of Microprocessors

DIGITAL TONE GENERATION TECHNIQUES 435
while there is a great variety of possible tranSItiOnS, the technique is not
general enough so that any arbitrary transition can be realized. One could
piecewise approximate an arbitrary transition by using a sequence of tables,
however.
Another method of dynamic spectrum variation using the table lookup
method actually amounts to a continuous Fourier series evaluation. One
would have a single table, which is actually a sine table, and program several
table pointers with pointer increments that are integer multiples of the
smallest increment. Then, using each pointer in sequence, the corresponding
samples would be fetched from the table, multiplied by a corresponding
amplitude factor, and the products added together to produce the output
sample. This is equivalent to treating each harmonic as a separate tone and
controlling its amplitude independently. Relative phase can be controlled by
temporarily adding a phase parameter to the pointer when table access is
performed but using the original pointer value when the increment is added.
The technique is not limited to exact harmonic frequencies either, since
the set of pointer increments need not be integer multiples. Most stringed
musical instruments in which the string is plucked or struck have upper
harmonics that are somewhat sharp with respect to the fundamental. Bells
and chimes have spectra that are decidedly inharmonic. For these and other
similar sounds, this is the only general technique available. Although
dynamic depth FM can also produce inharmonic spectra, only gross control is
possible; the details of the spectrum are pretty much left to chance.
While this technique can be time consuming for a large number of
harmonics, it is quite effective for a small number. Its primary strength over
faster Fourier techniques to be discussed is that amplitudes and phases of the
harmonics may be changed at any time and as rapidly as desired without
glitches and discontinuities in the composite waveform. In particular, a
hardware implementation of the technique can be extremely flexible and
effective as well as adequately fast for real-time tone generation.
Fourier Transformation
Fourier transforms are the cornerstone of many modern signalprocessing
techniques. They have a list of desirable mathematical properties
that seems never to end as well as many useful physical properties for synthesis
and analysis work. The main attractive feature about any kind of Fourier
operation is that it is a bridge between the time domain, which is concerned
with waveforms and sample values, and the frequency domain, which is concerned
with the amplitudes and phases of frequency components. The primary
need for such a bridge is that the human ear hears in the frequency
domain, while sound waves are stored, synthesized, and observed (via an
oscilloscope) in the time domain.