Effect of Wire Delay on the Design of Prefix Adders in Deep-Submicron Technology

Transcription

1 Effect of Wre Deay on the Desgn of Prefx Adders n Deep-Submcron Technoogy Zhun Huang and Moš D. Ercegovac Computer Scence Department Unversty of Caforna at Los Angees Los Angees, CA 995 {zhuang, mos}@cs.uca.edu Abstract Ths paper nvestgates the re deay effect on the desgn of prefx adders hen the technoogy moves from 25nm to 7nm. The smuaton s based on parameters from TRS 97 and uses an anaytca re deay mode that consders the fanout effect and the dstrbutve nature of re capactance and resstance. Smuaton resuts sho that re deay exceeds ogc deay and domnates the crtca path deay of prefx adders n many cases. For a gven technoogy, the re deay contrbuton ncreases steady as the adder dth ncreases. As the feature sze decreases, hoever, the re deay contrbuton decreases soy. The smuaton data aso mpy that there s tte need to consder re resstance. On the other hand, the effect of re coupng capactance pays a crtca roe n prefx adders performance.. Introducton Among varous bnary adder archtectures, a arge famy of prefx adders s partcuary attractve because a of them have mnmum ogc depth and very effcent mpementatons [8][9][4] (Brent-Kung prefx adder [2] s an excepton snce t aos the ogc depth of the structure to ncrease. An mportant probem n prefx adder desgn s ho to further mprove performance snce the prefx structure has aready had the mnmum ogc depth. Among those factors affectng the performance, the fanout and re ength are often the determnstc factors. As VLSI technoogy moves nto the deep sub-mcron (DSM doman, the rng probem becomes even more sgnfcant and, n many cases, domnates n both area and deay optmzaton [5][6][3]. Recenty, Choe and Sartzander [3] shoed that arger mutpers oud suffer from re deays f they ere unformy scaed don nto deep sub-mcron regon. In the technoogy roadmap gven by the semconductor ndustry, hoever, the meta aspect rato keeps ncreasng to avod the probems n unform scang. It s nterestng and meanngfu to ook nto the re effects on arthmetc crcuts under practca technoogy scang parameters. Based on technoogy parameters on the roadmap TRS 97 [3], ths paper nvestgates the re deay effect on the performance of prefx adders hen the technoogy moves from 25nm to 7nm. Three structures have been seected to cover the range of the prefx adder famy th mnmum ogc depth: Ladner-Fsher structure [], Knoes structure [8] and Kogge-Stone structure [9]. The range of datapath dth under consderaton s from 6 bts to 28 bts. The smuaton uses an anaytca re deay mode that consders the fanout effect and the dstrbutve nature of re capactance and resstance. The rest of the paper s organzed as foos. Secton 2 formuates the genera deay cacuaton n prefx adder structures. Secton 3 ntroduces an anaytca deay estmaton mode used for the re deay anayss. Secton 4 shos the smuaton resuts and the mpcatons. Concusons are gven n Secton Crtca path deay n prefx adders To key propertes, assocatvty and dempotency, make the prefx formuaton of addton very fexbe. Even for prefx adders th the same mnmum ogc depth, dfferent structures can be deveoped for varous area and speed requrements [8][4]. In ths study, three prefx structures th the same mnmum depth are chosen: Ladner-Fsher adder, Knoes adder and Kogge- Stone adder. They spread over the range of area-speed tradeoffs n prefx adder desgn [8]. These adder archtectures are shon n Fgure, 2 and 3, respectvey. A graphs are dran n the tradtona prefx descrpton stye: back nodes depct nodes performng computaton ogc and hte nodes represent nodes th no ogc or ony some buffers. The functons of computaton nodes are shon n Fgure 4.

