Page 17 - Analog and Digital Filter Design

1 4 Analog and Digital Filter Design

The principles behind digital filters are based on the relationship between the
time and frequency domains. Although digital filters can be designed without
knowledge of this relationship, a basic awareness makes the process far more
understandable. The relationship between the time and frequency domains can
be grasped by performing a practical test: apply a range of signals to both the
input of an oscilloscope and the input of a spectrum analyzer, and then compare
the instrument displays. More formally, Fourier and Laplace transforms are
used to convert between the time and frequency domains. A brief introduction
to these is given in chapter 3. Whole books are devoted to the Fourier and
Laplace transforms; references are given in the Bibliography.

All the designs described in this book have been either built by myself or sim-
ulated using circuit analysis software on a personal computer. As is the case in
all filter design books, not every possible design topology is included. However,
I have included useful material that is hard to find in other filter design books.
such as Inverse Chebyshev filters and filter noise bandwidth. I have researched
many filter design books and papers in search of simple design methods to
reduce the amount of mathematics required.

Chapters have been arranged in what I think is a logical order. A summary of
the chapters in this book follows.

Chapter 1 gives examples of filter applications, to explain why filter design is
such an important topic. A description of the limitations for a number of filter
types is given; this will help the designer to decide whether to use an active,
passive, or digital filter. Basic filter terminology and an overview of the design
process are also discussed.

Chapter 2 describes the frequency response characteristics of filters, both ideal
and practical. Ideally, filters should not attenuate wanted signals but give infi-
nite attenuation to unwanted signals. This response is known as a brick wall
filter: it does not exist, but approximations to it are possible. The four basic
responses are described (Le.. flat or rippled passband and smooth or rippled
stopband) and show how standard Bessel, Butterworth, Chebyshev, Cauer, and
Inverse Chebyshev approximations have one of these responses. Graphs describe
the shape of each frequency response.

A very important topic of this chapter is the use of normalized lowpass filters
with a 1 rad/s cutoff frequency. Normalized lowpass filters can be used as a basis
for any filter design. For example, a normalized lowpass filter can be scaled to
design a lowpass filter with any cutoff frequency. Also, with only slightly more
difficulty, the normalized design can be translated into highpass, bandpass, and
bandstop designs. Tables of component values for some normalized approxi-
mations are given. Formulae for deriving these tables are also provided, where
applicable.