Page 476 - DSP Integrated Circuits

11

PROCESSING ELEMENTS

11.1 INTRODUCTION

In this chapter we will discuss the implementation of basic arithmetic operations
such as addition, subtraction, and multiplication using fixedpoint arithmetic, as
well as compound operations such as multiple-constant multiplication, vector-multi-
plication, and multiplication involving complex values. The two latter types of arith-
metic operations can be implemented using so-called distributed arithmetic, which
will be discussed in detail in section 11.14. Arithmetic operations used in DSP algo-
rithms can be implemented with many different techniques. For example, the data
involved can be either bit-serial or bit-parallel processed. Arithmetic operations can
also be done by using different binary representations for the negative numbers.
These representations differ with respect to their implementation cost (space-time-
power). In this chapter we will therefore first review the types of number represen-
tations and arithmetic most commonly used in DSP algorithms.

11.2 CONVENTIONAL NUMBER SYSTEMS

Not all numbers were created equal.

A number system is defined by the set of values that each digit can assume and by
an interpretation rule that defines the mapping between the sequence of digits and
their numerical values. Conventional number systems are nonredundant, weighted,
1
positional number systems. In such number systems every number, x, represented
x
by a sequence of digits (XQX^, •••> w - P ^ as a unique representation.
The value of a number x, is

!• We will use bold face to denote the number and plain face to denote the digits in the
corresponding sequence of digits whenever both are used in the same context.