2 Fgure. Ladner-Fsher adder To get accurate re deay estmaton, the detaed ayout s usuay requred. Fortunatey, the graph representatons of the above prefx structures can be drecty mapped nto a ayout topoogy under structured custom desgn stye [8][4], hch makes re deay estmaton th reatvey hgh accuracy possbe. From the dot graphs, t can be seen that each crtca path goes from the top-rght corner to the bottom-eft corner. The crtca path can be dvded nto the severa ogc/re stages. Each ogc/re stage can be abstracted as some ogc drvng a dstrbuted RC chan modeng rng effect th spaced capactve oads, as shon n Fgure 5: V n C Tnt Ro R W C W C L R W 2 CW 2 C L 2 R W CW V out CL Fgure 5. One ogc/re stage a b Fgure 2. Knoes adder Fgure 3. Kogge-Stone adder ( g p ( + g, p ( g, p ( g, p ( g, p, + + ab g = g + p g g = p + = a b p = p p + c = g g Fgure 4. Back computaton nodes s p s c = p c For smpcty, e assume equa nput bt arrva tmes n the foong anayss. Under unt gate-deay mode th no consderaton of rng and fanout, these structures oud have the same mnmum ogc depth and hence have the same deay. Such a deay estmaton mode, hoever, can ony be usefu for coarse evauaton. oadays, the mpact of arge fanout and more mportanty, the mpact of re deay cannot be negected. The ogc has nput capactance and ntrnsc deay T nt C, output resstance Ro. The effect of dffuson capactance n the ogc output, C d, has been ncuded n R are the capactance and resstance T nt. ( W C and ( W of a re segment hch shoud be treated as a dstrbuted RC ne. ( C s the oad capactance at snk node. L The tota deay s the deay sum of each ogc/re stage. To carfy, e focus on the deay cacuaton of one ogc/re stage n Fgure 5. The deay has to parts: ogc deay and re deay,.e., T = T + T. The ogc tota ogc re deay s defned as the deay th zero re effects. By assumng a ( C W and ( R W to be zero, the ogc deay can be easy cacuated through a umped RC mode: T ogc = T nt +.7Ro ( CL ( = Tradtonay, the propagaton deay s defned as the tme perod from 5% pont of the nput to 5% pont of the output, hch generates a.7 factor []. The re deay ncudes the effects of dstrbuted ( C W and ( R W. The re capactance probem s further compcated f the re frngng capactance and coupng capactance are consdered. Before t s ustfed to make some smpfcaton, a possbe effects shoud be taken nto account. 3. Wre deay mode Emore [7] shoed that the foong cosed-form tme-constant expresson as a frst-order approxmaton s vad for a non-branched -stage RC chan n Fgure 6: τ = = R C C R (2 = = = =

3 V n R C R 2 C 2 R Fgure 6. RC chan To some extent, the RC chan n Fgure 6 s smar to the dstrbuted RC chan n our study. Accordng to [], a good approxmaton can st be made by combnng eghted resstve and capactve terms n the same ay as n Emore deay mode. The eght s decded as foos. Under step votage exctaton, the propagaton deay n a dstrbuted RC netork and the deay n a umped RC netork, are.4rc and.7rc, respectvey. Thus, the resstve and capactve terms are eghted by.4 hen they are dstrbuted and by.7 hen they are umped. o a re deay estmaton equaton can be formed for the re part n Fgure 5: T re =.7R ( C + o = W C R ( ( R ( ( ( ( W.4 CW +.7 CL +.7 CL + C (3 W = = + The frst tem n Equaton (3 s the pure effect of re capactance, hch s a umped RC mode. The second tem represents the effect of re resstance, hch s treated as a dstrbuted RC mode. The frst tem gros neary th the re ength he the second tem gros quadratcay th the re ength. ote that the re ength s represented by the number of re segments n Equaton (3. In many cases of prefx adder structures, each oad n one ogc/re stage has the same vaue and s unformy spaced, hch can hep smpfy Equaton (3. 4. Smuaton resuts Prefx adders are organzed n three parts: preprocessng part generatng g and p, prefx computaton part generatng c and post-processng part generatng s. Dfferent prefx structures ony dffer n the computaton parts. Thus, our smuaton ony cacuates the deay of the prefx computaton part. To make the smuaton data more practca and meanngfu, e do not use overy smpfed scang modes, such as unform scang and genera scang [2]. Instead, our smuaton uses re and devce parameters that are derved from TRS 97 [3]. The dervaton has been conducted by Cong and Pan [5][6]. We drecty borroed ther resuts for our purpose. The reated parameters are sted n Tabe. The devce here s a buffer, made up of