of Microprocessors

386 MUSICAL ApPLICATIONS OF MICROPROCESSORS
dis-cuss the general characteristics of various filter types so that the reader can
make an intelligent decision in choosing one.
The simplest type of low-pass filter is the single-pole R-C. U nfortunately,
its gentle cutoffslope of6 dB/octave is totally inadequate for an audio
DAC. Also its passband flatness is not very good.
In order to get sharper slopes and flatter passbands, several filter sections
may be combined together. There are two methods of combination called
parallel and cascade, which are shown in Fig. 12-13. In the parallel setup, the
same raw input signal is filtered by each of the sections and then their
outputs are combined together, not necessarily equally, in the mixer. In the
cascade arrangement, the signal passes through filter sections one after
another. Thus, any filtering action of the second stage is in addition to that
of the first stage and so forth.
With the cascade arrangement, it is easy to determine the total
amplitude response if the amplitude response of each section is known. The
filter gain at any given frequency is simply the product of the section gains at
that frequency. If gains are expressed in decibels (usually negative, since a
filter is designed to attenuate certain frequencies), then the overall decibel gain
is simply the sum of the section decibel gains.
The overall response of the parallel arrangement is considerably more
difficult to determine, since the phase response of the sections must also be
known. If the section outputs are out of phase, which is the usual case, then
their sum in the mixer will be less than the sum of their gains. Nevertheless,
there are certain advantages of the parallel arrangement. Also, for the types of
filters that will be discussed, any response curve that can be obtained with
one arrangement can be precisely duplicated using the same number of
sections of the same complexity wired in the other arrangement, although
the individual section responses will be different. Thus, for convenience, the
examples will use the cascade arrangement.
ltertttive R-C Low-Pass Filter
Returning to the simple R-C filter, Fig. 12-14 shows what can be done
by cascading these simple sections using a unity-gain buffer amplifier between
each section for isolation. The curves are all normalized so that the
-3-dB frequency is the same for each curve. For any individual curve, all of
the sections are identical. However, each curve requires sections with a
different cutoff frequency. As can be seen, adding more sections improves
cutoff slope, although passband flatness is affected only slightly. However,
even 32 sections does not give a very sharp cutofffor the first 50 dB, which is
the most important region for sample filtering. Using the 32-section filter,
the sample rate must be 5.45 times the - 3-dB frequency to be assured of
- 50-dB alias distortion. This type of filter is termed "iterative R-C" and is
used primarily where overshoot and ringing cannot be tolerated in the step
response.