of Microprocessors

DIGITAL FILTERING
497
INPUT ~8~~~~~---.-~_. ~ =
r + ~ +r
A
~1--+---"""'8>
I
F = TURNOVER FREQUENCY
o ~ _F_ WHERE Frl "'- UPPEA 90~ POINT AND
FH-L Fl = LOWER 90° POINT AND
FH- FL IS TRANSITION WIDTH
( - rrF)
A '" 2 cos (21TF) exp -0-
( -2rrF)
B = exp -0-
Fig. 14-11. Digital all-pass filter
smeared out in time. The effect is called dispersion and is quite analogous to
the effect of a prism on white light. Figures 14-lOC and D illustrate an
extreme case of dispersion. Usually, high frequencies are delayed least, which
means that the transient is converted into a quickly descending frequency
sweep having a "thunk"-like sound. Such an extreme case gives the auditory
illusion of the sound being sent over a lOO-foot stretched wire and in fact is
heard frequently on long-distance telephone circuits. Such a large amount of
dispersion requires many filter sections in combination to emulate.
Figure 14-11 shows a second order all-pass filter that is a useful building
block. Like all other second order digital filters, it is an adaptation of the
cannonical form; however, due to symmetry of the constants and unity gain,
three of the five multiplications can be bypassed. Examination of the constants
reveals that the feedforward and feedback paths are completely complementary.
Essentially, the zeroes cancel the poles to produce a"flat amplitude
response, but since phase shifts add rather than multiply, the frequencysensitive
phase shift is reinforced.
The phase and delay characteristics of the basic second order all-pass
filter section are shown in Fig. 14-12. At low frequencies, the phase shift is
near zero and at high frequencies it is 360° with a monotonic, though
nonlinear, transition between. Two parameters describe the filter. One is the
"turnover frequency" at which the phase shift is 180°. The other is called the
"transition width" and is related to the sharpness of the transition from 0° to
360° shift, quite analogous to the Q of a bandpass filter. The edges of the
transition zone are where the phase shift is 90° and 270°. The delay curve