of Microprocessors

DIGITAL FILTERING
495
OUTPUT
Fig. 14-9. Cannonical second order digital filter
amplitude response curve, while the feedforward paths cause response dips or
zeroes. Note that the same design equations for the filters given earlier can be
applied to the cannonical form with any unused constants set to zero. Because
of its generality, this structure is ideal for use in a genetal-purpose filtering
subroutine or hardware implementation as a digital filter module.
All-Pass Digital Filter
One important type of filter that has not yet been discussed is the
all-pass filter. As its name implies, it is not a frequency-discriminating filter
and in fact passes all frequencies equally well with unity gain. Instead, its
purpose is frequency-sensitive phase shift. All filters shift phase in varying
amounts as a natural consequence of frequency discrimination, but the allpass
filter exhibits phase shift alone. Consequently, an all-pass network can
be used to correct for the phase shift of a regular filter without disturbing its
amplitude response. In synthesis applications, however, it is used to introduce
a time-varying, frequency-sensitive phase shift to otherwise stationary
sounds, thereby imparting added richness to the sound.
Before looking at the filter itself, the relation between phase shift and
time delay should be understood. Either parameter can be plotted as a
function of frequency as in Fig. 14-10. One type of all-pass filter is a simple
delay line. As shown in Fig. 14-lOA, an ideal delay line has constant delay
independent of frequency. Its phase shift, however, is a linear function of
frequency as shown in Fig. 14-lOB. This can be explained by noting that at
very low frequencies the 500-1LSec delay is only a small fraction ofa cycle. At
500 Hz, which is a 2,000-J.Lsec period, the 500-J.Lsec delay becomes a quarter
cycle phase lag or -90°. At 1 kHz, the delay becomes a half cycle and so on.
At higher frequencies, the phase-shift magnitude continues to increase, but
it is customary to plot the principal value of the shift. Thus, the curve shifts
up to 180° leading and continues its decline from there.
In mathematical terms, the delay curve is the derivative (proportional
to slope) of the phase curve or, conversely, the phase curve is the integral