of Microprocessors

482 MUSICAL ApPLICATIONS OF MICROPROCESSORS
INPUT 0-""""' ....-.
(8)
C
AP'""""~~OUTPUT
F=_Ic
2"RloakC
GAIN= Rloak
Rgain
Fig. 14-1. Analog R-C filters. (A) Passive implementation. (B) Active implementation.
zero phase shift at all passband frequencies is asking exactly that. For example,
if the filter were presented the first three samples of a low-frequency yet
high-amplitude wave, it would have no way of "knowing" whether it really
was part of a low-frequency cycle or part of a high-frequency but lowamplitude
cycle without further data. Zero phase shift implies that such a
decision is made immediately and the samples either pass to the output or are
blocked. Linear phase shift, which implies constant time delay independent of
frequency, however, is readily available. Note that for filtering outside of real
time this constraint can be effectively overcome by delaying everything else
to match.
Digital filtering, like other aspects of digital signal processing, can be
highly mathematical. After all, a digital filter is nothing more than a
mathematical function that accepts a string of input numbers and provides a
string of output numbers. In this chapter, however, digital filters will be
discussed as a natural outgrowth of fundamental analog filtering circuits.
Later, an intuitive approach will be used to discuss filters with an arbitrary
amplitude response shape. It should be noted that many of the filtering
concepts mentioned here are equally applicable to analog filtering and some
in fact were not mentioned in the sections on analog filtering.
Digital Equivalents of Analog Filters
The simplest analog filters consist of just two components: a resistor
and a capacitor. These can be configured in two ways to provide single-pole
(6 dB/octave) low-pass or high-pass filters. For the moment, we will concentrate
on the low-pass circuit.
Back in Chapter 6, it was shown that the exact same R-C low-pass
response could be obtained with an op-amp and an R-C feedback circuit such
as in Fig. 14-1. Careful examination of this circuit reveals a standard analog
integrator with a "leak" resistor placed across the capacitor. The resistor
causes the capacitor charge to leak away and thereby puts an upper limit on
the very low frequency and dc gain of the circuit. In fact, the 3-dB attenuation
point is the frequency at which the capacitive reactance equals the leak