# 11

# Signed-Digit Number Operations

In a signed-digit (SD) number system, carry propagation can be limited to one position to the left during the digit-wise addition and subtraction. The addition time is independent of the word length since the chains of carry-propagations are eliminated.

## 11.1 CHARACTERISTICS OF SD NUMBERS

In a conventional number representation with an integer radix *r* > 1, each digit is allowed to have exactly *r* values: 0,1, · · · , *r* − 1. In an SD representation with the same radix *r*, each digit is allowed to have more than *r* values. In the method of addition described below, each digit is allowed to have *q* different values where

Obviously *q* is more than *r.* Redundancy in the number representation allows a method of fast addition/subtraction called *totally parallel* addition/subtraction.

In the totally parallel addition/subtraction, the signed-digit representations are required to have a unique representation of zero. Thus the magnitude of allowed digit values may not exceed *r* − 1, since otherwise we could let a digit equal to *r* and represent zero by *r* as well as 00.

Some characteristics of the SD numbers are as follows.

Let *A* = (*a*_{n−1} · · · *a*_{−k}) be an SD number, we have

- where
- Let
*p*=*max*{*i*|*a*≠ 0},_{i}*sign*(*a*_{n−1}· · ·*a*_{−k}) =*sign*(*a*) for (_{p}*a*_{n−1}· · ·*a*_{−k}) ≠ 0.

## 11.2 ...

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