Sinusoidal Frequency Modulation (FM)

Frequency Modulation (FM) is well known as the broadcast signal format for FM radio. It is also the basis of the first commercially successful method for digital sound synthesis. Invented by John Chowning [1], it was the method used in the the highly successful Yamaha DX-7 synthesizer, and later the Yamaha OPL chip series, which was used in all ``SoundBlaster compatible'' multimedia sound cards for many years. At the time of this writing, descendants of the OPL chips remain the dominant synthesis technology for ``ring tones'' in cellular telephones.

A general formula for frequency modulation of one sinusoid by another can be written as

$$x(t) = A_c\cos[\omega_c t + \phi_c + A_m\sin(\omega_m t + \phi_m)], \protect$$ (5)

where the parameters $(A_c,\omega_c,\phi_c)$ describe the carrier sinusoid, while $(A_m,\omega_m,\phi_m)$ specify the modulator sinusoid. Note that, strictly speaking, it is not the frequency of the carrier that is modulated sinusoidally, but rather the instantaneous phase of the carrier. Therefore, phase modulation would be a better term (which is in fact used). Potential confusion aside, any modulation of phase implies a modulation of frequency, and vice versa, since the instantaneous frequency is always defined as the time-derivative of the instantaneous phase. In this course, only phase modulation will be considered, and we will call it FM, following common practice.²

It is well known that sinusoidal FM has a harmonic spectrum with harmonic amplitudes given by Bessel functions of the first kind [1]. We will derive this in the next section.³