (c) Modulo (2 + 1) ultpler. Where, U ad L are the upper ad lower bts of the product (X Y), respectvely, (a) c 2 ad L <. (c) odulo (2 + 1) subtracts (L U). Table 1: Coparso betwee reverse coverters, odular adders ad odular ultplers for systes based o sets that provde DR = 3 87

4 Joural of Eergg Treds Coputg ad Iforato Sceces CIS Joural. All rghts reserved. Modul Set DR odd/ eve od # RC Crtcal Modular Adders Modular Multplers Delay Coplexty Chael Delay Coplexty Delay Coplexty {2 1, 2, 2 + 1} [4] 3 ay (2 + 1) {2 1 1, 2 1, 2 } [5] 3 1 ay (2 1) {2 1, 2, } [6] 3+1 ay (2 +1 1) Table 1 detals each of these sets, ts DR, possble values that ca be used ths set, the uber of ts odul, the crtcal chael that presets the logest delay, te ad hardware requreets for pleetg RCs, odular adders ad odular ultplers. a. Reverse Coverters Both RCs based o the sets [4] ad [5] are based o the CRT. Whereas [6], three archtectures of the RC were proposed, they deped o MRC, CRT ad CRT wth ROM. Durg our research ad coparsos, we have chose the secod archtecture that s based o the CRT. The coputed delay [5] s dfferet fro the reported oe, due to the reaso that t was calculated based o the aforeetoed cosderatos, (e.g. the carry look -ahead adder (CLA) was replaced by a CPA). Accordg to our study, the fastest coverter was [6] (for 3), whereas [4] was the oe wth the least cost (for 3). However, cosderg the balaced strategy, the RC based o [4] s the ost effcet oe. b. Resdue Arthetc Uts The RAU cludes odular adders, subtractors ad ultplers. Sce a subtractor ca be costructed by a adder ad a few verters, we have decded to copare just adders ad ultplers. The crtcal chaels of the studed sets are (2 + 1), (2 1) ad (2 +1 1) [4], [5] ad [6], respectvely. As show table 1, the superorty of the adders ad ultplers based o the set [5] s clear. Accordg to ths table, the crtcal chael aog all sets s (2 + 1); both arthetc copoets based o the set [4] have the logest delay. However, ths chael akes the RC spler. As etoed above, the RC based o the set [4] s the ost effcet oe. 5. MODULI SETS WITH 4N DYNAMIC RANGE I ths secto, we have vestgated fve sets; the three-odul sets {2 1, 2, } [7] ad {2 1, 2 + 1, } [8], ad the four-odul sets {2 1, 2, 2 + 1, }, {2 1, 2, 2 + 1, } [10] ad {2 /2 1, 2 /2 + 1, 2 + 1, } [20]. For splcty, we have deoted {2 1, 2, 2 + 1, } as [10-I] ad {2 1, 2, 2 + 1, } as [10-II]. [10-I] ad [20] ca oly be used wth eve values of, whereas, [10-II] ca oly be used wth odd oes. Ths presets a kd of ltato for usg such sets. All these sets have (4 + 1) DRs, except [8] that has 4 DR. Table 2 detals each of these sets, ts DR, possble values that ca be used ths set, the uber of ts odul, the crtcal chael that presets the logest delay, te ad hardware requreets for pleetg RCs, odular adders ad odular ultplers. a. Reverse Coverters The RC based o the set [7] depeds o the MRC. Whereas, the oe based o the set [8] depeds o the ew CRT-I. The RCs based o the sets [10-I] ad [10-II] have cobatoral archtectures. Each of the s parttoed to two ew parts. The frst part s coverted by usg the RC based o the set {2 1, 2, 2 + 1}. The the fal bary equvalet s calculated fro the result of the frst part ad the fourth resdue by usg the MRC. The four-odul set [20] depeds o the ew CRT-I. As show table 2, the RC based o the three-odul set [8] has the least delay ad coplexty. The reaso of