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FM Spectra
Using the expansion in Eq.
(4.7), it is now easy to determine
the spectrum of sinusoidal FM. Eliminating scaling and
phase offsets for simplicity in Eq.
(4.5) yields

(4.8) 
where we have changed the modulator amplitude
to the more
traditional symbol
, called the FM index in FM sound
synthesis contexts. Using phasor analysis (where phasors
are defined below in §4.3.11),^{4.11}i.e., expressing a realvalued FM signal as the real part of a more
analytically tractable complexvalued FM signal, we obtain
where we used the fact that
is real when
is real.
We can now see clearly that the sinusoidal FM spectrum consists of an
infinite number of sidebands about the carrier frequency
(when
). The side bands occur at multiples of the
modulating frequency
away from the carrier frequency
.
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