of Microprocessors

110 MUSICAL ApPLICATIONS OF MICROPROCESSORS
assumption that the wave did not change very much from sample to sample is
not necessary.
Of course, for sound and music a whole spectrum of frequencies must
be reproducible, not just single sine waves. Fortunately, a sampled arbitrary
waveform behaves just the same as a sampled sine wave. The spectrum of the
waveform is reproduced unaltered and then symmetrical pairs of copies
around each harmonic of the sampling frequency are also introduced. Figure
4-3 shows the result of a full 20-Hz to 20-kHz audio spectrum sampled at a
50-ks/s rate. Since there is no overlap between the desired spectrum and the
copies, a low-pass filter can once again be used to eliminate the copies.
The preceding argument is equally valid for conversion of sound
waveforms into numbers. The waveform is first sampled with a balanced
modulator or its equivalent and then the sample pulse amplitudes are measured
with an analog-to-digital converter (ADC),which is nothing more than
an ultrafast digital voltmeter. Each sample thus becomes a number that may
then be ptocessed by the computer. No information about the curves and
undulations of the waveform is lost provided that the spectrum of the
waveform has no frequency components above one-half of the sampling rate.
Dnfortunately, some natural sounds do have appreciable .energy beyond the
audible range so a low-pass filter is needed to prevent these high frequencies
from re~ching the sampler and ADC. Since these frequency components are
beyond the audible range anyway, the filter does not audibly affect the
sound. If signal frequencies higher than half the sample rate are allowed to
enter the sampler, their offspring in the copies will overlap the original
spectrum and will cause distortion. Such a situation is termed aliasing and
the resulting distortion is called alias distortion.
Any Sound Can Be Synthesized
Figure 4-4 shows a block diagram of a complete computerized audioto-digital
and back to audio again conversion system. If the two low-pass
filters are matched and have infinite attenuation beyond one-half of the
sample rate, the waveforms at points A and B are exactly alike! Since the
waveform at point A is not audibly different from the input, this is truly a
high-fidelity system. The computer could also be programmed to supply its
own stream of numbers to the DAC, and it therefore follows that the
computer can produce any sound. Note also that the system has no lowfrequency
limitation, thus allowing signals down to dc to be reproduced.
ANY
AUDIO
SIGNAL
LOW-PASS
FILTER
~
20 kHz
LOW-PASS
~
20 kHz
POINT A
POINT B
Fig. 4-4. Computerized digital audio system.