ths superorty s obvous, the fal odular adder s of (2 k 1) for. The secod best set regardg the RC s [20]. Accordg to [20], the odul wth ths set have sple ultplcatve verses. Ths cosderably reduces the coplexty of the RC, whch s obvous table 2. O the other had, the delay ad coplexty of the RCs, based o both four-odul sets [10-I] ad [10-II], are the greatest. Table 2: Coparso betwee reverse coverters, odular adders ad odular ultplers for systes based o sets that provde DR = 4 Modul Set DR odd/ eve od # RC Crtcal Modular Adders Modular Multplers Delay Coplexty Chael Delay Coplexty Delay Coplexty {2 1, 2, } [7] 4+1 ay ( ) {2 1, 2 + 1, } [8] 4 ay ( ) {2 1, 2, 2 + 1, } [10-I] {2 1, 2, 2 + 1, } [10-II] {2 /2 1, 2 /2 + 1, 2 + 1, } [20] 4+1 eve /2( 2 3 4) (2 + 1) odd ( ) eve ( )

5 Joural of Eergg Treds Coputg ad Iforato Sceces CIS Joural. All rghts reserved. Table 3: Coparso betwee reverse coverters, odular adders ad odular ultplers for systes based o sets that provde DR = 5 Modul Set DR odd/ eve od # RC Crtcal Modular Adders Modular Multplers Delay Coplexty Chael Delay Coplexty Delay Coplexty {2, 2 2 1, } [9] 5 eve ( ) {2 1, 2, 2 + 1, } [11] {2 1, 2, 2 + 1, } [12-I] {2 1, 2, 2 + 1, 2 2 (+1)/2 + 1, (+1)/2 + 1} [13] {2 1, 2, 2 + 1, 2 1 1, } [15] {2, 2 /2 1, 2 /2 + 1, 2 + 1, } [16] 5 ay ( ) ay ( ) odd eve l ( )/ (2 + 2 (+1)/2 + 1) ( ) eve ( ) b. Resdue Arthetc Uts Sce [10-I] has the crtcal chael wth the least agtude, ts RAUs are the fastest. I addto, the ultpler based o ths set has the least area requreets. However, the least cost adder s the oe based o the threeodul set [7]. O the other had, usg four odul [20] dd ot help speedg up the RAUs, sce the agtude of the crtcal odulo s greater tha those of other sets. Accordg to our results, usg two sall odul of the for (2 /2 ± 1) has ot beeft the syste, sce the agtude of the crtcal chael reas bg. 6. MODULI SETS WITH 5N DYNAMIC RANGE I ths secto, we have vestgated sx sets; the three-odul set {2, 2 2 1, } [9], the four-odul sets {2 1, 2, 2 + 1, } [11] ad {2 1, 2, 2 + 1, } [12-I], ad the fve odul sets {2 1, 2, 2 + 1, 2 2 (+1)/2 + 1, (+1)/2 + 1} [13], {2 1, 2, 2 + 1, 2 1 1, } [15] ad {2, 2 /2 1, 2 /2 + 1, 2 + 1, } [16]. The sets [9], [15] ad [16] ca oly be used wth eve values of, whereas, [13] ca oly be used wth odd oes. Each of the sets [9], [11], [13] ad [15] provdes 5 DR, whereas [12- I] ad [16] provde (5 + 1) ad (5 1) DRs, respectvely. Table 3 detals each of these sets, ts DR, possble values that ca be used ths set, the uber of ts odul, the crtcal chael that presets the logest delay, te ad hardware requreets for pleetg RCs, odular adders ad odular ultplers. a. Reverse Coverters The RC based o the set [9] depeds o the CRT. Ths RC has a very sple ad effcet structure cosstg of a (4-bt) CSA-EAC ad odulo (2 4 1) adder. The set [11] depeds o the ew CRT-I. I case of cobg ts frst ad thrd odul, we get the set [9]. However, the RC based o [11] has greater delay ad coplexty tha that of [9]. The RC based o the set [12-I] depeds o the ew CRT-II. The eory-less RC based o the set [13] depeds o the CRT. The set [15] s parttoed to two ew parts. The frst part s coverted by usg the RC based o {2 1, 2, 2 + 1, }. The, the fal bary equvalet s calculated fro the result of the frst part ad the ffth resdue (odulo ( )) by the MRC. The delay ad coplexty of ths RC do ot oly deped o the value of, but also