Frequency and phase modulation
If a sinusoid is given a frequency which varies slowly in time we hear it as
having a varying pitch. But if the pitch changes so quickly that our ears
can't track the change--for instance, if the change itself occurs at or
above the fundamental frequency of the sinusoid--we hear a timbral change.
The timbres so generated are rich and widely varying. The discovery by
John Chowning of this
possibility [Cho73] revolutionized the field of computer music.
Here we develop
usually called FM,
as a special case of waveshaping [Leb79]; the
treatment here is adapted from an earlier publication [Puc01].
The FM technique, in its simplest form, is shown in figure 5.8
A frequency-modulated sinusoid is one whose frequency varies sinusoidally, at
some angular frequency , about a central frequency , so
that the instantaneous frequencies vary between and
, with parameters controlling the frequency of
variation, and controlling the depth of variation. The parameters
, , and are called the
carrier frequency, the
modulation frequency, and the
index of modulation, respectively.
It is customary to use a simpler, essentially equivalent formulation in
which the phase, instead of the frequency, of the carrier sinusoid is
modulated sinusoidally. (This gives an equivalent result since the
instantaneous frequency is just the change of phase, and since the
sample-to-sample change in a sinusoid is just another sinusoid.) The
phase modulation formulation is shown in part (b) of the figure.
If the carrier and modulation frequencies don't themselves vary
in time, the resulting signal can be written as
parameter , which takes the place of the earlier parameter , is also
called the index of mosulation; it too
controls the extent of frequency variation relative to the carrier frequency
. If , there
is no frequency variation and the expression reduces to the unmodified,
Block diagram for frequency modulation (FM) synthesis: (a) the classic
form; (b) realized as phase modulation.
To analyse the resulting spectrum we can write,
so we can consider it as a sum of two waveshaping
generators, each operating on a sinusoid of frequency and
with a waveshaping index , and each ring modulated with a sinusoid of
frequency . The waveshaping function is given by
for the first term and by
for the second.
Returning to Figure 5.4, we can see at a glance what the
spectrum will look like. The two harmonic spectra, of the waveshaping outputs
have, respectively, harmonics tuned to
and each is multiplied by a sinusoid at the carrier frequency. So there
will be a spectrum centered at the carrier frequency , with
sidebands at both even and odd multiples of the modulation frequency ,
contributed respectively by the sine and cosine waveshaping terms above.
The index of modulation , as it changes, controls the relative strength of
the various partials. The partials themselves are situated at the frequencies
As with any situation where two periodic signals are multiplied, if there is
some common supermultiple of the two periods, the resulting product will repeat
at that longer period. So if the two periods are and , where
and are relatively prime, they both repeat after a time interval of
. In other words, if the two have frequencies which are both multiples
of some common frequency, so that
again with and relatively prime, the result will repeat at a frequency
of the common submultiple . On the other hand, of no common
submultiple can be found, or if the only submultiples are lower than
any discernable pitch, then the result will be inharmonic.
Much more about FM can be found in textbooks [Moo90, p. 316]
[DJ85] [Bou00] and research publications; some of the
possibilities are shown in the following examples.