Page 50 - Analog and Digital Filter Design

Time and Frequency Response 47

selected set of values, scale for the frequency, impedance, and frequency
response (lowpass, highpass. etc.) as required.
Active filters are designed using S-plane pole and zero locations, which
are described in more detail in Chapter 3. Basically, a table of pole and zero
locations are used in conjunction with simple equations to find component
values. They can be used in a similar way to tables of normalized component
values, like those used in passive filter design, by scaling for the frequency and
response required. The source or load impedance does not affect pole and zero
locations.

The process of denormalizing pole and zero locations, or passive component
values, is explained in Chapters 4, 5,6, and 7. These describe lowpass, highpass,
bandpass, and bandstop responses, respectively.

Normalized Lowpass Responses

I will now describe Bessel, Butterworth, Chebyshev, Inverse Chebyshev, and
Cauer responses, describing the frequency and time domain characteristics of
each response in turn. Tables of normalized passive filter component values and
the processes to denormalize them will be given in this chapter. Pole and zero
locations, and circuit designs relevant to active filters will be given in the next
chapter.

Bessel Response

The Bessel response is smooth in the passband, and attenuation rises smoothly
in the stopband. The stopband attenuation increases very slowly until the signal
frequency is several times higher than the cutoff point. Far away from the cutoff
point the attenuation rises at n x 6dB/octave, where n is the filter order and an
octave is the doubling of frequency. For example, a third-order filter will give
an 18 dB/octave rise in attenuation. The slow rise in attenuation near the cutoff
frequency gives it an excellent time domain response, but this is not very useful
in removing unwanted signals just outside the passband.

The natural cutoff frequency for the Bessel response is that which gives a one-
second delay. This is not a constant value, but depends on the filter order. To
make the design process simpler, the Bessel response can be scaled to give a
1
3 dB cutoff frequency at o= for all filter orders. To do this the frequency com-
ponents of the transfer function have to be scaled. This has been done for the
fifth-order Bessel filter, and the response curve is shown in Figure 2.4. In this
graph, the attenuation versus frequency plot is given with the frequency axis
normalized. Also in Figure 2.4 is the attenuation versus frequency plot of a