of Microprocessors

566 MUSICAL ApPLICATIONS OF MICROPROCESSORS
sample of the spectrum is represented by a cosine magnitude and a sine
magnitude. We have already seen how this form can be converted into
amplitude and phase form. The amplirude, which is always positive, can be
further co~verted into decibels if desired. Since the human ear has difficulty
in distinguishing amplitude differences much less than 1 dB, the decibel
amplitude data can be quantized to as few as 6 bits without serious degradation.
There may be an inclination to discard the phase data, since they have
little audible effect on the sound. However, the phase data give valuable
information about a sound parameter that is quite important-frequency.
The fundamental and harmonics of an arbitrary tone are unlikely to fall
precisely at the center frequencies of the analysis bands. The results of this
mismatch are twofold. First, since the bandwidth of the Hamming window
is four times the frequency spacing of the analysis, a given signal component
will show up strongly in as many as four adjacent frequency bands. Second,
the exact frequency of the signal is unknown. Even though the analysis seems
imperfect, resynthesis will yield a result essentially equal to the original
signal. It is the phase information that allows accurate reconstruction. We
will see later how phase can be utilized to precisely determine the component
frequencies.
Direct Spectral Modification
Spectral analysis, modification, and resynthesis via FFT comprise the
easiest method of implementing a filter with arbitrary amplitude and phase
characteristics. Often, it is the most efficient method as well, since the
computation effort is independent of the filter's response shape. Another
advantage is that time-varying filters are handled as easily as fixed ones, a
virtue not shared by the transversal method of arbitrary filter implementation.
'Basically, one takes the sequence of spectral frames and multiplies the
amplitude of each spectral component by the amplitude response of the filter
at the corresponding frequency. The resulting sequence of spectral frames is
then converted back into sound via FFT synthesis. When the spectral data is
in sine--cosine form, both components must be multiplied by the filter's
amplitude response.
One can also add two or more spectra together. Since the FFT used in
synthesis is a linear process, the result should be equivalent to individual
resynthesis and conventional mixing of the results. However, there are two
advantages to mixing in the frequency domain. First, there is no phase
cancellation among the combined spectra if just amplitude spectra are used.
Directly combining sine--cosine spectra, however, gives the typical amount
of interference among harmonics of the combined tones. The other advantage
of spectral combination is that only one resynthesis is necessary, thus reducing
computation effort.