The Mathematics of FM

• The amplitudes of the spectral components produced by FM can be analytically determined from the trigonometric identity:
  $$\sin(\theta + a \sin \beta ) = J_0(a) \sin\theta + \sum_{k=1}^{\i... ... J_k(a) \left[ \sin(\theta + k \beta) + (-1)^k \sin(\theta - k \beta) \right],$$
  where $J_k(x)$ is a Bessel function of the first kind and order $k$ given by
  $$J_k(x) = x^k \sum_{m=0}^\infty \frac{(-1)^m x^{2m}}{2^{2m+k} m!(k+m)!}$$   for integer $$k.$$

  Figure 11: Bessel functions of the first kind vs. modulation index (I).

  

• Each spectral component amplitude is proportional to $J_k(I)$, values of which are plotted in Fig. 11 below.

• 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.