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
![$\displaystyle x(t) = \cos[\omega_c t + \beta\sin(\omega_m t)], \protect$](http://www.dsprelated.com/josimages_new/mdft/img538.png) |
(4.8) |
where we have changed the modulator amplitude
![$ A_m$](http://www.dsprelated.com/josimages_new/mdft/img539.png)
to the more
traditional symbol
![$ \beta $](http://www.dsprelated.com/josimages_new/mdft/img21.png)
, called the
FM index in FM sound
synthesis contexts. Using
phasor analysis (where
phasors
are defined below in §
4.3.11),
4.11i.e., expressing a real-valued FM
signal as the real part of a more
analytically tractable complex-valued FM signal, we obtain
where we used the fact that
![$ J_k(\beta)$](http://www.dsprelated.com/josimages_new/mdft/img518.png)
is real when
![$ \beta $](http://www.dsprelated.com/josimages_new/mdft/img21.png)
is real.
We can now see clearly that the sinusoidal FM spectrum consists of an
infinite number of side-bands about the carrier frequency
![$ \omega_c$](http://www.dsprelated.com/josimages_new/mdft/img484.png)
(when
![$ \beta\neq 0$](http://www.dsprelated.com/josimages_new/mdft/img546.png)
). The side bands occur at multiples of the
modulating frequency
![$ \omega_m$](http://www.dsprelated.com/josimages_new/mdft/img495.png)
away from the carrier frequency
![$ \omega_c$](http://www.dsprelated.com/josimages_new/mdft/img484.png)
.
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