EE241 - Spring 2000 Advanced Digital Integrated Circuits. Carry-Skip Adder
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1 EE4 - Srng 000 Advanced Dgtal Integrated Crcuts Lecture 6 Adders B. Nkolc Carry-Sk Adder 0 G 0 G G G C,0 C o,0 C o, C o, FA FA FA FA C o, 0 G 0 G G G B= o C,0 C o,0 C o, C o, FA FA FA FA Multlexer Co, Idea: If (0 and and and = ) then C o = C 0, else kll or generate. MacSorley, roc IRE /6 Lehman, Burla, IRE Trans on Com, /6 Byass B. Nkolc
2 Carry-Sk Adder Bt 0- Bt 4-7 Bt 8- Bt -5 Setu Setu Setu Setu C,0 Carry roagaton Carry roagaton Carry roagaton Carry roagaton Sum Sum Sum Sum Crtcal ath For N-bt adder wth k-bt grous N t d = k ( k ) t RCA t SKI ( k ) t RCA B. Nkolc t d Carry-Sk Adder Crtcal ath delay wth constant grous N = k ( k ) trca tski t rle adder byass adder 4..8 N B. Nkolc
3 Carry-Sk Adder Varable Grou Length t d = c cn c Oklobdza, Barnes, Arth 85 B. Nkolc Carry-Sk Adder Varable Block Lengths B. Nkolc
4 Manchester Carry Chan Imlement wth ass-transstors Imlement G wth ull-u, kll (delete) wth ull-down Use dynamc logc to reduce the comlexty and seed u V DD φ 0 4 C,0 G 0 G G G G 4 φ Klburn, et al, IEE roc, 959. B. Nkolc Szng Manchester Carry Chan Dscharge Transstor R R R R 4 R R 6 Out M C C M 0 M M M M 4 C C C 4 C 5 C 6 Seed N t = 0.69 C R = = Area k Seed (normalzed by 0.69RC) k Area (n mnmum sze devces) B. Nkolc 4
5 Manchester Chan wth Carry-Sk 0 C,0 G 0 G G G B C o, B Delay model: B. Nkolc TL wth SA-F/F Imlementaton Matsu, JSSC /94 B. Nkolc 5
6 roagate and Generate Sgnals Defne new varable whch ONLY deend on A, B Generate (G) = AB roagate () = A B Delete = A B Can also derve exressons for S and C o based on D and B. Nkolc Carry Lookahead Adder A 0,B 0 A,B A N-,B N-... C,0 0 C, C,N- N-... Wenberger, Smth, 958. B. Nkolc 6
7 Lookahead Adder Looakahead Equatons oston : oston : c c = g c = g = g = g ( g c ) g c c Carry exsts f: - generated n stage - generated n stage and roagated through - roagated through both and B. Nkolc Lookahead Adder Unrollng of carry recurrence can be contnued If unrolled to level k, resultng n two-level AND-OR structure AND Fan-In = k, OR Fan-In = k k transstors n the MOS stack Lmts k to -4 B. Nkolc 7
8 8 B. Nkolc Lookahead Adder VDD 0 G G G G 0 C,0 Co, B. Nkolc Block Lookahead 4 = c g g g g c Fourth bt carry: g g g g G, =, =,, 4 = c G c Block generate and block roagate:
9 9 B. Nkolc Block Lookahead Can create grous of grous, or suer-grous : G G G G G * = * = Delay s N c t d log = B. Nkolc Block Lookahead From Oklobdza
10 Lookahead Examle Generate roagate Multle Outut Domno (MODL) B. Nkolc Lookahead Examle 4-bt grou generate 4-bt grou roagate B. Nkolc 0
11 64-b Lookahead Examle B. Nkolc Lookahead Examle B. Nkolc
12 Modfed CLA Nan, CICC 9 B. Nkolc Modfed CLA B. Nkolc
13 Condtonal Sum Adders s0 = x y s = x y c0 = x y c = x y Sklansky, Trans on Com 6/60 B. Nkolc Condtonal Sum Adders B. Nkolc
14 TG Condtonal Sum Condtonal Sum Adder Condtonal Cell -way MUXes Rothermel, JSSC 89 B. Nkolc TG Condtonal Sum Seral connecton of transmsson gates Chan length = log n Sgnal roagaton B. Nkolc 4
15 DL Condtonal Sum CLA Condtonal carry select B. Nkolc DL Condtonal Sum Block Condtonal Sums B. Nkolc 5
16 Carry-Select Adder Setu,G Carry roagaton "" "" Carry roagaton C o,k- Multlexer C o,k Sum Generaton Carry Vector B. Nkolc Carry Select Adder: Crtcal ath Bt 0- Bt 4-7 Bt 8- Bt -5 Setu Setu Setu Setu Carry Carry Carry Carry "" "" Carry "" "" Carry "" "" Carry "" "" Carry C,0 Multlexer C o, Multlexer C o,7 Multlexer C o, Multlexer C o,5 Sum Generaton Sum Generaton Sum Generaton Sum Generaton S 0- S 4-7 S 8- S -5 B. Nkolc 6
17 Lnear Carry Select Bt 0- Bt 4-7 Bt 8- Bt -5 Setu Setu Setu Setu () () Carry Carry Carry Carry "" Carry "" (5) (5) Multlexer C,0 "" Carry "" Carry "" Carry "" "" "" (5) (5) (5) (6) (7) (8) Multlexer Multlexer Multlexer (9) Sum Generaton Sum Generaton Sum Generaton Sum Generaton S 0- S 4-7 S 8- S -5 (0) B. Nkolc Square Root Carry Select Bt 0- Bt -4 Bt 5-8 Bt 9- Bt 4-9 Setu () Setu Setu Setu Carry () Carry Carry Carry "" "" Carry "" "" Carry "" "" Carry "" "" Carry () () (4) (5) (6) (4) (5) (6) (7) Multlexer Multlexer Multlexer Multlexer C,0 Sum Generaton Sum Generaton Sum Generaton Sum Generaton S 0- S-4 S5-8 S 9- (7) Mux (8) Sum S 4-9 (9) B. Nkolc 7
18 Logarthmc Lookahead Adders A 0 F A A A A 4 A 5 A 6 A 7 A 0 t N A A A A 4 A 5 A 6 A 7 F t log (N) B. Nkolc Tree Adders G G G = = g m m l m g l m more sgnfcant l less sgnfcant Start from the nut, G, and contnue u the tree -bt grous, then 4-bt grous, ( g, ) ( g, ) = ( g g ) ( g, ) =, m m l l m m l m l Kogge, Stone, Trans on Com, 7 B. Nkolc 8
19 Brent-Kung Adder (G 0, 0 ) (G, ) C o,0 Co, C o, C o,4 (G, ) C o, C o,5 (G, ) (G 4, 4 ) (G 5, 5 ) C o,6 (G 6, 6 ) (G 7, 7 ) C o,7 t add log (N) Brent, Kung, Trans on Com, /8 B. Nkolc Tree Adders B. Nkolc 9
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