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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 [15], 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

$\displaystyle x(t) = A_c\cos[\omega_c t + \phi_c + A_m\sin(\omega_m t + \phi_m)], \protect$ (4.5)

where the parameters $ (A_c,\omega_c,\phi_c)$ describe the carrier sinusoid, while the parameters $ (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 book, only phase modulation will be considered, and we will call it FM, following common practice.4.8

Figure 4.14 shows a unit generator patch diagram [44] for brass-like FM synthesis. For brass-like sounds, the modulation amount increases with the amplitude of the signal. In the patch, note that the amplitude envelope for the carrier oscillator is scaled and also used to control amplitude of the modulating oscillator.

Figure 4.14: Unit generator diagram for simple FM brass synthesis.
\includegraphics{eps/fmug}

It is well known that sinusoidal frequency-modulation of a sinusoid creates sinusoidal components that are uniformly spaced in frequency by multiples of the modulation frequency, with amplitudes given by the Bessel functions of the first kind [15]. As a special case, frequency-modulation of a sinusoid by itself generates a harmonic spectrum in which the $ k$ th harmonic amplitude is proportional to $ J_k(\beta)$ , where $ k$ is the order of the Bessel function and $ \beta $ is the FM index. We will derive this in the next section.4.9



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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8.
Copyright © 2014-04-06 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA