of Microprocessors

DIGITAL TONE GENERATION TECHNIQUES
459
., ,
~
....,: '0" . '.' '. .
'.i
(AI
(8)
Fig. 13-21. Generating an arbitrary frequency sine wave with the FFT. (A)
Record boundaries with a 256-Hz wave. (8) Phase correction
between records.
tially in time to produce a continuous string of samples suitable for further
processing or output to a DAC. Generally, the record size is chosen once and
remains constant throughout the synthesis but a dynamically variable record
size is conceivable. If the synthesis result is intended for human consumption,
record sizes in the lO-msec to 50-msec range provide the best tradeoff
between frequency and time resolution consistent with human perceptual
capabilities.
In order to get acquainted with some of the problems and their
solution, let's first consider the task of generating a sine wave of an arbitrary
but constant frequency using the FFT. For the purpose ofillustration, we will
assume a sample rate of 10 ks/s = 0.1 msec/sample, a record size of 256 =
25.6 msec, and a sine wave frequency of 200 Hz. The first problem that will
be noted is that 200 Hz is not an exact harmonic of 1/25.6 msec = 39.0625
Hz. The closest harmonic is the fifth, which is 195.3 Hz, an error of about
one-third semitone. If FFT synthesis is to be useful, a way must be found to
produce such intermediate frequencies accurately.
Figure 13-21A illustrates the problem. Shown is an exact sampled
200-Hz waveform and the record boundaries for 256-sample records. Obviously,
the phase of the desired wave with respect to the record boundaries is
different for each record. In fact, the phase advances by
(200 - 5 X 39.0625)/39.0625 = 0.12 cycle every record period. Since the
spectrum fed to the FFT includes phase, one can increment the phase of the