to cascaded nverters th stage rato of :5. The devce parameters are obtaned through HSPICE smuaton. The re capactances are obtaned usng a 3D capactance sover FASTCAP [] for a re of 2 mnmum dth and th to parae neghborng res of 2 mnmum spacng. V out C Tabe : Devce/Wre parameters [5] Tech Tnt Ro Cg Rs Ca Cf Cx Tech(µm: technoogy feature sze; Tnt(ps: ntrnsc devce deay; Ro(kΩ: output resstance of a mnmum devce; Cg(fF: nput capactance of a mnmum devce; Rs(Ω/ : sheet resstance of re; Ca(fF/µm 2 : unt area capactance of re; Cf(fF/µm: unt frngng capactance of re; Cx(fF/µm: unt coupng capactance of re In our study, each ogc devce on the crtca path mpements + g = g + p g (Fgure 4. We assume each devce s physcay composed of one AD-OR gate and one nverter of 5 mnmum sze. The ntrnsc + deay from g to g n such a devce s approxmated as 7 6Tnt. The oad capactance s the AD-nput capactance n an AD-OR gate, hch s about.5cg. The tota re capactance of a re of dth W and of ength L can be estmated as [4]: C = C W L + C L + C L (4 a f The custom desgner can change the re spacng to reduce the coupng effect. For smpcty, e assume each channe re to be of 2 mnmum dth and of 2 mnmum spacng th neghborng res f exst. Among the three structures consdered, Ladner-Fsher structure has ony one re each channe and hence no coupng effect he both Knoes and Kogge-Stone structures exhbt coupng effects. Because of the array topoogy of prefx adder structures, re unts can be defned to smpfy the smuaton. A mnmum re unt s defned as a transverse channe re segment beteen to adacent bts. In our study, the re unt s of ength 6λ and of dth 4λ, here λ s tradtonay defned as haf of the technoogy feature sze. Based on the above anaytca mode and TRS 97 technoogy parameters, the effects of re resstance and capactance on the prefx adders have been smuated and the resuts are dscussed n the foong. 4. Wre deay contrbuton In DSM doman, re deay pays an mportant roe n crcuts performance. Fgure 7 shos the smuaton x

4 resuts on re deay contrbuton to the tota crtca path deay. In many case, re deay exceeds ogc deay and domnates crtca path deay. As the feature sze decreases, hoever, the re deay contrbuton decreases soy! When the feature sze becomes ess than.5um, the ogc deay and re deay of the crtca path n prefx adders scae fary e th the technoogy advancement. There are to maor reasons for such good scang. Frst, the technoogy roadmap has made great effect to reduce the rng effect hen the technoogy scaes don. Second, the argest re ength n these adders (e.g.,.28mm n 28-bt prefx adders s st much ess than a typca goba re ength (e.g., 2cm and the quadratc part n re deay Equaton (3 s very sma. WreDeay / TotaDeay Wre Deay Contrbuton.25u.8u.5u.3u.u.7u Technoogy LF(6b LF(32b LF(64b LF(28b KS(6b KS(32b KS(64b KS(28b Fgure 7. Wre deay contrbuton For a gven technoogy, the re deay contrbuton ncreases steady as the adder dth ncreases. Such a steady ncrease mpes that more effort shoud be put on re desgn as datapath becomes der. A typca effort exampe s to appy sophstcated bufferng technque. The re deay mpact on Ladner-Fsher adder s much ess than the mpact on Kogge-Stone adder. Ths s because Ladner-Fsher adder has the east parae res n the channes and hence the east rng effect. In fact, Ladner- Fsher adder and Kogge-Stone adder are to extrema n respect to re deay contrbuton. 4.2 The mpact of re resstance As shon n re deay Equaton (3, the second part that represents re resstance mpact ncreases quadratcay th the re ength. In the desgn of goba res, such as cock and bus, ths quadratc part has posed a very dffcut probem to system desgners. In prefx adder desgn, hoever, our smuaton shos that the re resstance deay s ess than % of the tota re deay under any stuaton. Tabe 2 sts the smuaton resut of such re resstance deay percentages n Ladner-Fsher adder, hch has the argest percentages among the three prefx structures under study. Tabe 2: re-resstance deay / re deay (%.25µ.8µ.5µ.3µ.µ.7µ 6bt bt bt bt Therefore, there s tte need to consder the effect of re resstance n prefx adder desgn and a umped deay mode s suffcenty accurate. By negectng the quadratc part n the re deay equaton (3, the tota deay can be expressed n a smpe form: Ttota = Tnt +.7Ro ( ( CL + ( CW (5 = 4.3 Comparson of three prefx adder structures The comparsons of the overa deay of three prefx structures are shon n Fgure 8 and 9. In dea cases here there s no re coupng capactance, Ladner- Fsher structure oud not be a good choce n terms of deay. In ths case th no coupng effect, the re deays of three structures are about the same and the dfference n tota deays comes from the ogc deays. Because the ogc gates n Ladner-Fsher structure have the argest fanout oads, Ladner-Fsher adder exhbts the argest ogc deay and aso the argest tota deay. Deay (ns Deay Comparson Wth o Cx effect.25u.8u.5u.3u.u.7u Technoogy LF(6b LF(64b Kno(6b Kno(64b KS(6b KS(64b LF(32b LF(28b Kno(32b Kno(28b KS(32b KS(28b Fgure 8. Deay comparson thout coupng capactance effect (dea case

