of Microprocessors

394 MUSICAL ApPLICATIONS OF MICROPROCESSORS
12-11 showed that the rolloff is about 4 dB for 100% width and less than 1
dB for 50% width at one-half the sample frequency. With reasonably good
filters, this would amount to no more than 2 dB of loss at the top of the
frequency range. For the utmost in fidelity, it may be desirable to
compensate for the rolloff, particularly if the pulse width is equal to the
sample period (stairstep DAC output). Correction for this effect can best be
accomplished in the low-pass filter. Although it is mathematically possible to
design a filter with other than flat passbands, it is a very involved process.
If a Butterworth filter is being used, a slight rise in response just before
cutoff can be effected by raising the Qs of the lowest Q stages. This action
does not have much effect on the cutoff characteristics and in fact can be
expected to raise the 50-dB cutoff point no more than a couple of decibels.
Correcting a Chebyshev or elliptical filter is best accomplished by adding
another re$onant low-pass section with a resonant frequency somewhat beyond
the main filter cutoff frequency. Q factors in the range of 1 to 2 for the
added section will provide a gentle but accelerating rise in the otherwise flat
passband, which then abruptly cuts off as before. Any resonant peak in the
extra section will occur so far into the main filter stopband that its effect will
be completely insignificant.
Figure 12-22A shows the typical performance of such a compensation
stage. The frequency scale has been normalized so that 1. 0 represents the
cutoff frequency of the main low-pass filter; thus, compensation accuracy
above 1. 0 is of no concern. Curve A shows the amplitude response of a DAC
with stair-step output (100% pulse width) at a sample rate 2.5 times the
main filter cutofffrequency and followed by a deglitcher with a time constant
that is 10% of the sample period. Since a finite deglitcher time constant also
represents a slight high frequency loss, it is appropriate to compensate for it
at the same time. Curve B is the amplitude response of the compensation
stage, and curve C is the composite, compensated response.
The short BASIC program in Fig. 12-22B can be used to iteratively
design compensation filters. The variables FO and QO are the frequency and Q
parameters of the compensation filter normalized to the main filter cutoff
frequency. FI is the sample rate, also normalized. TO is the DAC output
pulse width, and TI is the time constant of the deglitcher, both as fractions
of the sample period. For track-and-ground deglitchers, set T1 to zero. To
use the program, set the FI, TO, and TI variables to match the DAC
operating conditions and "guess" at values for FO and QO. Then run the
program and note the shape of the compensated curve either by manually
plotting it or adding plotting statements to the program. The goal is
accurate compensation up to 1.0 with a gradual rolloff for higher frequencies
as in Fig. 12-22A. Good starting values are those used to produce the curves
in A; PI = 2.5, TO = 1.0, T1 = 0.1, FO = 1.58, and QO = 1.45.
Fig. 12-22. (A) Two-pole sin(X)/X compensator performance. (B)
program to evaluate compensation filters.
BASIC