EE241 - Spring 2000 - University of California,...

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UC Berkeley EE241 B. Nikoli c

EE241 - Spring 2000Advanced Digital Integrated Circuits

Lecture 16

Adders

Carry-Skip Adder

FA FA FA FA

P0 G1 P0 G1 P2 G2 P3 G3

Co,3Co,2Co,1Co,0Ci,0

FA FA FA FA

P0 G1 P0 G1 P2 G2 P3 G3

Co,2Co,1Co,0Ci,0

BP=PoP1P2P3

Idea: If (P0 and P1 and P2 and P3 = 1)then Co3 = C0, else “kill” or “generate”.

Bypass

MacSorley, Proc IRE 1/61Lehman, Burla, IRE Trans on Comp, 12/61

Carry-Skip Adder

CarryPropagation

Bit 0-3 Bit 4-7 Bit 8-11 Bit 12-15

Critical Path

( ) ( ) RCASKIPRCAd tktkN

tkt 121 −+

−+−=

For N-bit adder with k-bit groups

Carry-Skip Adder

( ) SKIPRCAd tkN

−+−= 212

Critical path delay with constant groups

ripple adder

bypass adder

Carry-Skip Adder

Variable Group Length

Oklobdzija, Barnes, Arith’85

321 cNcctd ++=

Carry-Skip Adder

Variable Block Lengths

Manchester Carry Chain

Kilburn, et al, IEE Proc, 1959.

•Implement P with pass-transistors•Implement G with pull-up, kill (delete) with pull-down•Use dynamic logic to reduce the complexity and speed up

Sizing Manchester Carry Chain

M0 M1 M2 M3 M4MC

Discharge Transistor

1 2 3 4 5 6

tp 0.69 Ci Rjj 1=

1 1.5 2.0 2.5 3.0k

Speed (normalized by 0.69RC) Area (in minimum size devices)

Manchester Chain with Carry-Skip

Delay model:

PTL with SA-F/F Implementation

Matsui,JSSC 12/94

Propagate and Generate Signals

Define 3 new variable which ONLY depend on A, B

Generate (G) = AB

Propagate (P) = A ⊕ B

Delete = A B

Can also derive expressions for S and Co based on D and P

A0,B 0 A1,B 1 AN-1 ,BN-1...

Ci,0 P0 Ci,1 P1Ci,N-1 PN-1

Carry Lookahead Adder

Weinberger, Smith, 1958.

Lookahead Adder

1−+= iiii cpgc

Looakahead Equations

( )1111

−+++

iiiiii

cppgpg

cpgpgcpgc

Position i:

Position i + 1:

Carry exists if:- generated in stage i + 1- generated in stage i and propagated through i + 1- propagated through both i and i + 1

Lookahead Adder

• Unrolling of carry recurrence can be continued• If unrolled to level k, resulting in two-level AND-OR

structure• AND Fan-In = k + 1, OR Fan-In = k + 1• k + 1 transistors in the MOS stack• Limits k to 3 -4

Lookahead AdderVDD

Block Lookahead

1123123

1232334

−++++++

+++++++

iiiiiiiii

iiiiiii

cppppgppp

gppgpgcFourth bit carry:

iiiiiiiiiiii gpppgppgpgG 1231232333, ++++++++++ +++=

iiiiii ppppP 1233, ++++ =

13,3,4 −+++ += iiiiii cPGc

Block generate and block propagate:

Block Lookahead

Can create groups of groups, or ‘super-groups’:

jjjjjjjjjjj GPPPGPPGPGG 123123233*

+++++++++ +++=

jjjjj pPPPP 123*

Delay is Nctd log1=

Block LookaheadFrom Oklobdzija

Lookahead Example

Multiple Output Domino (MODL)

Generate Propagate

4-bit group generate 4-bit group propagate

Lookahead Example

64-b Lookahead Example

Lookahead Example

Modified CLA

Naini, CICC’92

Modified CLA

Conditional Sum Adders

Sklansky,Trans on Comp6/60

iii yxs ⊕=0

iii yxs ⊕=1

iii yxc ⋅=0

iii yxc +=1

Conditional Sum Adders

TG Conditional SumConditional CellConditional Sum Adder

2-way MUXes

Rothermel, JSSC 89

TG Conditional Sum

l Serial connection of transmission gates l Chain length = 1+log2n

Signal propagation

DPL Conditional Sum

CLA“Conditional carry select”

DPL Conditional Sum

Block Conditional Sums

Carry-Select Adder

"0" Carry Propagation

"1" Carry Propagation

Multiplexer

Sum Generation

Co,k-1 Co,k+3

Carry Vector

Carry Select Adder: Critical Path

"0" Carry

"1" Carry

Multiplexer

Sum Generation

"0" Carry

"1" Carry

Multiplexer

Sum Generation

"0" Carry

"1" Carry

Multiplexer

Sum Generation

"0" Carry

"1" Carry

Multiplexer

Sum Generation

Bit 0-3 Bit 4-7 Bit 8-11 Bit 12-15

S0-3 S4-7 S8-11 S12-15

Co,15Co,11Co,7Co,3Ci,0

Linear Carry Select

"0" Carry

"1" Carry

Multiplexer

Sum Generation

"0" Carry

"1" Carry

Multiplexer

Sum Generation

"0" Carry

"1" Carry

Multiplexer

Sum Generation

"0" Carry

"1" Carry

Multiplexer

Sum Generation

Bit 0-3 Bit 4-7 Bit 8-11 Bit 12-15

S0-3 S4-7 S8-11 S12-15

(5)(6) (7) (8)

(5) (5) (5)(5)

Square Root Carry Select

"0" Carry

"1" Carry

Multiplexer

Sum Generation

"0" Carry

"1" Carry

Multiplexer

Sum Generation

"0" Carry

"1" Carry

Multiplexer

Sum Generation

"0" Carry

"1" Carry

Multiplexer

Sum Generation

Bit 0-1 Bit 2-4 Bit 5-8 Bit 9-13

S0-1 S2-4 S5-8 S9-13

(4) (5) (6) (7)

(3) (4) (5) (6)

S14-19

Bit 14-19

Logarithmic Lookahead Adders

A6A5A4A3A2A1

tp∼ log2(N)

tp∼ N

Tree Adders

lmG ppP ⋅=

lmmG gpgG ⋅+=

m – more significantl – less significant

Start from the input P, G, and continue up the tree2-bit groups, then 4-bit groups, …

( ) ( ) ( )lmlmmllmm ppgpgpgpgpg ⋅⋅+=•= ,,,),(

Kogge, Stone, Trans on Comp,’73

Brent-Kung Adder(G0,P0)(G1,P1)

(G2,P2)

(G3,P3)

(G4,P4)(G5,P5)

(G6,P6)

(G7,P7)

Co,1Co,2

tadd ∼ log2(N)Brent, Kung, Trans on Comp, 3/82

Tree Adders

EE241 - Spring 2000 - University of California,...

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