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Dy Vrt Trc r D Crcuts K-C Wu, C-T Hs, S-C C Dprtt CS, Nt Ts Hu Uvrsty, Hscu, Tw Ax@tuc.cs.tu.u.tw, s@tuc.cs.tu.u.tw, scc@cs.tu.u.tw ABSTRACT Fctrs y vrt, suc s prcss vrt s cts, y cus uctur cp t vt t pr-spc t cstrt. I ts ppr, w prps v r-sytss tcqu t trt y vrt r crcuts. Nt tt t scs s crtc pts r zr; y y t t ts zr-sc s w wrs t prrc crcut. Our sc s t crs t scs s t crtc r y pp rut uxry su-crcut t t r crcut. T uxry su-crcut c cus crtc pts t c s pts r prcpt pts [7] s s t prv t cpty y vrt trc. Exprt rsuts r vry cur. 1. Itruct Crcut y vc tcs cs crsy sstv t prcss vrt s [1][2]. Ts ctrs wu cus crcut s prrc t uctut, t wrst cs, t vt y ccur. Espcy r prrc ss, t crtc rs r t pt crcut sty. As rsut, t y vrt pr cs crtc ssu r crcuts. I ts ppr, w prps r-sytss t t trt y vrt r crcut. T scs s crtc pts r zr; y y t t ts zr-sc s w wrs t prrc crcut. T r y vrt trc [3] s rut y t ccpt scs. T utrs prps r-sytss tcqu wc crprts Trp Mu Rucy (TMR) structur t crcut s tt t crcut c trt v r y ucrtty. Tr rsuts sw tt y 40% r vr, crt r vrt trc c cv r sttc CMOS crcuts. U sttc crcuts, yc crcuts prt tw pss: t pr-cr ps t vut ps. T v t rc pr, crcuts rqur c Fur 1: A xp strt y vrt trc r crcut. ss, xcpt prry puts, t v y rs trsts ur t vut ps. Du t t rs-trst-y prprty, y vrt trc crcuts s uc sr t ccps t tt sttc crcuts. I tr wrs, rcty ppy t s t s [3] t crcuts y v ucssry r r. Our sc y vrt trc s t c (trt) crcut wt rut uxry su-crcut, wc s ustrt t w xp. A crcut Fur 1 pts c uct = (++c)+ = ++c+. Fr y trc, w cstruct uxry crcut pt F 2 = (+) = + rt w utput = +F 2. Bcus t -st pprty cvrs tt F 2 (.., F 2 ), t w uct = +F 2 s tc t t r uct. W w sw tt pp crcut s t c t uctty t r crcut ut s t ct y trc. Csr put pttr (,, c,, ) = (1, 0, 0, 1, 0) wc ucs trsts prpt crtc (t) pts t. Sc t F 2 v y rs trsts t OR t, wcvr rs s rrvs rr w t t OR t s utput. I tr wrs, t rr rrv tr r F 2 trs t utput vu st t t. Dy vrt trc s csquty cv cus t rrv ss w t uc t w crcut s y. I t t prps w structur r crcuts, ts ppr s rvs v trs wc w us t crry ut sr t ts r [3]. W s sw tt y vrt trc c pp t t tr s prry utputs t cv ttr rsuts. F 2

T xprts sw tt t r vr ur t s ss t tt r [3]. As, w v prr Mt-Cr xprts. T y t s v s prty sty uct sr t [6]. T rsuts sw tt ut 77% sps t r crcut C6288 c t crt y rqurt; wvr, ut 99% sps t crrsp r-sytsz crcut c stsy t s rqurt. W wu t t tt tr v svr stus [4][5][7] ttpt t ptz t sttstc t rsuts y t sz tcqus. Ts ts rqur sttstc s t v r ts. W t tt sttstc struts ts sttstc crrts ts r v prcsy, ts [4][5][7] c r ct t rut structurs s scr y ts ppr. O t tr, ts ppr s t ssu y sttstc t s. Trr, ur t w ttr w ccurt sttstc s r t