The amplitudes of the spectral components produced by FM can be analytically determined from the trigonometric identity:
where is a Bessel function of the first kind and order given by
The time-varying output of a simple FM algorithm is
(2)
where
is referred to as the “modulation index”.
Note the difference between the phase argument of Eq. (2) and the previously mentioned equation for the instantaneous carrier frequency, Eq. (1), as
.
If
, then the instantaneous frequency is given by
Since this expression must equal that of Eq. (1), we have
The Matlab script FM2.m
demonstrates the difference between using the standard FM algorithm, Eq. (2), to calculate an FM synthesis result versus phase increment approaches (including an instantaneous frequency as in Eq. (1)).
Each spectral component amplitude is proportional to , values of which are plotted in Fig. 11 below.
Figure 11:
Bessel functions of the first kind vs. modulation index (I).
Lower frequency sidebands will wrap around 0 Hz and interfere with the other components (either constructively or destructively).
For any given modulation index, individual components of the spectrum (including the carrier frequency) may be missing.