Page 45 - Analog and Digital Filter Design

42 Analog and Digital Filter Design

known and the type of filter has been selected, the appropriate normalized fre-
quency response curve can be used to find the filter order required. The lowest
filter order to achieve the desired stopband attenuation is usually chosen because
the filter will be simpler and lower cost.

For example, if the design must pass DC and signals up to a frequency F1, but
attenuate signals above frequency F2, a lowpass filter is required. The ratio of
F2 to F1 will be, by necessity, greater than one. Let’s say the frequency ratio is
2.0: the passband must be flat and the attenuation required is 40dB. Graphs
showing the attenuation versus frequency for a number of filter orders are given
in this chapter; use the Butterworth response graph shown in Figure 2.10 to find
the required filter order. Since the plot is normalized, the frequency axis is equal
to the stopband to passband ratio in the final (denormalized) filter. For 40dB
attenuation at o = 2 (the frequency ratio is equal to two), the required filter order
is seven (n = 7).

In this example a Butterworth response was used because it has a smooth
passband. Other responses also have a smooth passband: these are the Bessel
and the Inverse Chebyshev. The Bessel response does not have sufficient atten-
uation at a frequency ratio of two, no matter how high the filter order. The
Inverse Chebyshev response would require a fifth-order filter, to give 40 dB atten-
uation, but practically would be more difficult to make. This chapter will provide
the designer with the information so that he or she can make the correct
approach.
The most popular responses will be described in more detail later. In Chapter
1, the passband and stopband of a filter were described. The frequencies that
are intended to pass through the filter with very little loss determine the pass-
band. Those frequencies where a certain level of attenuation is required deter-
mine the stopband. There are four basic responses that can be made from the
combination of flat or rippled passband and smooth or rippled stopband.
I will show how standard Bessel, Butterworth, Chebyshev, Cauer, and Inverse
Chebyshev approximations have one of these responses. Graphs will be used to
describe the shape of each frequency response.

The subject of Inverse Chebyshev filters will be covered in some detail. In par-
ticular I will show how to obtain a more practical 3dB cutoff point, rather than
have the filter normalized at the stopband. I will also give tables for third- and
fifth-order passive filters.

Practical filters are characterized by passband, stopband, and skirt response, as
shown in Figure 2.1. The passband is the region where the loss is less than at
the cutoff point. If the cutoff point is at, say, 1 dB, then all frequencies at which
the loss is lower than 1 dB are in the passband. The stopband is the region of