5 Deay (ns Deay Comparson Wth Cx Effect.25u.8u.5u.3u.u.7u LF(6b LF(64b Kno(6b Kno(64b KS(6b KS(64b Technoogy Fgure 9. Deay comparson th coupng capactance effect LF(32b LF(28b Kno(32b Kno(28b KS(32b KS(28b Hoever, the dea case s sedom acceptabe f the chp area and cost are consdered. When the coupng effect s taken nto account, the concuson s reversed: Ladner-Fsher structure s the best choce under a technooges and datapath dth cases. It has about 2% to 3% ess deay. Thus, the effect of re coupng capactance pays a crtca roe n the desgn choce. To acheve the best performance, desgners coud keep re spacng arge enough to negect coupng effect. Fgure 8 and 9 aso sho that the deay of Knoes structure and that of Kogge-Stone structure are amost the same n a crcumstances. Because Knoes structure has much oer re compexty, t s preferabe hen desgners need to make choce beteen Knoes structure and Kogge-Stone structure. 5. Concusons The re deay effect on the desgn of prefx adders n deep-submcron technoogy has been studed. The smuaton s based on practca technoogy parameters from TRS 97 and uses an anaytca re deay mode that consders the fanout effect and the dstrbutve nature of re capactance and resstance. Smuaton resuts sho that re deay exceeds ogc deay and domnate the crtca path deay of prefx adders n many cases. For a gven technoogy, the re deay contrbuton ncreases steady as the adder dth ncreases. As the feature sze decreases, hoever, the re deay contrbuton decreases soy. The smuaton data aso mpy that there s no need to consder re resstance n prefx adder desgn and hence a umped deay mode s suffcenty accurate as a frst-order estmaton. On the other hand, the effect of re coupng capactance pays a crtca roe n prefx adder performance. Wthout coupng effect, Knoes and Kogge-Stone structures are better. When the coupng effect s taken nto account, Ladner- Fsher structure s the best choce under a technooges and datapath dths. Acknoedgment: Ths research has been supported n part by the SF Grant MIP References [] H.B. Bakogu, Crcuts, Interconnectons, and Packagng for VLSI, Addson-Wesey Pubshng Company, 99. [2] R.P.Brent and H.T. Kung, A reguar ayout for parae adders, IEEE Trans. Computers, Vo. C-3, o. 3, pp , Mar [3] G. Choe, E.E. Sartzander, Interconnecton effects n fast mutpers, Proc. 33rd Asomar Conf. on Sgnas, Systems, and Computers, pp , 999. [4] J. Cong, L. He, C.-K. Koh and Z. Pan, Goba nterconnect sze and spacng th consderaton of coupng capactance, Proc. Int. Conf. on Computer Aded Desgn, pp , 997. [5] J. Cong, Chaenges and opportuntes for desgn nnovatons n nanometer technooges, n SRC Workng Paper, Dec [6] J. Cong and D.Z. Pan, Interconnect deay estmaton modes for synthess and desgn pannng, Proc. Asa and South Pacfc Desgn Automaton Conf., pp.97-, 999. [7] E. Emore, The transent response of damped near netorks th partcuar regard to deband ampfers, Journa of Apped Physcs, pp.55-63, Jan [8] S. Knoes, A famy of adders, Proc. 4th IEEE Symp. on Computer Arthmetc, pp.3-34, 999. [9] P.M Kogge and H.S. Stone, A parae agorthm for the effcent souton of a genera cass of recurrence equatons, IEEE Trans. Computers, Vo. C-22, o. 8, pp , Aug [] R.E. Ladner and M.J. Fscher, Parae prefx computaton, Journa of ACM, Vo. 27, o. 4, pp , Oct. 98. [] K. abors and J. Whte, Fastcap: A mutpoe acceerated 3-D capactance extracton program, IEEE Trans. Computer-Aded Desgn of Integrated Crcuts and Systems, pp , ov. 99. [2] J. Rabaey, Dgta Integrated Crcuts: A Desgn Perspectve, Prentce-Ha, 996. [3] Semconductor Industry Assocaton, atona Technoogy Roadmap for Semconductors, 997. [4] R. Zmmermann, Bnary Adder Archtectures for Ce- Based VLSI and ther Synthess, Ph.D. thess, Hartung- Gorre Verag Konstanz, 998.