of Microprocessors

116 MUSICAL ApPLICATIONS OF MICROPROCESSORS
distinct loss of bass is the result. Also, any power line hum in the recording
will be reproduced at a much more audible 120 Hz.
As mentioned earlier, the low-pass filter following the DAC is critical.
One might think that if the sample rate is above 40 ks/s a filter is not needed
at all, since all unwanted frequencies are above the audible range anyway.
However, since these frequencies are so strong, they can easily mix with the
bias oscillator in an audio tape recorder and result in audible beat notes.
Thus, a filter of at least moderate effectiveness is required with high sample
rates. The filter becomes more critical at the lower sample rates 'used for
experimentation because these unwanted frequencies are audible and sound
terrible. Sharp cutoff low-pass filter design will be discussed in detail in
Chapter 12.
Computation of Sound Waveforms
Now that it has been established that computers can generate sound of
the highest quality directly, the next topic of interest is the techniques used
for computing sound waveforms. Although one could theoretically sit down
at a computer console and type in sample values, just about anything that
could be done without recourse to a pocket calculator would sound like white
noise on playback. The problem is similar to the one experienced when
drawing waveforms directly onto movie film. Of course, this does not even
address the difficulty of providing tens of thousands of samples for every
second of sound output.
In this section, a general overview of practical methods of sound
waveform computation will be presented. A more detailed description of
these techniques and others along with example programs will be given in
Section III.
Sin and Other Built-in Functions
An obvious method of generating a single sine wave with a computer is
to use the sin function found in nearly all high-level programming
languages. The general expression for the Nth sample of a sine wave of
frequency F and the amplitude A is Sn = Asin(2'TrFTIFs), where T is time in
seconds and Fs is the sample rate in samples per second. Since time and
sample number are directly proportional and the sample rate is constant,
considerable computer time may be saved by defining a constant K = 2'TrIFJ
and using the form Sn = Asin(KnF). Samples for the sine wave are computed
simply by defining values for the variables A and F and then evaluating the
expression for sample numbers 0, 1, 2, up to M, where MIFJ is the duration
in seconds desired for the wave.
Of course, during the calculations the values of A and F can change.
Often separate expressions that are also evaluated every sample time determine
the value of A and F to be used in the "tone generation" expression