Phase modulation (also known as FM

Patch E08.phase.mod.pd, shown in Figure 5.15, shows how true frequency modulation (part a) differs from phase modulation (shown in part b). To accomplish phase modulation, the carrier oscillator is split into its phase and cosine lookup components. The signal is of the form

$$x[t] = \cos( \omega_c n + a \cos(\omega_m n))$$

where $\omega_c$ is the carrier frequency, $\omega_m$ is the modulation frequency, and $a$ is the index of modulation--all in angular units.

Figure 5.15: Frequency modulation (a) and phase modulation (b) compared.

We can predict the spectrum by expanding the outer cosine:

$$x[t] = \cos( \omega_c n ) \cos (a \cos(\omega_m n)) - \sin( \omega_c n ) \sin (a \cos(\omega_m n))$$

Plugging in the expansions from example 5.5.6 and simplifying yields:

$$x[t] = {J_0}(a) \cos( \omega_c n )$$

$$+ {J_1}(a) \cos( (\omega_c+\omega_m) n + {\pi\over2}) + {J_1}(a) \cos( (\omega_c-\omega_m) n + {\pi\over2})$$

$$+ {J_2}(a) \cos( (\omega_c+2\omega_m) n + \pi) + {J_2}(a) \cos( (\omega_c-2\omega_m) n + \pi)$$

$$+ {J_3}(a) \cos( (\omega_c+3\omega_m) n + {{3\pi}\over2}) ... ...}(a) \cos( (\omega_c-3\omega_m) n + {{3\pi}\over2}) + \cdots$$

So the components are centered about the carrier frequency $\omega_c$ with sidebands extending in either direction, each spaced $\omega_m$ from the next. The amplitudes are functions of the index of modulation, and don't depend on the frequencies. Figure 5.16 shows some two-operator FM spectra, measured using Patch E09.FM.spectrum.pd.

Figure 5.16: Spectra from phase modulation at three different indices. The indices are as multiples of $2\pi$ radians.

Although FM is thus just a form of ring modulated waveshaping, a confusing name change has taken place: it is FM's carrier frequency that appears as the ring modulation frequency. Keeping this in mind, we can use the strategies described in section 5.2 to generate particular combinations of frequencies. For example, if the FM carrier frequency is half the FM modulation frequency, you get a sound with odd harmonics exactly as in the octave dividing example (5.5.2).

Returning to the spectra shown in Figure 5.16, we can imagine superposing several of these (sharing a fundamental (modulating) frequency but with carriers tuned to different harmonics) to build designer spectra with energy peaks, called formants, at chosen locations, imitating the production of the human voice. This was used to great effect by Chowning [Cho89] in his piece, Phoné.

Frequency modulation need not be restricted to purely sinusoidal carrier or modulation oscillators. One well-trodden path is to effect phase modulation on the FM spectrum itself. There are then two indices of modulation (call them $a$ and $b$) and two frequencies of modulation ($\omega_m$ and $\omega_p$) and the waveform is:

$$x[n] = \cos(\omega_c n + a \cos(\omega_m n) + b \cos(\omega_p n))$$

To analyze the result, just rewrite the original FM series above, replacing $\omega_c n$ everywhere with $\omega_c n + b \cos(\omega_p n)$. The third positive sideband becomes for instance:

$${J_3}(a) \cos( (\omega_c+3\omega_m) n + {{3\pi}\over2} + b \cos(\omega_p n))$$

This is itself just another FM spectrum, with its own sidebands of frequency

$$\omega_c+3\omega_m + k \omega_p , k = 0, \pm 1, \pm 2, \cdots$$

having amplitude ${J_3}(a) {J_k}(b)$ and phase $(3+k)\pi / 2$ [Leb77]. Patch E10.complex.FM.pd (not shown here) illustrates this by graphing spectra from a two-modulator FM instrument.

Since early times [Sch77] researchers have sought combinations of phases, frequencies, and indices that manage to imitate familiar instrumental sounds; this became a major industry with the introduction of the Yamaha DX7 synthesizer.