Continued..& Multiplier

Transcription

1 CS222: Computer Arthmetc : Adder Contnued..& Multpler Dr. A. Sahu Dept of Comp. Sc. & Engg. Indan Insttute of Technology Guwahat 1

2 Outlne Adder Unversal Use (N bt addton) RppleCarry Adder, Full Adder, Adder/Substrator, Algorthmc Adder Scheme Reducng Adder Delay Rpple Carry Adder, Manchester Adder Carry Skp Adder, Carry Select Adder Carry Look Ahead Adder Carry Save Adder (Multple Operand Adder) Multplcaton Algorthms Shft and Add, Sequental Algorthms Array Multpler wth RCA/CSA 2

3 Adder Unversal Use Adder : A = B C Substractor: A = B ( C), 2 s complement Compare : C = A> B? 1 : 0 (A B>0) : 1 : 0 Specal case of compare wth 0 Multply Dvde Mod Floatng pont: Add/sub/mul 3

4 N Bt Rpple Carry Adder: Seres of FA Cells To add two n bt numbers A n-1 B n-1 A 2 B 2 A 1 B 1 A 0 B 0 FA C n... n FA FA FA C 0 S n-1 S 2 S 1 S 0 Adder delay = Tc * n Tc =(C n to C out delay) ofafa A B C out FA C n Sum 4

5 Mathematcally: C & S C =FunC ( x 1,.., x 0, y 1,.., y 0, c n ) S =FunS (x, y, c ) = ( x y c ) mod 2; 5

6 Rpple Carry Adder Analyss x C n =0 y P K P P P G P P P K C Case X Y X Y C 1 Comment Kll (K =1) Kll/Stop C n /C out =0 Propagate P = C Propagate C n C C out =C n Generate (G =1), Generate C out, C out =1 6

7 Propagate, Generate & Kll Case 1 (Kll): k = x y = (x y ) Case 2 (Propagate): p = x XOR y Case 3 (Generate): g = x y Then c 1 = g p c = x y (x XOR y ) c Alternatve t ( (smpler) expresson: c 1 = g a c Snce a = k, we call t "alve" 7

8 Reducng Adder Delay Rpple Carry Adder: N(t c )max(t c,t s ) Reducng Carry Delay t c : Manchester Swtch Changng glnear factor to smaller N/k or logn Carry Look Ahead, Carry Skp, Carry Select, Condtonal Sum Adder Includng a competton sgnal: addton always may not be the worst case. Changng number representaton: Carry Saved Adder 8

9 Swtched Carry Rpple (Manchester) Adder Idea: Fast crcut to propagatng carry chan x y G P K C C

10 Chan Control Swtch Module x y Carry Control GKP g k 1 0 p C 1 C 10

11 Manchester Adder (MRCA) Delay: t sw (n 1)*t p Here t p < t c, so total delay s less Y n 1 X n 1 Y X Y 1 X 1 Y0 X0 CC CC CC CC C n S n 1 S S 1 S 0 11

12 Carry Skp Adder Smaller modfcaton to Rpple Carry Adder Reduce worst case delay by reducng the number of FA cell through carry has to Propagate Dvde n bts n to (n/m) groups of m bts If sum of group s 2 m 11 then carry s propagated Carry s propagated when t propagated by all the bts of the groups P j = p j0 p j1 p j2..p jm 1 C n,j1 = C out, j. P j C n,j P j 12

13 Carry Skp Example P=0 P=1 P=1 P=0 P=1 P= C P C j1 C j 13

14 CSK Module C If Pj=1: smply skp the group j, C n,j1 =C n,j C x j y j P j m m C n,j1 M0M 0 U X 1 C out M bt RCA Group J m C n Module n 1 m m Module m m s j Module 1 m m Module 0 m m CSK Module CSK Module CSK Module CSK Module m m m m 14

15 Carry Skp Path skpped 15

16 Carry Select Adder For a group Sum & Carry s already calculated Smply select based on carry FA FA FA FA FA FA FA FA 16

17 Carry Look Ahead Adder (CLA) Basc Idea: Compute Several Carres Smultaneously and use t As Rpple Carry or MultLevel CLA A3 B3 A2 B2 A1 B1 A0 B0 FA FA FA FA S 3 S2 S 1 S0 C 4 P3 G3 C3 P2 G2 C2 P1 G1 C1 P0 G0 4 Bt Carry Look Ahead Generator 17

18 CLG CLG 1 B A G where C P G C = = C P P... P P... G P = G B A G where C P G C 4) nputs (# of for Avalable P C S = 4) nputs for (# of Avalable 18

19 CLG (Carry Lookahead Generator) C 0 P 1 G 1 P 2 C 1 G 2 C 2 P 3 C 3 G 3 P 4 C 4 G

20 RCA wth CLA unts As Carry are generated wth CLG t takes less tme Module n 1 m m Module m m Module 1 m m Module 0 m m CLA Module CLA Module CLA Module CLA Module m m m m 20

