Synthesis of Arithmetic Circuits: FPGA, ASIC and Embedded Systems

7.1 BASE CONVERSION

Given the representation of a number in a specific system, conversion operations consist of finding the representation of this number in another system. Arithmetic algorithms deal with a diversity of systems such as base-B for naturals or signed systems for integers; the most common signed systems are sign-magnitude, B's complement, excess-k, or signed-digit systems such as Booth coding. Finally, special attention is paid to floating-point representations for computer applications. Redundant systems are also important in arithmetic operations, among others in multiplication (Booth algorithm, Chapter 5) or division (SRT, Chapter 6). Booth coding and redundant base-B coding are generally related to specific algorithms; the conversion techniques are therefore developed in the sections devoted to the respective algorithms. Floating-point conversion is reviewed in Chapter 16. The most classic problem to deal with is the base conversion for base-B unsigned representations: given a number by its representation in base B₁, find its corresponding representation in base B₂.

Let the base-B₁ (source system) representation of a natural number x be given by

weighted sum of powers of B. The problem at hand is to compute the base-B₂ (target system) representation of x

A first ...