**Tom Kelliher, CS 240**

**Feb. 27, 2012**

Pay attention!!! A future homework assignment will cover this material, which isn't in the textbook.

Read 4.6-4.7.

Addition limits.

- Carry lookahead addition.
- Signed digit representations.

Introduction to VHDL.

- Now, we demonstrate a feasible
adder.
- Recall:
- Carry generate: .
- Carry propagate:
.

- Carry generate: .

Restricting the carry computation circuitry to a tree structure:

- Leaves: Four-bit carry lookahead adders.
- Non-Leaves: Four-bit carry lookahead group units.

- Design a four-bit full carry lookahead adder.
Block diagram:

Block generate, propagate.

- What is the fan-in?
- What is the delay model from inputs to outputs?

- Design a 4-Group carry lookahead unit.
Block diagram:

Use of block generates, propagates.

- What is the fan-in?
- What is the delay model from inputs to outputs?

Total gate delays for ripple-carry adder.

Gate delays for cascaded and full carry lookahead adders.

- Consider the digit set of the maximally redundant signed digit
representation for radix :
- For radix 2 we have:
.
Radix 4: .

- For some values, there are multiple representations. For example:
(radix 2).
- This redundancy can be exploited so that we can design constant time signed
digit adders.

- Idea: Ensure that a carry propagates no further than two bit positions.
- Circuit sketch:
- Stage 1 adder addition table:
**Addend + Augend****Carry****Sum**0 1 0 0 0 1 0 1 2 1 0 - Stage 2 adder addition table:
**Addend + Augend****Carry****Sum**0 0 0 0 1 1 2 1 0 - Final stage addition table:
**Addend + Augend****Carry****Sum**0 0 0 0 1 0 1

Thomas P. Kelliher 2012-02-24 Tom Kelliher