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L23 – Adder Architectures

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

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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 2012 - Joanne DeGroat, ECE, OSU3

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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 2012 - Joanne DeGroat, ECE, OSU4

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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 2012 - Joanne DeGroat, ECE, OSU5

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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 2012 - Joanne DeGroat, ECE, OSU6

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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 2012 - Joanne DeGroat, ECE, OSU7

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And the synthesis Resources used Combinational LUTs – 8 Pins - 14 9/2/2012 – ECE 3561 Lect 9 Copyright 2012 - Joanne DeGroat, ECE, OSU8

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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 2012 - Joanne DeGroat, ECE, OSU9

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When done for a 5-bit adder Resources LUTs – 12 Pins - 71 9/2/2012 – ECE 3561 Lect 9 Copyright 2012 - Joanne DeGroat, ECE, OSU10

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

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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 2012 - Joanne DeGroat, ECE, OSU12

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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 2012 - Joanne DeGroat, ECE, OSU13

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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 2012 - Joanne DeGroat, ECE, OSU14

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

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

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

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

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

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

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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 2012 - Joanne DeGroat, ECE, OSU21

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