of Microprocessors

DIRECT COMPUTER SYNTHESIS METHODS 117
given above. Thus, amplitude envelopes, amplitude modulation, and frequency
modulation are all very easily accomplished.
Other waveshapes may be easily computed even though a built-in
function for the desired shape is not provided by the computer's programming
language. A sawtooth wave, for example, is defined by the expression
where the mod function gives only the remainder when the expression to its
left is divided by the expression to its right. A triangle wave may be formed
from sawtooth samples by ~pplying
Sn TRI = 2A TR ! ( ISiJSAI\" I - A~w )
One must be careful when directly computing complex waveshapes
with sharp corners and vertical lines because alias distortion can occur. When
these waves are computed, they are computed perfectly with all of their
harmonics up to infinity. However, those harmonics that exceed half the
sample rate will be reflected back down into the audible spectrum and cause
distOrtion. The problem is generally not severe as long as these waveforms are
held to low frequencies, but it does prevent use of these shapes as fundamental
basis waves such as is done in the voltage-controlled synthesizer.
Fourier Series
Probably the most flexible method of generating waveshapes by computer
is to use the Fourier series. Individual sine wave harmonic (or specific
nonharmonic) frequencies can be compuced at selected phase angles and
amplitudes and summed together to produce a composite waveform. Any
problem with alias distOrtion due to high harmonics is circumvented by
simply omitting those that exceed half the sample rate from the calculation.
Of course, the amplitudes of individual components may be continuously
varied for a changing spectrum shape. The effect of a filter with any arbitrary
frequency response may be simulated by multiplying each harmonic
amplitude by the filter response curve amplitude at the corresponding frequency.
Simultaneous Sounds
Simultaneous sounds are easily done by computing the string of samples
for each sound and then simply adding the strings of samples together to
get one string for the composite sound. Fortunately, the same effect is
realized if one sample for each sound is computed and then the samples are