of Microprocessors

DIGITAL-TO-ANALOG AND ANALOG-TO-DIGITAL CONVERSION 385
INPUT
OUTPUT
(A)
INPUT
OUTPUT
(8)
Fig. 12-13. Methods of filter section combination. (A) Parallel method. (B)
Cascade method.
kHz, the interval from 5 kHz to 7 kHz is just short of one-half an octave
using the formula: Octaves = 1. 443LN(Fh/Ff), where Fh and PI are the
uppet and lower frequencies, respectively. Thus, the cutoff slope would have
to be 95 dB to 100 dB/octave, a very high figure indeed. If the sample rate
were raised 25% to 15 ks/s, the filter could go all the way to 10 kHz before it
must attenuate 50 dB. This gives a cutoff slope of only 50 dB/octave, a
much, much easier filter to implement. In most cases, these requirements
could be relaxed somewhat. It is unlikely that one would want to shatter
glass with a maximum-amplitude, maximum-frequency tone and then worry
about small fractions of a percent of alias distortion.
Both of the preceding examples assumed that the digital synthesis
system never tried to generate frequencies higher than 5 kHz. As will be seen
later, it may be difficult to meet that constraint and still make good use of
frequencies close to 5 kHz. If a 6-kHz tone was actually synthesized in the
15-ks/s system, the simple filter would pass its alias at 8 kHz with an
attenuation of only 33 dB. On the other hand, if one used the filter designed
for a 12-ks/s system with a 15-ks/s sample rate, it would be permissible to
synthesize frequencies as high as 8 kHz without exceeding the - 50-dB alias
rejection requirement. Note that 8 kHz is actually above one-half the sample
rate. Its alias frequency is therefore lower than the signal frequency, but since
that is 7 kHz, the filter attenuates it adequately. The conclusion, then, is
that a good filter can either reduce the required sample rate, simplify the
synthesis computations, or some of both.
Sharp Low-Pass Filter Design
Sharp low-pass filter design is itself an interesting topic that has filled
many books, usually with quite a bit of mathematics. Here we will just