L23 – Adder Architectures

Adders  Carry Lookahead adder  Carry select adder (staged)  Carry Multiplexed Adder  Ref: text Unit 15 9/2/2012 – ECE 3561 Lect 9 Copyright Joanne DeGroat, ECE, OSU2

The carry lookahead adder  The generation of all outputs is a direct function of the inputs. The carry out is a function of all inputs. The msb sum output is also a function of all inputs.  Time of addition is the shortest. 9/2/2012 – ECE 3561 Lect 9 Copyright Joanne DeGroat, ECE, OSU3

The equations  Create two signals called propagate and generate, P i and G i  P i (A,B) = A i + B i or A i  B i  G i (A,B) = A i · B i  G is the generate function – when both A and B are a 1, there is a carry generated  P is the propagate function – when a 1 it will propagate the carry in. When both inputs are 1, the G function generates a carry so either form of the P function works. 9/2/2012 – ECE 3561 Lect 9 Copyright Joanne DeGroat, ECE, OSU4

The equations  C 1 = G0 + P0 C0  C2 = G1 + P1 C1  C2 = G1 + G0 P1 + C0 P0 P1  C3 = G2 + P2 C2  C3 = G2 + G1 P2 + G0 P1 P2 + C0 P0 P1 P2  C4 = G3 + P3 C3  C4 = G3 + G2 P3 + G1 P2 P3 + G0 P1 P2 P3  + C0 P0 P1 P2 P3 9/2/2012 – ECE 3561 Lect 9 Copyright Joanne DeGroat, ECE, OSU5

And the sum output  The sum is the conventional equation  Sum (i) = A(i)  B(i)  C(i)  If the XOR form of propagate is used  Sum (i) = P(i)  C(i)  Note on CLA – the logic required to implement it exponentially increases with the number of bits to be added. 9/2/2012 – ECE 3561 Lect 9 Copyright Joanne DeGroat, ECE, OSU6

The VHDL – 4-bit carry lookahead  ENTITY cla IS PORT (a,b : IN bit_vector (3 downto 0); cin : IN bit; sum : OUT bit_vector (3 downto 0); cout : OUT bit); END cla; ARCHITECTURE one OF cla IS SIGNAL P,G : bit_vector (3 downto 0); SIGNAL C : bit_vector (4 downto 0); BEGIN P <= A OR B; G <= A AND B; C(0) <= cin; C(1) <= G(0) OR (P(0) AND C(0)); C(2) <= G(1) OR (G(0) AND P(1)) OR (C(0) AND P(0) AND P(1)); C(3) <= G(2) OR (G(1) AND P(2)) OR (G(0) AND P(1) AND P(2)) OR (C(0) AND P(0) AND P(1) AND P(2)); C(4) <= G(3) OR (G(2) AND P(3)) OR (G(1) AND P(2) AND P(3)) OR (G(0) AND P(1) AND P(2) AND P(3)) OR (C(0) AND P(0) AND P(1) AND P(2) AND P(3)); sum <= P XOR C(3 downto 0); END one; 9/2/2012 – ECE 3561 Lect 9 Copyright Joanne DeGroat, ECE, OSU7

And the synthesis  Resources used Combinational LUTs – 8 Pins /2/2012 – ECE 3561 Lect 9 Copyright Joanne DeGroat, ECE, OSU8

Growth of LUTs  For a 6 bit unit C5 = G4 + P4 C4 C5 = G4 + G3 P4 + G2 P3 P4 + G1 P2 P3 P4 + G0 P1 P2 P3 P4 + C0 P0 P1 P2 P3 P4 C6 = G5 + P5 C5 C6 = G5 + G4 P5 + G3 P4 P5 + G2 P3 P4 P5 + G1 P2 P3 P4 P5 + G0 P1 P2 P3 P4 P5 + C0 P0 P1 P2 P3 P4 P5 9/2/2012 – ECE 3561 Lect 9 Copyright Joanne DeGroat, ECE, OSU9

When done for a 5-bit adder  Resources LUTs – 12 Pins /2/2012 – ECE 3561 Lect 9 Copyright Joanne DeGroat, ECE, OSU10

And for a 6-bit CLA  Resources LUTs – 17 Pins – 20 9/2/2012 – ECE 3561 Lect 9 Copyright Joanne DeGroat, ECE, OSU11

The carry select adder  The basic idea is to group the add. The group of bits is added assuming a 0 carry in and in a parallel adder assuming a 1 carry in. Once the correct carry arrives the valid result is chosen. 9/2/2012 – ECE 3561 Lect 9 Copyright Joanne DeGroat, ECE, OSU12

Carry select metrics  Speed is better than ripple carry adder but only by the staging factor.  Area growth is linear and a constant factor more than ripple.  Will see this at end of lecture. 9/2/2012 – ECE 3561 Lect 9 Copyright Joanne DeGroat, ECE, OSU13

The Carry Multiplexed adder  An extension of the Carry select Speed is on the order of full carry lookahead # gates used is linear growth and ~3x that of a ripple adder of the same bit with. As speed is about as fast as the add can be completed and growth is linear this area has the best area time metric of all adders. This is the adder architecture used in all modern computer architectures. 24 patents exist for CMA architectures. 9/2/2012 – ECE 3561 Lect 9 Copyright Joanne DeGroat, ECE, OSU14

Carry Multiplexed Adder  The basic concept 9/2/2012 – ECE 3561 Lect 9 Copyright Joanne DeGroat, ECE, OSU15

Goals in the CMA  No redundant operations. 9/2/2012 – ECE 3561 Lect 9 Copyright Joanne DeGroat, ECE, OSU16

Moving to higher order  Still just add twice – presumed 0 and 1 9/2/2012 – ECE 3561 Lect 9 Copyright Joanne DeGroat, ECE, OSU17

Adder performance – Area metric  Number of gates 9/2/2012 – ECE 3561 Lect 9 Copyright Joanne DeGroat, ECE, OSU18

The time metric  Speed 9/2/2012 – ECE 3561 Lect 9 Copyright Joanne DeGroat, ECE, OSU19

Area Time Metric  Area Time Product 9/2/2012 – ECE 3561 Lect 9 Copyright Joanne DeGroat, ECE, OSU20

Lecture summary  The adder was a simple ripple carry adder.  Other Architectures Carry Lookahead Carry select Carry multiplexed 9/2/2012 – ECE 3561 Lect 9 Copyright Joanne DeGroat, ECE, OSU21