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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 [1], it was the method used in
the the highly successful Yamaha DX7 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

(5) 
where the parameters
describe the
carrier sinusoid, while
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 timederivative 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}
Subsections
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