of Microprocessors

466 MUSICAL ApPLICATIONS OF MICROPROCESSORS
CARRIER
FREQUENCY
CONTROL
r
INCREMENT
ADDRESS
I TABLE I SINE
ADD POINTER TABLE -.
I
LOOKUP
OUTPUT
SAMPLES
INCREMENT
ADDRESS
MODULATION 2 TABLE 2 SINE
FREQUENCY .. POINTER TABLE ~ MULTIPLY ,----
CONTROL
2 LOOKUP
DEVIATION
CONTROL
Fig. 13-25. FM synthesis calculations
power relative to table lookup and Fourier synthesis techniques. In its pure
form, only one table, which contains a sine wave, is required. For each
sample, as few as three additions, two table lookups, and one multiplication
are required. This remains true even when the modulation index is changed
for dynamic timbre variation.
Figure 13-25 is a "block diagram" of these calculations, which can
easily be translated into a program or dedicated hardware. Note that, unlike
the analog case, controls for the three variables (carrier frequency, modulation
frequency, and deviation amount) are kept separate rather than combined to
provide the more useful pitch frequency, ratio, and modulation index
controls. Otherwise, the combination calculations would be done for every
output sample, which substantially increases computing effort. Typically,
the more useful "source" controls would be translated into these "object"
controls at a slower control sample rate, which may be one-tenth of the audio
rate or less. Flexibility is also increased because the same synthesis routine
could serve for constant deviation or constant modulation index synthesis as
the pitch frequency is varied.
Note that if the carrier frequency is relatively low and the deviation
amount is large, it is possible for table pointer 1 to have a negative increment
specified, i.e., a negative frequency. This is no problem for digital synthesis,
provided signed arithmetic is being used, and simply results in the sine table
being scanned backward. In actual use, it has been found that interpolation
noise in the sine table lookups is much more detrimental in FM synthesis
than in direct table-scanning synthesis. This is because errors in lookup ofthe
modulating wave are compounded when they modulate the carrier wave.
Therefore, either the sine table must be large or linear interpolation in a
smaller table used. One should shoot for at least 16-bit accuracy in the table
lookup results, which means effective table sizes of 64K points for no
interpolation or 0.5 to lK for linear interpolation. Since the table content is
fixed as 1 cycle of a sine wave, symmetry can be used to reduce table size by a