of Microprocessors

DIGITAL TONE GENERATION TECHNIQUES
437
-I RECORD
(AI
(B1
Fig. 13-9. Actual and assumed waveforms used with discrete Fourier transform.
(A) Actual wave to be analyzed. (B) Assumed wave that will
be analyzed.
record is of less consequence. Techniques are available to "tail out" the ends
of the record to minimize the error even further.
When a large but nonintegral number of cycles of a truly periodic
waveform is transformed, each harmonic of the actual waveform becomes a
group of harmonics of the assumed waveform. An example of this is shown in
Fig. 13-10 in which 5.31 cycles of a waveform containing fundamental,
second, and third harmonics in equal proportions was made into a record and
Fourier transformed. The three clusters of transform harmonics correspond to
the individual harmonics of the actual waveform. High-frequency energy
above the third waveform harmonic is due to the discontinuity caused by the
nonintegral number of cycles in the record.
When a section of changing sound is marked off into a record, the
spectrum reported by the Fourier transform is the average spectrum during
the time interval represented by the record. Thus, if one wishes to use
Fourier transformation to track a changing spectrum, the variation contours
will themselves be sampled curves with the sample period equal to the record
period. As a result, there is a tradeoff between long record size for minimum
"periodicity error" and maximum frequency resolution, and a short record
size for following rapidly changing sounds. One technique that is often
useful is to overlap successive records rather than arrange them end to end.
This way one can obtain a high "spectrum sample rate" while using adequately
long records.
Fourier transform synthesis is also complicated by record orientation.
There are actually two problems. One is that an integral number of cycles of