CDA3101 Recitation Section 5

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Presentation transcript:

CDA3101 Recitation Section 5 Ripple Carry vs Carry Lookahead Adder Boolean Multiplication (Booth's Algorithm)

Boolean Addition - One Bit Adder Takes two 1-bit numbers, and a carry-in, and computes the sum, with carry-out. CIN A Y B COUT ADD

Boolean Addition - One Bit Adder CIN A Y B COUT Logic Circuit ADD Truth Table A B CIN SUM (Y) COUT 1

Ripple Carry - N Bit Adder CIN Y COUT A0 B0 Y0 Put several 1-bit adders end-to-end, connected via carry-in and carry-out Time Complexity in number of bits? Each adder must wait for previous adder's carry: O(N) A B CIN Y COUT A1 B1 Y1 A B CIN Y COUT A2 B2 Y2 Carry

Carry Lookahead Adder - N Bit Adder (CLA) Can we do better? If we could predict the carry, output at each adder, we wouldn't need to wait. For convenience, define: Pi = Ai + Bi Gi = Ai Bi Now from the truth table: COUTi = Gi + Pi CINi COUTi = Gi + Pi COUTi-1 A B CIN COUT 1

Carry Lookahead Adder - N Bit Adder (CLA) The boolean algebra expression for Ci is expanded as follows. COUT0 = G0 + P0 COUT0-1 COUT1 = G1 + P1 COUT0 = G1 + P1 G0 COUT2 = G2 + P2 COUT1 = G2 + P2 (G1 + P1 G0) COUT3 = G3 + P3 COUT2 = G3 + P3 (G2 + P2 (G1 + P1 G0)) COUT4 = G4 + P4 COUT3 = ...

Carry Lookahead Adder - N Bit Adder (CLA) Tradeoff - more hardware, lower latency. Cost to build 16 bit CLA is prohibitively expensive. Intermediate solution connects Ripple Carry Adders together using lookahead logic. Complexity - O(logN)

Pencil and Paper Multiplication Algorithm A series of partial products followed by a summation step. Problem: Does not work with negative numbers. 0010 x 1101 ------- 0000 0011010

Pencil and Paper Multiplication Algorithm

Solution: Booth’s Algorithm  A 0, Q-1 0 M Multiplicand Q Multiplier Count n START = 10 = 01 Q0,Q-1 = 00 = 11 A A - M A A + M Arithmetic shift right: A, Q, Q-1 Count Count - 1 No Yes Count = 0 ? END 10

Problem 1 - Booth's Multiplication Algorithm Use Booth's Algorithm to find the product 6x5 in signed binary notation. 6 = 0110 5 = 0101

Problem 1 - Booth's Multiplication Algorithm Operation A Q Q-1 M Count Subtract 0000 0101 0110 4 Shift 1010 Add 1101 0010 1 3 0011 0001 1001 2 1011 1100 Done 1110 Answer: 00011110 = 30

Problem 2 - Booth's Multiplication Algorithm Use Booth's Algorithm to find the product -3x9 in signed binary notation. -3 = 11101 9 = 01001 *Why are we using 5 bits now, instead of 4?

Problem 2 - Booth's Multiplication Algorithm Operation: A Q Q-1 M Count Subtract 00000 01001 11101 5 Shift 00011 Add 00001 10100 1 4 11110 11111 01010 3 10101 2 00010 Done 00101 Answer: 1111100101 = -27

Booth's Algorithm Simulator http://www.ecs.umass.edu/ece/koren/arith/simulat or/Booth/