of Microprocessors

438 MUSICAL ApPLICATIONS OF MICROPROCESSORS
~ • • /\ ,1"1., /.... .~~ ", r
~ 0 I-':"' ...:;---:""";":"---:-";-f .• -':--+Co.-"~---,;._-;-, --'---."' ....'.'--~+...;~...--, ......:i-',/ .....-'--'c., ...i'--':-+-~...+-J-
§;: ....i 'w' \./ _ \./ :
,
:\
I/'i'" ACTUAL WAVEFORM
r rI' HARMONICS
TIME
.; END OF
RECORD
5.31 CYCLES
TRANSFORM "HARMONICS"
FREOUENCY
Fig. 13-10. Discrete Fourier transform of 5.31 cycles of a waveform containing
equal proportions of fundamental, second, and third harmonics
the synthesized waveform is just as unlikely to span the record as in the
analysis case. The other problem is that if the spectrum changes significantly
from one synthesized record to the next, there is likely to be a discontinuity
between the two records when they are spliced together. These synthesis
problems may be overcome by record overlapping and time-variable interpolation
between the overlapping records.
The frequency domain data produced and used by the discrete Fourier
transform bear a very specific relationship to the corresponding sample record.
In particular, the number of harmonics (including the zeroth or dc
component) in the frequency domain representation is exactly one-half the
number of samples in the time domain record plus one. It is easily seen that
any more than this would not make any sense. For example, let's assume a
record size of 200 samples taken at a sample rate of 10 ks/s. The record
duration, therefore, is 20 msec, which is also the period of the assumed
periodic waveform. A 20-msec period is a 50-Hz frequency for the assumed
wave, so harmonics would fall at 50 Hz, 100 Hz, 150 Hz, etc. It is easy to
determine, then, that there are 100 harmonics at or below one-half the
sample rate and, when the dc component is included, the total becomes 10 1,
which is one-half the record size plus one. This also shows that the frequency
resolution of the analysis is 50 Hz or simply the reciprocal of the record
duration.
Each harmonic in turn consists of two components. Although harmonics
have been characterized by amplitude and phase so far, the Fourier