of Microprocessors

56 MUSICAL ApPLICATIONS OF MICROPROCESSORS
although the phase axis is usually linear. Actually, for simple filter circuitry,
the shape of the phase response is rigidly tied to the shape of the amplitude
response. For use as a sound modifier, the phase response of a filter is usually
ignored.
Theoretically, the shape of an amplitude response curve may be anything
desired. However, like the principle that any waveshape may be built
from sine shapes, amplitude response curves may be built from a small class
of basic shapes. Figure 2-4 shows some of these..
Shape A is called a low-pass response because the lower frequencies are
passed without attenuation (reduction in amplitude), while the higher frequencies
are reduced considerably. Although several parameters may be
necessary to fully specify such a shape, twO are of primary importance in
sound modification applications. One is the cutoff frequency or the frequency
above which the attenuation really begins to increase. By convention, the
cutofffrequency is the frequency at which the amplitude response is 3 dB less
(one-half the power output or 0.7071 times as much voltage) than it is at
very low frequencies. The other parameter is the cutoff slope. In a practical
filter of minimal or moderate complexity, the slope of amplitude decrease
beyond the cutoff frequency approaches an asymptote, which is a straight
line. Cutoff slope is usually stated in decibels per octave, particularly for
musical purposes. Actually, most simple filter circuits have cutoffslopes that
are multiples of 6 dB/octave. Thus, a simple low-pass filter might have a
slope of 6, 12, 18, etc., dB/octave.
Shape B is called a high-pass response for the same reason A was termed
low-pass. The parameters of the high-pass response are also similar.
Shape C is called a bandpass response. This is because in the general case
a small band of frequencies are passed and the others, both higher and lower,
are rejected. Two parameters are generally used to characterize bandpass
responses, although four are required for completeness. The frequency corresponding
to the top of the peak is variously termed the center frequency,
natural frequency, resonant frequency, or pole frequency. The natural and
resonant frequencies are actually very slightly different from the true center
frequency, but for musical purposes they are all identical.
The width of the curve can also be specified in different ways. One
common method calls for measuring the frequencies of the two 3-dB down
points, subtracting them, and calling the result the bandwidth in hertz. In
music, it is more useful to specify bandwidth in octaves, thus the term" 1/3
octave bandpass filter" is frequently encountered. A formula for the octave
bandwidth in terms of the lower cutoff, FL, upper cutoff, FU J
and center
frequency, PC is BW = 10g2 [1 + (FU -PL)/FC]. A final method, which only
applies to a certain but very common class of bandpass filters, is the quality
factor or Q. Q is defined by the relation: Q = FC/(FV - FL). The significance
of Q will be studied in greater detail later.