Carry-Save Addition

Tom Kelliher, CS 315

Apr. 5, 1999

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Assignment

Read 1.2.4.

From Last Time

Segmented prefix problem.

Outline

1. Brief review of carry lookahead addition.

2. Carry-Save addition.

Coming Up

Multiplication and convolution.

Previous Results

Carrylookahead addition:

1. Add two k-bit numbers in bit steps.

2. Suppose we wish to add N k-bit numbers as quickly as possible.
   1. What is the organization of the adder?

   2. Do the adders become wider as we progress?

   3. Running time:

      

      Are these really equivalent?

      Can we do better?

Carry-Save Addition

1. Idea: Add three k-bit numbers in a single bit step, producing two k+1-bit numbers.

2. Implementation: Use a binary full adder for each bit position, producing a low order bit and a carry bit for each bit position.

3. The final sum is the sum of the low order bits and the shifted carry bits.

4. Example:

   

5. Time required to reduce the sum of N k-bit numbers to the sum of two -bit numbers: bit steps.

   Proof:

   1. After the first step of CSA, how many numbers are we left with? (At most . Why?)

   2. After the second step?

   3. After the jth step:

      

   4. After steps, we're left with numbers, so we need steps to reduce to two -bit numbers.

6. Use a carry lookahead adder to reduce the two-number redundant representation to a single, non-redundant number.

7. Example: Draw the Wallace tree for N = 9.