of Microprocessors

576 MUSICAL ApPLICATIONS OF MICROPROCESSORS
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(B)
Fig. 16--16. Homomorphic spectral analysis. (A) Conventional linear scale
spectrum. (B) Log scale spectrum.
spectrum shape is due to multiplication of the excitation function spectrum
and the system function response curve. Our goal is to separate the two
spectra based on differences in their variability or "frequency content" (remember
the spectral curve is being considered as a time sequence) by filtering.
A filter, however, can only separate components that have been added,
not multiplied, together. Thus, directly filtering the raw spectrum can lead
to incorrect, though not necessarily useless, results.
The problem is solved by remembering from high school math that the
product of two numbers is equal to the antilog of the sum of their logarithms.
Thus, if the amplitude spectrum is converted to decibels as in Fig. 16-16B,
then the resultant shape is the sum of the excitation decibel spectrum and the
filter response curve in decibels. The two curves may now be separated by
filtering using a high-pass to recover the excitation spectrum and a low-pass
to obtain a clean system function response shape. In music synthesis, the