o two ore varables l ad ; where, l deotes the uber of the levels of a CSA tree wth ((/2) + 1) puts, ad = 4, 9 12 or 5 8 for = 6k 2, 6k or 6k + 2, respectvely [15]. The set [16] s also parttoed to two ew parts, {2 (2 /2 1)(2 /2 + 1)(2 + 1), }. The frst part s coverted by usg the ew CRT-I. The the result of the frst part s cobed wth the ffth resdue by usg the MRC. Accordg to table 3, the ost effcet RC, ters of delay ad coplexty, s the oe based o the three-odul set [9]. Cotrary, the ost effcet RC s the oe based o the fve-odul set [15]. b. Resdue Arthetc Uts As show table 3, the adder wth the least delay s the oe based o [13]. At frst glace, odulo ( ) adder should have less delay tha that of (2 + 2 (+1)/2 + 1), but t does ot for the followg reasos, the structure of ( ) adder depeds o a (2 k +1) adder, ts legth s ( +2) bts. Whereas the structure of (2 + 2 (+1)/2 + 1) adder depeds o a geeral odular oe, ts legth s (+1) bts. Sce the utlzed odulo (2 k +1) adder has a slar structure as that of the geeral odular oe, as etoed secto 3, thus, the delay of (2 + 2 (+1)/2 + 1) adder s less tha that of ( ). I case of usg aother structure of (2 k +1) adder durg our study, the delay of (2 + 2 (+1)/2 + 1) ay be loger tha that of ( ), however, the area cosupto wll be dfferet too. The cost of the adder based o [13] s the greatest oe. The least cost adder s the oe based o the fve-odul set [16]. Cocerg the ultplers, the oe based o the fveodul set [15] was superor over the other oes. I cotrast to the geeral odular adders, geeral odular ultplers have cosderably loger delay tha specal odular oes. 7. MODULI SETS WITH 6N DYNAMIC RANGE I ths secto, we have vestgated four sets, {2 1, 2 + 1, 2 2 2, } [14], {2 1, 2 2, 2 + 1, } [12-II], {2 + 1, 2 1, 2 2, } [21] ad {2 2+1, , 2 + 1, 2 1} [22] that cosst of four odul. 89

6 Joural of Eergg Treds Coputg ad Iforato Sceces CIS Joural. All rghts reserved. Table 4: Coparso betwee reverse coverters, odular adders ad odular ultplers for systes based o sets that provde DR = 6 Modul Set DR odd/ eve od # RC Crtcal Modular Adders Modular Multplers Delay Coplexty Chael Delay Coplexty Delay Coplexty {2 1, 2 + 1, 2 2 2, } [14] {2 1, 2 2, 2 + 1, } [12-II] 6+1 ay ( ) ay ( ) {2 + 1, 2 1, 2 2, } [21] 6+1 ay ( ) {2 2+1, , 2 + 1, 2 1} [22] 6+1 ay ( ) Durg our research we dd ot coe across a odul set of ore tha four odul that provdes a DR of 6-bts. All these sets provde (6 + 1) DR, except [12-II] that provdes 6 DR. Table 4 detals each of these sets, ts DR, possble values that ca be used ths set, the uber of ts odul, the crtcal chael that presets the logest delay, te ad hardware requreets for pleetg RCs, odular adders ad odular ultplers. a. Reverse Coverters The RC based o the set [14] depeds o the ew CRT-II. Accordg to [12], the authors of [14] used odul (2 2 2, ) order to have sple ultplcatve verses ad get rd of the ROM tables ad ultplcatos requred by the ew CRT-II. However, the use of these two odul results creasg the delay ad coplexty of the RC. The RC based o the set [12-II] depeds o the ew CRT-II. Accordg to the authors, ths set s referred to as a coverso fredly set, whch s evdet table 4. I [21], the four-odul set was parttoed to two ew odul subsets, ad the MRC was used for the coverso process. The RC [22] s based o the ew CRT-I. As show table 4, the fastest RCs are the oes based o [12-II] ad [22]. Moreover, the