CHAPTER 11 Introduction to the Design of Discrete Filters..........................................
When in doubt, don’t.
Benjamin Franklin (1706–1790)
Printer, inventor, scientist, and diplomat11.1 Introduction
In this chapter we introduce the design of discrete filters. This material complements the theory of
analog and discrete filtering presented in previous chapters, and in particular provides continuity to
the introduction to analog filter design from Chapter 6.
Filtering is an important application of linear time-invariant (LTI) systems. According to the eigen-
function property of LTI systems (Figure 11.1) the steady-state response of a discrete-time LTI system
to a sinusoidal input—with a certain frequency, magnitude, and phase—is also a sinusoid of the
same frequency as the input, but with the magnitude and the phase affected by the response of the
system at the input frequency. Since periodic as well as aperiodic signals have Fourier representations
consisting of sinusoids of different frequencies, the frequency components of any signal can be mod-
ified by appropriately choosing the frequency response of an LTI system or filter. Filtering can thus be
seen as a way to change the frequency content of an input signal.
The appropriate filter is specified using the spectral characterization of the input and the desired spec-
tral characteristics of the output of the filter. Once the specifications of the filter are set, the problem
becomes one of approximation, either by a ratio of polynomials or by a polynomial (if possible).
After establishing that the filter resulting from the approximation satisfies the given specifications, it
is then necessary to check its stability (if not guaranteed by the design method) in the case of the filter
being a rational approximation, and if stable we need to figure out what would be the best possible
way to implement the filter in either hardware or software. If not stable, we need either to repeat the
approximation or to stabilize the filter before its implementation.
In the continuous-time domain, filters are obtained by means of rational approximation. In the
discrete-time domain, there are two possible types of filters. The first is the result of rational
Signals and Systems Using MATLAB®. DOI: 10.1016/B978-0-12-374716-7.00015-6
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