Quantized Filter Analysis - Introduction to Digital Signal Processing and Filter Design - page 354

CHAPTER 7

Quantized Filter Analysis

7.1 INTRODUCTION

The analysis and design of discrete-time systems, digital filters, and their realiza-
tions, computation of DFT-IDFT, and so on discussed in the previous chapters
of this topic were carried out by using mostly the functions in the Signal Pro-
cessing Toolbox working in the MATLAB environment, and the computations
were carried out with double precision. This means that all the data representing
the values of the input signal, coefficients of the filters, or the values of the unit
impulse response, and so forth were represented with 64 bits; therefore, these
numbers have a range approximately between 10 − 308 and 10 308 and a precision
of

= 2 . 22 × 10 − 6 . Obviously this range is so large and the precision with
which the numbers are expressed is so small that the numbers can be assumed to
have almost “infinite precision.” Once these digital filters and DFT-IDFT have
been obtained by the procedures described so far, they can be further analyzed
by mainframe computers, workstations, and PCs under “infinite precision.” But
when the algorithms describing the digital filters and FFT computations have
to be implemented as hardware in the form of special-purpose microprocessors
or application-specific integrated circuits (ASICs) or the digital signal processor
(DSP) chip, many practical considerations and constraints come into play. The
registers used in these hardware systems, to store the numbers have finite length,
and the memory capacity required for processing the data is determined by the
number of bits—also called the wordlength —chosen for storing the data. More
memory means more power consumption and hence the need to minimize the
wordlength. In microprocessors and DSP chips and even in workstations and PCs,
we would like to use registers with as few bits as possible and yet obtain high
computational speed, low power, and low cost. But such portable devices such as
cell phones and personal digital assistants (PDAs) have a limited amount of mem-
ory, containing batteries with low voltage and short duration of power supply.
These constraints become more severe in other devices such as digital hearing
aids and biomedical probes embedded in capsules to be swallowed. So there is a

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