coplexty of ther RCs s very slar too. O the other had, the RC wth the logest delay ad largest coplexty s the oe based o [14]. b. Resdue Arthetc Uts The crtcal odulo the set [14] has alost the greatest agtude aog all other sets ad ts arthetc crcuts are geeral odular oes, therefore, ther delay ad cost are the greatest. Accordg to table 4, the fastest ad the least coplex odular adder s [21]. The fastest ultpler s oe based o [12-II] ad [22]. The least coplex ultpler s the oe based o [21]. The effcecy of usg geeral odul as [14] s clear. Its RC ad RAUs have the logest delays ad the hghest coplextes. 8. COMPARING ALL SETS TOGETHER I our prevous paper [23], we have preseted a detaled coparso betwee dfferet odul sets based o (delay coplexty) rato for each copoet accordg to three precse DRs (12 bt, 24 bt ad 60 bt). I ths paper, we copared dfferet odul sets based o the delay of each copoet. Thus, we ca study how the odul uber affects tg perforace of the overall syste. Moreover, our prevous paper [23], the lttle dffereces the DRs were gored, e.g. 3, 3 1 ad were cosdered as oe category (3). For exaple, for DR = 12 bt, the accurate DRs of [4], [5] ad [6] for = 4 are 12, 11 ad 13 bts, respectvely. Ths presets soe cosstecy durg the coparso. Therefore, ths paper, for ehacg ad coparg a ore farly way we have take to accout these lttle dffereces ad coputed the approxated delays for each set. However, ths dd ot chage the results ether affect the estated ost effcet ad effcet sets for each DR. The a cocetrato durg the coparsos was tg perforace. The delays of the RCs, odular adders ad odular ultplers for each of the three precse DRs, (12 -bt, 24-bt ad 60-bt) are show Fg. 3, 4 ad 5, respectvely. 3-odul sets 4-odul sets 5-odul set Fg. 3: The delays of each of the basc copoets based o the odul sets for DR = 12 bt (edu DR) 90

7 Joural of Eergg Treds Coputg ad Iforato Sceces CIS Joural. All rghts reserved. 3-odul sets 4-odul sets 5-odul set Fg. 4: The delays of each of the basc copoets based o the odul sets for DR = 24 bt (large DR) I each set, has bee chose order to provde the requred DR. For exaple, for DR = 12-bt, was (4, 3, 3, 2) for sets wth DR (3, 4, 5, 6). However, soe cases there was soe cosstecy (e.g. for sets wth DR = 5, = 3 provdes a DR greater tha the requred 12-bt). Nevertheless, as aforeetoed before, we have dealt wth ths ssue ad estated the approxate delay for the requred DR. Not all sets are llustrated the graphs, due to the reaso that soe of these sets ca oly be used wth eve values of ([9], [10-I], [20], [15] ad [16]) or odd values of ([10-II] ad [13]), whch does ot ft the chose value of order to acqure the requred DR. Fgure 3 shows the delays of the RCs, odular adders ad odular ultplers based o each of the sets [4], [5], [6], [7], [8], [10-II], [11], [12-I], [14], [12-II], [21], [22] ad [13], for DR = 12-bt (edu DR). I order to acqure ths DR, was chose (4, 3, 3, 2) for sets wth DR (3, 4, 5, 6), respectvely. The cossteces were dealt wth as etoed above. Accordg to Fg. 3, the adder wth least delay was the oe based o the fve-odul set [13]. However, the uexpected thg was that the secod fastest adder s the oe based o the three-odul set [5]. The fastest ultplers were the oes based o the three-odul set [5] ad fourodul set [22]. Whereas, the slowest oes were based o the three-odul set [8] ad the four-odul set [14]. Cocerg