of Microprocessors

476 MUSICAL ApPLICATIONS OF MICROPROCESSORS
(H)
l A AAu r
rVl/\J\T V
lL~
(Il
(J I
Fig. 13-31. (cont.) + 56X 3 - 7X. (H) Eighth-order Chebyshev polynomial, Y
128XB - 256X 6 + 160X4 - 32X2 + 1. (I) Combination of fourthand
seventh-order Chebyshev polynomials, Y = 0.5 [T 4 (X) +
T 7 (X)]. (J) Combination of first eight Chebyshev polynomials, Y =
0.28 [T,(X) + 0.72T2(X) + 0.36T 3 (X) + 0.57T 4 (X) + 0.42T 5 (X) +
0.25T 6 (X) + 0.16T 7 (X) + 0.12T B (X)]. (K) Contrived transfer function,
Y = 0.42X + X3 exp (X - 1) sin (14 'ITX).
N will produce only the Nth harmonic of the input sine wave. Note that
when the curve describes an odd function (the shape to the left ofthe Y axis is
the negative image of the shape to the right) only odd harmonics are created.
Likewise, when it is an even function (the shape to the left is a positive image
of the shape to the right), only even harmonics are created.
Figure 13-311 and J shows that two or more Chebychev polynomials
can be added together to get a controlled mixture of harmonics in direct
proportion to the amount of each polynomial included. Note that, in this
case, the resulting polynomial must be scaled to restrain the output waveform
amplitude to unity. In Fig. 13-31K, an arbitrary mixture of polynomial,
exponential, and trignometric functions is used to create the curve. Its
spectrum has harmonics out to the 30th and a desirable gently rolling
spectral envelope. Thus, it is apparent that nonlinear waveshaping can
produce any arbitrary amplitude spectrum just as direct waveform table
scanning could. In fact, one could consider the process as simply scanning a
waveform table with a sine wave rather than a sawtooth wave as in the direct
procedure.
The preceding analysis only holds when the input sine wave is of unit
amplitude. If the amplitude is reduced, the spectrum will change. Generally,
the upper harmonics will decline more rapidly than the lower ones as the