Carry Lookahead Addition

Tom Kelliher, CS 240

Feb. 27, 2002

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Announcements

Speaking of lookahead: Exam I 3/5.

Assignment

Read 4.6.

From Last Time

MIPS ALU.

Outline

1. Preliminaries, ``constant time addition.''

2. Four-bit full carry lookahead adder.

3. Four-big Group carry lookahead unit.

4. Building large adders.

Coming Up

Multiplication.

Carry Lookahead Addition

1. Why bother with speeding up addition?

2. Divide and conquer approach to addition: , where .

3. Can we pre-compute the carries?

4. Carry generate: .

5. Carry propagate: .

6. Illustration: look at p, g, and c for 11010101 and 01110011.

7. Some carry equations:
   1. 

   2. 

   3. 

8. A bit of recursion:
   1. Base case:

   2. Recursive step:

9. Why this isn't feasible. The Winograd lower bound.

Carry Lookahead: The Big Picture

Restricting the carry computation circuitry to a tree structure:

What does this buy us?

Four-Bit Carry Lookahead Adder

1. Design a four-bit full carry lookahead adder.

   Block diagram:

   

   Block generate, propagate.

2. What is the fan-in?

3. What is the delay model from inputs to outputs?

4-Bit Group Carry Lookahead Unit

1. Design a 4-Group carry lookahead unit.

   Block diagram:

   

   Use of block generates, propagates.

2. What is the fan-in?

3. What is the delay model from inputs to outputs?

16-Bit Carry Lookahead Adders

Cascaded and full carry lookahead.

32-Bit Carry Lookahead Adders

Cascaded and full carry lookahead.