the RCs, the fastest oe s [6] ad the slowest oe s based o the four-odul set [10-II]. Cosderg the delay of the three copoets, t s obvous that the three-odul set [6] ad the four-odul set [22] are the ost effcet, sce the copoets based o these sets have relatvely sall delays. Thus, for DR = 12 bt, we see that the uber of odul does ot affect the overall speed of the syste. Fgure 4 shows the delays of the RCs, odular adders ad ultplers based o each of the sets [4], [5], [6], [7], [8], [10-I], [20], [11], [12-I], [14], [12-II], [21], [22] ad [13], for DR = 24-bt (large DR). I order to acqure ths DR, was chose (8, 6, 5, 4) for sets wth DR (3, 4, 5, 6), respectvely. The cossteces were dealt wth as etoed above. Fgure 4 shows that the delay treds of the basc copoets are slar to those DR = 12 bt. The fastest adder s the oe based o the fve-odul set [13]. The fourodul set [10-I] has the fastest ultpler ad oe of the best adders. However, ts RC s the worst. The fastest RC s the oe based o the three-odul set [6]. Cosderg the delay of the three copoets, we ca say that the ost effcet odul sets for ths DR = 24-bt are the three-odul sets [4] ad [6], ad the four-odul sets [12-II] ad [22], sce the copoets based o these sets have relatvely sall delays. Fgure 5 shows the delays of the RCs, odular adders ad ultplers based o each of the sets [4], [5], [6], [7], [8], [10-II], [9], [11], [12-I], [14], [12-II], [21], [22], [15] ad [16], for DR = 60-bt (very large DR). I order to acqure ths DR, was chose (20, 15, 12, 10) for sets wth DR (3, 4, 5, 6), respectvely. The cossteces were dealt wth as etoed above. Fgure 5 shows that the fastest adder ad ultpler are the oes based o the fve-odul set [15]. However, ther RC s the slowest oe. The fastest RC s the oe based o the three-odul set [6]. Ths set also has reasoably fast adder ad ultpler. 3-odul sets 4-odul sets 5-odul set Fg. 5: The delays of each of the basc copoets based o the odul sets for DR = 60 bt (very large DR) 91

8 Joural of Eergg Treds Coputg ad Iforato Sceces CIS Joural. All rghts reserved. Cosderg the delay of the three copoets, we ca say that the ost effcet odul sets for ths DR = 60-bt are the three-odul sets [4] ad [6], ad the four-odul sets [12-II] ad [22], as the copoets based o these sets have relatvely sall delays. Sce the ost copetet sets are ot the fve-odul oes for all three DRs, we coclude that the uber of odul does ot affect o the overall delay of the syste cosderg all ts copoets. There s o pot for choosg a fveodul set f the overall tg perforace wll be worse tha that based o three or four-odul sets. I our prevous paper [23], the ost effcet sets for edu ad large DRs, ters of (te coplexty) rato, were the three odul sets [5] ad [6]. I ths paper, we foud out that the ost effcet sets for the sae DRs, ters of delay, are the three odul set [4], [6] ad the four-odul set [12-II] ad [22]. Aga, [23], the ost copetet set wth the relatvely best (D C) ratos for very large DR was the four odul set [12-II]. I ths paper, the ost effcet sets for the sae DR are the three-odul sets [4], [6], ad the four odul sets [12-II] ad [22]. Aother portat ssue to cosder s that fveodul sets show better tg perforace edu ad large DRs tha that very large DR. Although ther RAUs were of the best oes for the very large DR, ther RCs were the worst. Accordg to our research, the uexpected ssue we have ascertaed s that fve-odul sets do ot show ay superorty over other sets takg to accout the three copoets of RNS (odular adders, odular ultplers ad RCs). 