Frequency 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 frequency modulation, 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 part (a). A frequency-modulated sinusoid is one whose frequency varies sinusoidally, at some angular frequency $\omega_1$, about a central frequency $\omega_2$, so that the instantaneous frequencies vary between $(1-r)\omega_2$ and $(1+r) \omega_2$, with $\omega_1$ controlling the frequency of variation. It is customary to use a simpler, essentially equivalent formulation in which the phase of the sinusoid is modulated sinusoidally (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 signal can be written as

$$x[n] = \cos(r \cos(\omega_1 n)) + \omega_2 n )$$

The parameter $r$, called the index of modulation, controls the extent of frequency variation (relative to the carrier frequency $\omega_2$). If $r=0$, there is no frequency variation and the expression reduces to an unmodified sinusoid:

$$x(t) = \cos(\omega_2 t)$$

We call $\omega_1$ the modulation frequency and $\omega_2$ the carrier frequency.

Figure 5.8: Block diagram for frequency modulation (FM) synthesis: (a) the classic form; (b) realized as phase modulation.

To analyse the resulting spectrum we can write,

$$x(t) = \cos(\omega_2 t) * \cos(r \cos(\omega_1 t))$$

$$- \sin(\omega_2 t) * \sin(r \cos(\omega_1 t)) ,$$

so we can consider it as a sum of two waveshaping generators, each operating on a sinusoid of frequency $\omega_1$ and with a waveshaping index $r$, and each ring modulated with a sinusoid of frequency $\omega_2$. The waveshaping function $f$ is given by $f(x) = \cos(x)$ for the first term and by $f(x) = \sin(x)$ 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

$$\cos(r \cos(\omega_1 t))$$

and

$$\sin(r \cos(\omega_1 t))$$

have, respectively, harmonics tuned to

$$0, 2\omega_1, 4\omega_1, \cdots$$

and

$$\omega_1, 3\omega_1, 5\omega_1, \cdots$$

and each being multiplied by a sinusoid at the carrier frequency. So there willl be a spectrum centered at the carrier frequency $\omega_2$, with sidebands at both even and odd multiples of the modulation frequency $\omega_1$, contributed respectively by the $\sin$ and $\cos$ waveshaping terms above. The index of modulation $r$, as it changes, controls the relative strength of the various partials. The partials themselves are situated at the frequencies

$$\omega_2 + m \omega_1$$

where

$$m = \cdots -2, -1, 0, 1, 2, \cdots$$

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 $k \tau$ and $m \tau$, where $k$ and $m$ are relatively prime, they both repeat after a time interval of $km\tau$. In other words, if the two have frequencies which are both multiples of some common frequency, so that $\omega_1=k\omega$ and $\omega_2=m\omega$, again with $k$ and $m$ relatively prime, the result will repeat at a frequency of the common submultiple $\omega$. On the other hand, of no common submultiple $\omega$ 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 examples.