v. Fr xp, vry w cps r uctur r w tcs, suc s 90 65, s t s cut t tr r t vry sttstc s r w tcs. I t t prcss vrt, y vrt u t s ssus, suc s IR rp, s t sy t. 2. Dy Vrt Trc Dupx D Systs Lt crcut ur csrt r y vrt trc. A upx syst s sw Fur 2, wc cssts t r crcut, ts upct, OR t t t utputs s ts puts. Tr r tw prprts r upx syst. Frst, t prrs t s uct s t r crcut. Scy, trst trv pt s trvs t rrr pt. Sc crcut s y 0-t-1 trsts, t OR t s utput trsts r 0 t 1 w t rr trst rrvs. Ay y crs wc s pstp tr r w t ct t vtu t rsut. Trr, c (xcpt t OR t) t Or crcut Dupct crcut F 2 Fur 2: A upx syst. upx syst s t sc. St, tr s r t 100% r vr r upx syst, t upx syst prctc. I t rr ts ppr, w w xp w t ruc w t v r y vrt trc. Lt us rst t r y vrt trc. T r y vrt trc c qut wt t sst sc ts/wrs crcut. A crcut s s v t y trc [3] t sc c t/wr s t st t. Gv y trc vu t crcut, ur ctv s t r-sytsz t crcut s tt vry t/wr t w crcut c trt t st y vrt t. T sc c t/wr upx syst s t (., t = ), wc s vr-prtctv r t y vrt prs. I r, y trc 10%~20% t r crcut y s suct r ur csrt prcss vrt s cts. 3. R-sytss r Dy Vrt Trc T ccps v y trc vu t wtut t uc r vr, prctc sc rt r upx syst s prps. Our r-sytss stps r s ws. (1) B wt upx syst Fur 2. (2) Rv y s wrs r r ruct w t t trc vu. W w scuss w t prr wr rv ct. Assu t r crcut pts t c uct, t rut uxry crcut pts F 2, t ctv utput s = +F 2. I t -st F 2 s su-st tt (.., F 2 ), w c prsrv t r uctty (= +F 2 = ). W tr r y pss B ucts wc stsy F 2, w y csr wr rv t ruc t -st F 2. Br prst ur trs, w scr wt wrs c rv t t F 2 t w. L 1: A rct put wrs t OR ts r rut c sutusy rv. Pr: Ott. Lt t y trc vu t. A pt s c t -crtc pt ts y s rtr t t t rqurt us t. I tr wrs, t y crt t -crtc pt w cus t pt s y t xc t

p x y q s t r w 5 w 6 w 3 w 4 w 2 w 1 p 1 p 2 w Fur 4: N s t AND-cvr t -crtc pts p 1 p 2. t -tr: {,, } t -s put: w 1 ( -> ), w 2 ( -> ), w 3 ( -> ), w 4 (r -> ), w 5 (p -> ), w 6 (q -> ) Fur 3: T r crcut. t rqurt. I t, w sy tt s t -crtc t s t -crtc pt. O c s tt t sc t -crtc s sr t t. Csr t xp Fur 3, wr t y c t s 1. Supps t y trc t s 1 t t rqurt r s 6. I ts xp, t t rqurt s qu t t t t st pt. Pt {s------} s t -crtc pt cus ts y s 6, rtr t r - t (6-1=5). I ct, t pts Fur 3 r t -crtc pts. N s t -crtc cus t s t -crtc pt. Bss, t s {,,,,,,,, } r t -crtc s. Nt tt t scs ts t -crtc s r 0, sr t t. A s t t -tr t s t -crtc t -crtc pts t prry utput ust pss tru t. A wr 1 -> 2 s t -s put 1 s t t -crtc ut 2 s t -crtc. I t s xp, s {,, } r t -trs cus t -crtc pts t t prry utput ust pss tru ts s. Wr w 1 ( -> ) s t -s put cus s t t -crtc ut s. Sry, wrs {w 2, w 3, w 4, w 5, w 6 } r s t -s puts. Tr 1 [3]: A t -s put wr w t OR t -tr c rv (rpc y -ctr vu,.. c 0) wtut vt t