9. CONCLUSIONS AND FUTURE WORK Ths paper preseted a detaled study o the effect of odul uber o the overall speed of the RNS. The study cossts of two a parts, the frst oe copares betwee dfferet odul sets that provde sae DR (3, 4, 5 ad 6). Te ad hardware requreets of the RCs ad RAUs based o each set are preseted ad descrbed a uversal way usg ut gate odel. The secod part s devoted for coparg tg perforace of the aforeetoed copoets based o the studed sets, for three specfc dyac rages; edu (12 - bts), large (24 -bts) ad very large (60 -bt). Ths part of the study was carred out order to fd-out the ost effcet set for each dyac rage regardless the uber of ts odul, ad the dyac rage t was tally suggested for. Accordg to the cooly kow ssue, whch says, as the odul uber creases the speed of the RAUs creases ad the RC becoes slower ad ore coplex, we expected to have a kd of lear relatoshp betwee the uber of odul ad the delay of the aforeetoed copoets. However, the results our study lead to the fact that the odul uber does ot have such a essetal role the overall speed of the RNS, sce tg perforace of the copoets based o fve-odul sets were ot better tha those based o three ad four-odul sets. Our ext step wll be dedcated for studyg the effect of the odul uber o the hardware coplexty of the overall syste RNS. ACKNOWLEDGMENTS Ths research was supported by the Mstry of Idustry ad Trade of the Czech Republc uder the MPO ČR č. FR-TI3/485 project ad Prospectve applcatos of ew sesor techologes ad crcuts for processg of sesor sgals, No.FEKT-S project. REFERENCES [1] M. Lu, Arthetc ad Logc Coputer Systes. Joh Wley & Sos, Ic., [2] A. Ood ad B. Prekuar, Resdue Nuber Systes: Theory ad Ipleetato. Iperal College Press, [3] K. Nav, A. S. Molahosse ad M. Esaeldoust, How to teach resdue uber syste to coputer scetsts ad egeers, IEEE Tras. o Educato, vol. 54, pp , February [4] S. J, Pestrak, A hgh-speed realzato of a resdue to bary uber syste coverter, IEEE Tras. o Crcuts ad Systes-II: Aalog ad Dgtal Sgal Processg, vol. 42, pp , October [5] W. Wag, M. N. S. Sway, M. O. Ahad ad Y. Wag, A hgh-speed resdue-to-bary coverter for three-odul (2 k, 2 k 1, 2 k 1 1) RNS ad a schee for ts VLSI pleetato, IEEE Tras. o Crcuts ad Systes-II: Aalog ad Dgtal Sgal Processg, vol. 47, pp , Deceber [6] P. V. A. Moha, RNS-to-bary coverter for a ew three-odul set {2 +1 1, 2, 2 1}, IEEE Tras. o Crcuts ad Systes-II: Express Brefs, vol. 54, pp , Septeber [7] A. S. Molahosse, K. Nav ad M. K. Rafsaja, A ew resdue to bary coverter based o xed-radx coverso, 3rd Iteratoal Coferece o ICTTA, pp. 1 6, Aprl [8] W. Wag, M. N. S. Sway, M. O. Ahad ad Y. Wag, A study of the resdue-to-bary coverters for the three-odul sets, IEEE Tras. o Crcuts ad Systes-I: Fudaetal Theory ad Applcatos, vol. 50, pp , February [9] A. Harr, K. Nav ad R. Rastegar, A ew hgh dyac rage odul set wth effcet reverse coverter, Coputers & Matheatcs wth Applcatos Joural, vol. 55, pp , February [10] P. V. A. Moha ad A. B. Prekuar, RNS-to-bary coverters for two four-odul sets {2 1, 2, 2 + 1, } ad {2 1, 2, 2 + 1, }, IEEE Tras. o Crcuts ad Systes-I: Regular Papers, vol. 54, pp , Jue

