Quantized Filter Analysis - Introduction to Digital Signal Processing and Filter Design - page 355

great demand for designing digital filters and systems in which they are embed-
ded, with the lowest possible number of bits to represent the data or to store the
data in their registers. When the filters are built with registers of finite length and
the analog-to-digital converters (ADCs) are designed to operate at increasingly
high sampling rates, thereby reducing the number of bits with which the samples
of the input signal are represented, the frequency response of the filters and the
results of DFT-IDFT computations via the FFT are expected to differ from those
designed with “infinite precision.” This process of representing the data with a
finite number of bits is known as quantization , which occurs at several points
in the structure chosen to realize the filter or the steps in the FFT computation
of the DFT-IDFT. As pointed out in the previous chapter, a vast number of
structures are available to realize a given transfer function, when we assume infi-
nite precision. But when we design the hardware with registers of finite length to
implement their corresponding difference equation, the effect of finite wordlength
is highly dependent on the structure. Therefore we find it necessary to analyze
this effect for a large number of structures. This analysis is further compounded
by the fact that quantization can be carried out in several ways and the arithmetic
operations of addition and multiplication of numbers with finite precision yield
results that are influenced by the way that these numbers are quantized.
In this chapter, we discuss a new MATLAB toolbox called FDA Tool avail-
able 1 for analyzing and designing the filters with a finite number of bits for the
wordlength. The different form of representing binary numbers and the results of
adding and multiplying such numbers will be explained in a later section of this
chapter. The third factor that influences the deviation of filter performance from
the ideal case is the choice of FIR or IIR filter. The type of approximation chosen
for obtaining the desired frequency response is another factor that also influences
the effect of finite wordlength. We discuss the effects of all these factors in this
chapter, illustrating their influence by means of a design example.
7.2 FILTER DESIGN-ANALYSIS TOOL
An enormous amount of research has been carried out to address these problems,
but analyzing the effects of quantization on the performance of digital filters
and systems is not well illustrated by specific examples. Although there is no
analytical method available at present to design or analyze a filter with finite
precision, some useful insight can be obtained from the research work, which
serves as a guideline in making preliminary decisions on the choice of suitable
structures and quantization forms. Any student interested in this research work
should read the material on finite wordlength effects found in other textbooks
[1,2,4]. In this chapter, we discuss the software for filter design and analysis
that has been developed by The MathWorks to address the abovementioned
1 MATLAB and its Signal Processing Toolbox are found in computer systems of many schools and
universities but the FDA Tool may not be available in all of them.