rqurt t y trc. Pr: Ott. T t r crcut Fur 3 s xp. A upx syst c cstruct y tw upcts ( ) t r crcut. Accr t Tr 1, w c rv wr w 1 y rpc t wt c 0 sc t s t -s put t OR t -tr. W sy tt s trstv ut wr w tr s pt r wr w t t. I t, Fur 4, tw pts y cvr, wc s c t cvr t tw pts. W t t AND-cvr tw t -crtc pts ts tw pts cvr s AND t. Fr xp Fur 4, t -crtc pts p 1 p 2 cvr AND t s s t AND-cvr p 1 p 2. W v t w tr. Tr 2: Lt wr w t -s put t OR t. I tr s AND-cvr t -crtc pts wr w s trstv ut, wr w c rv wtut vt t y trc. Pr: Ott. Fr xp, wr w 3 ( -> ) c rv ccr t Tr 2. Frst, wr w 3 s t -s put t OR t. T trstv ut s w 3 csst {,,, }, wc OR t s t y cvr t -crtc pts. Sc tr s AND-cvr w 3 s trstv ut, wr w 3 c rv. I ct, wrs {w 3, w 5, w 6 } stsy t ct Tr 2 s wrs {w 3, w 5, w 6 } r rv. T rsutt crcut tr rv wrs w 1 (y Tr 1), w 3, w 5, w 6 s sw Fur 5. W c s pt s sr s [3] t urtr ruc t r vr. Tw ss wc pt t 2 2 2 2 2 2 Fur 5: T r-sytsz crcut tr wr rv. 2

quvt B uct ut t t t -crtc pts c sr. Fr xp Fur 5, t utput tt 2 v t s uctty. W c sr t utput ss 2, s strt Fur 6. Supps quvt ss r w t sr wtut vt t rqurt t y trc. T crcut s sw Fur 6. A pt s s t prcpt pt y c (crs r crs) t pt s y c vr ct t crcut y [7]. Tr 3: Atr wr rv ccr t Tr 1 Tr 2, t -crtc pts r tr s pts r prcpt pts [7]. Pr: Ott. T tut r ts tr s s ws. Our wr rv trs urt tt tr wr rv, crcuts st v t s utput vu wvr t -crtc pts r r ctvt. Trr, y y crt t -crtc pt w t ct crcut s y. I Fur 6, t pts r tr s pts r prcpt pts. W w scuss scs s tr r-sytss. Csr Fur 6. T st tru pt pss tru s {----} ws pt t s 5. Assu t t rqurt s 6. Trr, s t sc 1 t r-sytsz crcut Fur 6 w t s t sc 0 t r crcut Fur 3. I tr xp, sc pts pss tru r s pts, t sc s t. Gry, tr r-sytss, pts ws ys r rtr t 5 c tr s pts r prcpt pts; tt s, t st tru pt t r-sytsz crcut s t y 5. As rsut, t sc c t r-sytsz crcut s t st 1 (= t ). 4. Dy Trc Itr Ss T r-sytss ty scr prvusy s pp t prry utputs ws rrv t s suscpt t y vrt. W c s py tc pprc t prtct t rrv t tr ss r y vrt. Dy trc tr ss c v t vt r ruct. I Fur 7, csr t crcut wc s st t s s tt Fur 3 xcpt Fur 7 s AND t. B s Fur 7 rprst t -crtc s t -crtc pts. I t r-sytss tcqu s pp t y t prry utput, y t -s put wr w 1 c rv t rsutt crcut s sw Fur 8. I ts xp, t r vr s r cus tr s y rv t -s put wr. W w sw tt y py y trc tr s w, t r pty c ruc. T rsut tr prr r-sytss w s sw t ry cs Fur 9. Accr t Tr 3, pt st {s---} cs tr s pt (st) r prcpt pt (st). Tus, pt {s----x---} pss tru s pt (st) {s---} s s s pt. Sry, tr pts pss tru {t---}, {s 2-2 - 2 }, {t 2-2 - 2 } r tr s pts r prcpt pts. Ts s t c Fur 7: A xp strt y trc tr ss. w w 1 2 2 2 2 2 Fur 6: T r-sytsz crcut tr s sr. Fur 8: T r-sytsz crcut y ppy y trc t prry utput Fur 7.