21 2 Levels of look ahead no rpplng a 4..7 of carry c c 0 0 P 0 a 0..3 b s 0..3 G c 4 P 4 b s 4..7 G 4 C 4..7 a c 8 s 8 11 P 8 b s G L A unt c 12 a b s c 16 P 12 G 12

22 Group propagate & generate c 1 = p 0 c 0 g 0 c 2 = p 1 c 1 g 1 = p 1 p 0 c 0 p 1 g 0 g 1 c 3 = p 2 c 2 g 2 = p 2 p 1 p 0 c 0 p 2 p 1 g 0 p 2 g 1 g 2 c 4 = p 3 c 3 g 3 = p 3 p 2 p 1 p 0 c 0 p 3 p 2 p 1 g 0 p 3 p 2 g 1 p 3 g 2 g 3 P 0 = p 3 p 2 p 1 p 0 G 0 = p 3 p 2 p 1 g 0 p 3 p 2 g 1 p 3 g 2 g 3 c 4 = P 0 c 0 G 0

23 Group propagate & generate P = p 3 p 2 p 1 p G = p 3 p 2 p 1 g p 3 p 2 g 1 p 3 g 2 g 3 c 4 = P 0 c 0 G 0 c 8 = P 4 P 0 c 0 P 4 G 0 G 4 c 12 = P 8 P 4 P 0 c 0 P 8 P 4 G 0 P 8 G 4 G 8 c 16 = P 12 P 8 P 4 P 0 c 0 P 12 P 8 P 4 G 0 P 12 P 8 G 4 P 12 G 8 G 12

24 Summery: Two operand Addtons Speedng up addton Rpple carry adder (carry propagate): O(n) Carry C look ahead: O(log n) What about Mult Operand Adder A=N 0 N 1 N 2..N n Where we requre: possbly n multply or advance computng places 24

25 Addng multple operands a3 b3 a2 b2 a1 b1 a0 b0 A B E F e3 e2 e1 e0 Tradtonal Tradtonal adder Adder Tradtonal Tradtonal adder f3 f2 f1 f0 Adder Tradtonal Tradtonal adder Adder S s5 s4 s3 s2 s1 s0

26 Carry save addton b3 e3 f3 b2 e2 f2 b1 e1 f1 b0 e0 f0 A B E F Carry Save Adder a3 a2 a1 a0 s'4 c'3 s'3 c'2 s'2 c'1 s'1 c'0 s'0 Carry save adder Carry Save Carry save adder Adder C' S' Tradtonal Tradtonal adder Adder S s5 s4 s3 s2 s1 s0

27 Multplcaton: paper pencl method A B x x x AxB

28 Multply: Shft & Add Decmal number : 15x20=300, 10x205x20 =300 Bnary number: 1111 X X X X X10100 Sft3(10100) sft2(10100) sft1(10100)sft0(10100) ft0(10100) 1111X X100 Sft5(1111)sft2(1111) f Multplcaton of N bt number, N shft, N Add, f bt s zero don t add Specal addton 28

29 Shft add multpler 0 shft 0 A B 0 sh hft 1 B 1 n 1 = 0 A B = A B 2 shft 2 B 2 shft 3 B 3 A x B

30 Shft add multpler (sequental) n 1 B = A A B 2 = 0 step1: =0; s=0 step1: =0; s=0 step1: =0; s=0 do { do { do { step2: step2: step2: s = A B x 2 f (B ) s = A f (B 0 ) s = A A = 2xA A = 2xA } whle ( < n) B = B/2; } whle ( < n) } whle ( < n)

31 Sequental shft add multpler 1 s A 2n step1: =0; s=0 do { 2n step2: B B f (B 0 ) s = A 0 A = 2xA control B = B/2; 2n } whle (< n)

32 Sequental shft add multpler 2 s n A n B n control step1: =0; s=0 do { step2: f (B 0 )s H = A B 0 step3: s = s/2 B = B/2; } whle ( < n)

33 Sequental shft add multpler 3 A n n s control n step1: =0; s=0 B do { step2: f (s 0 ) s H = A step3: s = s/2; } whle ( < n)

34 Smple Speedng up n 1 A B = = A. B = 0 A B A. B A. B 0 1 n 1 N Addton can be done n parallel n Log(N) steps usng N Adder A.B A.B A.B A.B n 1 34

35 Shft add multpler 0 shft 0 A B 0 sh hft 1 B 1 n 1 = 0 A B = A B 2 shft 2 B 2 shft 3 B 3 A x B

36 Array mult wth carry propagate p13 0 p12 p03 p11 p02 p10 p01 p00 pj = a. bj 0 p23 p22 p21 p20 p33 p32 0 p31 p30 0 s7 s6 s5 s4 s3 s2 s1 s0

37 Array multpler wth carry save pj = a. bj p23 p22 p13 0 p21 p12 p03 p20 p11 p02 0 p10 p01 p00 p33 p32 p31 p s7 s6 s5 s4 s3 s2 s1 s0

38 Summary Speedng up addton Rpple carry adder (carry propagate): O(n) Carry look ahead: O(log n) Speedng up array multpler Usng carry propagate adders: ~ 3 n d Usng carry save adders: ~ 2 n d (here d = delay of 1 bt adder)

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