of Microprocessors

474 MUSICAL ApPLICATIONS OF MICROPROCESSORS
the table content or time-varying interpolation along a sequence of waveform
tables. The more specific term, "nonlinear waveshaping synthesis," is
actually nothing more than subjecting a sine wave to a nonlinear distortion
process such as that described in analog form at the beginning of Chapter 2.
By simply varying the amplitude of the input sine wave, the spectrum of the
output waveform will vary in a complex fashion. What makes the technique
much more useful in digital form is that very precise arbitrary transfer
functions are easily implemented, using, guess what, a table. By contrast,
the analog equivalent is generally limited to clipping and rectification
operations or, at best, piecewise-linear approximations to curves using a
small number of segments.
Fig. 13-30 shows a standardized representation of a nonlinear transfer
function. As with Fig. 2-1, the input sine wave is applied in the X direction,
and corresponding Y points on the curve represent the output. For easy and
consistent application in digital circuitry, the input signal is restricted to the
range of - 1 to + 1, and the output signal is similarly restricted. Since the
input is always a sine wave, it is usually not shown while the output and its
spectrum are shown. It is not necessary for the curve to pass through the
origin, but, ifit doesn't, there will be an output dc offset for no input, which
could lead to audible envelope thumps and other problems in subsequent
processing. Thus, it may be desirable to shift the curve up or down so that it
does pass through the origin. Using this representation, it becomes easy to
put the shape into a table with the X values representing successive table
entries and the Y values representing the value of the entries. Although this
requires the curve to be a single-valued function, analog hysteresis effects
could probably be simulated by using two tables and keeping track of the
slope of the input sinusoid.
While any kind of curve can be used for a distorting function, even one
drawn by hand, mathematicians like to use polynomials to describe arbitrary
curves. With such a description, they can then predict what the output
spectrum will be without actually simulating the process on a computer and
then using the Fourier transform. Explaining that process here won't be
attempted, but some of the properties of polynomial curves are easily
understood. One desirable property is that the highest harmonic present in
the output is equal to the highest power of X used to construct the curve.
This aids tremendously in predicting and avoiding alias distortion.
One specific class of polynomials that has other desirable acoustic
properties when used to construct nonlinear distorting functions is called
Chebychev polynomials of the first kind. Figure 13-31A-H shows equations for
these polynomials up to the eighth order, their corresponding distortion
curve, the output waveform when the input is a sine wave of unity
amplitude, and the harmonic spectrum of the output. Note that, in effect,
each ofthese polynomials multiplies the frequency of the input sine wave by a
factor equal to its order. Or, in other words, a Chebychev polynomial of order