s t s 2 t 2 Fur 9: T r-sytsz crcut y ppy y trc tr s w Fur 7. s s pts r prcpt pts r t t -crtc pts. Csquty, s {,,,,, } Fur 9 c t -trs. W w ctu t prr y trc t prry utput, t -s put wrs t tr OR t -trs {,, } c rv. T r-sytsz crcut s sw Fur 10. T tt r vr t r-sytsz crcut Fur 10 s 7 (ts), wrs tt t r-sytsz crcut Fur 8 wtut y trc tr ss s 9 (ts). 5. Exprt Rsuts 2 2 2 2 2 x W v pt t r-sytss t [3] urs SIS vrt, s xprt st MCNC ISCAS crs. T 1 prvs t cprs t r vr tw t t [3] urs. T 2 strts t vts y vrt trc r crcuts. W rst ptz crcut wt scrpt.y, t us y trc vus 10% 15% t r crcut y t r-sytsz crcut. T t rqurt r c x w w Fur 10: T r-sytsz crcut y ppy y trc t prry utput Fur 9. 2 2 r-sytsz crcut s st t t y t crrsp r crcut. I T 1, cu sts t t r crcut. Cus tw tr prv t r y t r crcut, rspctvy. Cus ur v sw t rsuts r 10% y trc cus sx t sv r 15% y trc. Fr v y trc rqurt, w r-sytsz t r crcut y t t [3] urs vuy. T w cpr t r vr tw t r-sytsz crcuts. Fr xp, crcut C880 s t r 3569 t y 33.6. W t y trc s 3.36 (= 10%*33.6), t r-sytsz crcut y [3] s t r vr 31.2% t r crcut s r. O t tr, t r-sytsz crcut y ur t s t r vr 14.9% t r crcut s r. W t y trc s 5.04(=15%*33.6), t r-sytsz crcut y [3] s t r vr 53.1% w y ur t s t r vr 25.4%. O t vr, r 10% (15%) y trc, t r vr r [3] urs s 29% (48%) 12% (21%), rspctvy. Bt t ts usuy sp w scs r c crcut. W v s prr Mt-Cr xprts t strt t ct y vrt trc T 2. I t xprt, t y t s v s prty sty uct sr t [6]. W t 1000 sps r crcut 1000 sps ts r-sytsz crcut wt 10% y trc r Mt-Cr xprts. Atr tt, t y c sp s ccut. I T 2, cu sts t t r crcut. Lt r t y t crcut. Cu tw sws t ur crcut sps ws ys r sr t {0.8* r } r t r crcut, cu tr sws t ur r ts r-sytsz crcut. Sry, cus ur t v rprt t urs sps ws ys r sr t {0.9* r }, {1.0* r }, {1.1* r }, {1.2* r } r t t r crcut ts r-sytsz crcut. T crcut t481 s xp. Lt t t rqurt 1.1* r (1.1*34.1 = 37.51), 815 sps t r crcut t t t rqurt w 963 sps t r-sytsz crcut stsy t s rqurt. I Fur 11, w rw t strut curvs Mt-Cr sps crcut t481 ts r-sytsz crcut wt 10% y trc. T str vt

