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

2. 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

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

4. 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

5. 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

6. 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 = ...

7. 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)

8. 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

9. Pencil and Paper Multiplication Algorithm

10. Solution: Booth’s Algorithm 
A , Q
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

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

12. 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: = 30

13. 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?

14. 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: = -27

15. Booth's Algorithm Simulator
or/Booth/