9 Joural of Eergg Treds Coputg ad Iforato Sceces CIS Joural. All rghts reserved. [11] B. Cao, C. H. Chag ad T. Srkatha, A effcet reverse coverter for the 4-odul set {2 1, 2, 2 + 1, }, IEEE Tras. o Crcuts ad Systes-I: Fudaetal Theory ad Applcatos, vol. 50, pp , October [21] L. Sousa ad S. Atão, MRC-based RNS reverse coverters for the four-odul sets {2 + 1, 2 1, 2, } ad {2 + 1, 2 1, 2 2, }, IEEE Tras. o Crcuts ad Systes II: Express Brefs, vol. 59, pp , Aprl [12] A. S. Molahosse, K. Nav, C. Dadkhah, O. Kavehe ad S. Tarch, Effcet reverse coverter desgs for the ew 4-odul sets {2 1, 2, 2 + 1, } ad {2 1, 2 + 1, 2 2, } based o ew CRTs, IEEE Tras. o Crcuts ad Systes-I: Regular Papers, vol. 57, pp , Aprl [13] A. A. Hasat, VLSI pleetato of ew arthetc resdue to bary decoders, IEEE Tras. o VLSI Systes, vol. 13, pp , Jauary [14] W. Zhag ad P. Sy, A effcet desg of resdue to bary coverter for four odul set (2 1, 2 + 1, 2 2 2, ) based o ew CRT-II, Iforato Sceces Joural, vol. 178, pp , March [15] B. Cao, C. H. Chag ad T. Srkatha, A resdue-tobary coverter for a ew fve-odul set, IEEE Tras. o Crcuts ad Systes-I: Regular Papers, vol. 54, pp , May [16] A. S. Molahosse, C. Dadkhah ad K. Nav, A ew fve-odul set for effcet hardware pleetato of the reverse coverter, IEICE Electrocs Express, vol. 6, pp , July [17] A. S. Molahosse, K. Nav, O. Hashepour ad A. Jalal, A effcet archtecture for desgg reverse coverters based o a geeral three-odul set, Elsever Joural of Systes Archtecture, vol. 54, pp , October [18] S. Tarch ad K. Nav, Iproved odulo 2 +1 adder desg, World Acadey of Scece, Egeerg ad Techology, vol. 39, July [19] A. A. Hasat, New eoryless, od (2 ± 1) resdue ultpler, Electrocs Letters, vol. 28, pp , Jauary [20] A. S. Molahosse, F. Teyour ad K. Nav, A ew four-odulus RNS to bary coverter, Proc. of 2010 IEEE Iteratoal Syposu o Crcuts ad Systes (ISCAS), pp , Jue [22] R. Alabada, A. Alabada, A. Bolhasa, S. Z. Hosse ad A. Golsorkhtabar, A ovel hgh dyac rage 4-odule set {2 2+1, , 2 + 1, 2 1} wth effcet reverse coverter ad revew provg odular ultplcato s dyac rage wth ths odule set, Iteratoal Coferece o Coputer Coucato ad Iforatcs (ICCCI), pp. 1 6, Jauary [23] D. Youes ad P. Steffa, A coparatve study o dfferet odul sets resdue uber syste, Iteratoal Coferece o Coputer Systes ad Idustral Iforatcs (ICCSII), pp. 1 6, Deceber AUTHOR PROFILES D. Youes receved her B.Sc. ad M.Sc. degrees Coputer ad Autoatc Cotrol Egeerg fro Tshree Uversty, Lataka, Syra Curretly, she s a Ph.D. studet at the Faculty of Electrcal Egeerg ad Coucato at Bro Uversty of Techology, Bro, Czech Republc. Her a research terests s resdue uber syste based buldg blocks for applcatos dgtal sgal processg. She s a part te structor the Departet of Mcroelectrocs. She s author ad co-author of several research atoal ad teratoal coferece ad joural papers. P. Steffa receved hs B.Sc. ad M.Sc. degrees Mcroelectrocs fro Bro Uversty of Techology, Bro, Czech Republc 2000 ad 2003, respectvely. He also receved hs Ph.D. degree Optzato of croelectroc systes for sart crosystes fro Bro Uversty of Techology Sce 2011, he s Assocate Professor at the Departet of Mcroelectrocs, Faculty of Electrcal Egeerg ad Coucato, Bro Uversty of Techology. Hs research terests are techcal area easureet, processg ad dgtzato of sesors sgals. Curretly, he s teachg courses of dgtal crcuts ad croprocessor techology. He s author ad co-author of over tha 100 research atoal ad teratoal coferece ad joural papers. 93