T 1: Cprs tw [3] ur t Or crcut 10% y 15% y trc trc Crcut [3] Ours [3] Ours Ar Dy Ar Ar Ar Ar vr vr vr vr (%) (%) (%) (%) C432 1568 35.9 6.0 2.5 6.0 2.5 C880 3569 33.6 31.2 14.9 53.1 25.4 C1355 5854 35.8 29.3 10.4 47.1 20.4 C1908 6080 48.2 35.0 16.7 54.5 26.3 C2670 11281 62.2 23.2 11.4 34.2 16.8 C3540 19170 85.4 30.2 15.0 45.0 22.4 C5315 29937 90.0 38.1 18.6 59.5 29.4 C6288 46379 161.7 39.2 19.3 59.2 29.2 pr 12941 28.6 33.1 14.0 51.3 22.3 rt 5915 35.7 30.8 11.8 50.9 19.2 t481 4927 34.1 6.9 1.1 11.4 2.4 9sy 1317 21.4 45.3 10.9 74.1 21.9 u2 3855 50.7 39.8 19.0 57.0 27.2 px6 5220 56.2 18.5 7.2 63.8 24.4 px7 1836 26.9 32.9 12.5 48.0 17.4 Av. 29.3 12.3 47.7 20.5 t r crcut s 2.39 w tt t r-sytsz crcut s 1.92. T curvs s rv tt t r-sytsz crcut s r st t vr t t r crcut. 6. Ccuss W v prps rwr t r-sytsz v crcut r t y trc. Our t s wt upx syst; w t pt wr rv s sr t ruc t r vr t y trc structur. Tw v trs r urtr r ruct r prst. Exprt rsuts strt t vts y vrt trc r r-sytsz crcut. Rrcs [1] K. Br, G. Grtu, M. Lusr, I. Scstr, C. Hws, Dct-s y tst rsstv vs-ctcts, crtc vut, Prc. Itrt Tst Crc, pp. 467-476, Spt. 1999. [2] M. A. Brur, C. Gs, S. Gupt, Nw vt tst prs r prrc p su-cr VLSI crcuts, Tutr Nts, VLSI Tst Sypsu, Apr 1997. [3] S-C C, C-T Hs, K-C Wu, R-sytss r y vrt trc, Prc. Ds Autt Crc, pp. 814-819, Ju 2004. [4] Su H C, Bpu C. Pu, Kus Ry, Nv sz rt r y prvt ur T 2: Sttstc rsuts Mt-Cr xprts 0.8* r 0.9* r 1.0* r 1.1* r 1.2* Crcut r O R O R O R O R O R C432 0 0 1 0 125 280 663 852 977 1000 C880 0 0 2 117 265 819 907 996 999 1000 C1355 0 0 0 0 0 7 68 445 583 862 C1908 0 0 0 0 1 0 248 389 964 976 C2670 0 0 8 14 150 186 515 680 893 965 C3540 0 0 5 2 66 50 361 386 855 910 C5315 0 0 7 1 96 62 346 570 796 992 C6288 0 0 0 0 35 553 769 999 1000 1000 pr 0 0 0 0 1 13 162 386 753 912 rt 0 0 0 17 75 299 486 706 877 976 t481 1 0 29 19 373 463 815 963 984 1000 9sy 0 0 0 0 3 3 138 249 639 926 u2 0 0 0 5 75 259 551 825 940 1000 px6 248 398 368 504 470 601 564 718 693 836 px7 4 1 71 47 264 298 622 711 780 890 Av. 17 27 33 48 133 260 481 658 849 950 O: Or crcut R: R-sytsz crcut prcss vrt tr tcy, Prc. Ds Autt C., pp. 454-459, Ju 7-11, 2004. [5] E.T.A.F. Jcs M.R.C.M. Brr, Gt sz us sttstc y, Prc. DATE, pp. 27-30, 2000. [6] J-J Lu, A. Krstc, L-C. W, Kw-T C, Fs-pt-wr sttstc t yss ct pt sct r y tst t vt, Prc. Ds Autt Crc, pp. 566-569, Ju 2002. [7] Sr R, Sr B. K. Vruu, Jt W, A ty t prv t y t prsc prcss vrts, Prc. Ds Autt C., pp. 448-453, Ju 7-11, 2004. [8] Axr S, Fuct t ptzt, Prc. Itrt Crc Cputr-A Ds, pp. 539-543, Nv. 1999. Fur 11: T strut curvs Mt-Cr sps crcut t481 ts r-sytsz crcut.

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