System Design Flow and Fixed-point Arithmetic - Digital Design of Signal Processing Systems: A Practical Approach - page 81

3
System Design Flow and
Fixed-point Arithmetic
Pure mathematics is, in its way, the poetry of logical ideas.
Albert Einstein
3.1 Overview
This chapter describes a typical design cycle in the implementation of a signal processing
application. The first step in the cycle is to capture the requirements and specifications (R&S)
of the system. The R&S usually specify the sampling rate, a quantitative measure of the system's
performance, and other application-specific parameters. The R&S constrain the designer to explore
different design options and algorithms to meet them in the most economical manner. The algorithm
exploration is usually facilitated by MATLAB , which is rich in toolsets, libraries and functions.
After implementation and analysis of the algorithm inMATLAB , usually the code is translated into
higher level programming languages, for example, C/C þþ or C#.
This requires the chapter to focus on numbering systems. Representing signed numbers in two's
complement format is explained. In this representation, the most significant bit (MSB) has negative
weight and the remainder of the bits carry positive weights. Although the two's complement
arithmetic greatly helps addition and subtraction, as subtraction can be achieved by addition, the
negative weight of the sign bit influences multiplication and shifting operations. As the MSB of
a signed number carries negative weight, the multiplication operation requires different handling for
different types of operand. There are four possible combinations for multiplication of two numbers:
unsigned-unsigned, signed-unsigned, unsigned-signed and signed-signed. The chapter describes
two's complement multiplication to highlight these differences. Scaling of signed number is
described as it is often needed while implementing algorithms in fixed-point format.
Characteristics of two's complement arithmetic from the hardware (HW) perspective are listed
with special emphasis on corner cases. The designer needs to add additional logic to check the corner
cases and deal with them if they occur. The chapter then explains floating-point format and builds the
rationale of using an alternate fixed-point format for DSP system implementation. The chapter also