Carry Lookahead and Signed-Digit Addition

Tom Kelliher, CS 240

Feb. 26, 2010

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

Assignment

Read 4.6-4.7.

From Last Time

Addition limits.

Outline

  1. Carry lookahead addition.

  2. Signed digit representations.

Coming Up

Introduction to VHDL.

Carry Lookahead Addition

  1. Now, we demonstrate a feasible ${\rm O}(\log n)$ adder.

  2. Recall:
    1. Carry generate: $g_i = a_i b_i$.

    2. Carry propagate: $p_i = a_i \oplus b_i$.

Carry Lookahead: The Big Picture

Restricting the carry computation circuitry to a tree structure:

\begin{figure}\centering\includegraphics[width=6in]{Figures/carryTree.eps}\end{figure}

Four-Bit Carry Lookahead Adder

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

    Block diagram:

    \begin{figure}\centering\includegraphics[]{Figures/carrylookahead.eps}\end{figure}

    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:

    \begin{figure}\centering\includegraphics[]{Figures/lookaheadunit.eps}\end{figure}

    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

Total gate delays for ripple-carry adder.

Gate delays for cascaded and full carry lookahead adders.

Signed Digit Representations

  1. Consider the digit set of the maximally redundant signed digit representation for radix $r$: $\{\overline{r-1}, \overline{r-2}, \ldots, \overline{1}, 0, 1, \ldots, r-1 \}$

  2. For radix 2 we have: $\{\overline{1}, 0, 1\}$.

    Radix 4: $\{\overline{3}, \overline{2}, \ldots, 3\}$.

  3. For some values, there are multiple representations. For example: $3 = 011 = 10{\overline 1}$ (radix 2).

  4. This redundancy can be exploited so that we can design constant time signed digit adders.

Constant Time Radix 2 Signed Digit Adder

  1. Idea: Ensure that a carry propagates no further than two bit positions.

  2. Circuit sketch:

    \begin{figure}\centering\includegraphics[width=6in]{Figures/sdAdder.eps}\end{figure}

  3. Stage 1 adder addition table:
    Addend + Augend Carry Sum
    $\overline{2}$ $\overline{1}$ 0
    $\overline{1}$ $\overline{1}$ 1
    0 0 0
    1 0 1
    2 1 0
    Goal: Ensure sums are $\geq 0$ to eliminate -2 as a possible starting sum in the next stage.

  4. Stage 2 adder addition table:
    Addend + Augend Carry Sum
    $\overline{1}$ 0 $\overline{1}$
    0 0 0
    1 1 $\overline{1}$
    2 1 0
    Goal: Ensure sums are $\leq 0$ to eliminate 2 and -2 as possible starting sums in the next stage.

  5. Final stage addition table:
    Addend + Augend Carry Sum
    $\overline{1}$ 0 $\overline{1}$
    0 0 0
    1 0 1


Thomas P. Kelliher 2010-02-25
Tom Kelliher