Fourier Transform leads to Bessel Function

A frequency modulated waveform can be written as 

or, in terms of Bessel functions, as

where Jn(Mf) refers to the amplitude of the nth sideband with modulation index, Mf.

In the post graph, the horizontal axis, x, is the modulation index, Mf.

The values of the Bessel functions can be evaluated by simply taking the Fourier transform of the frequency modulated equation. A few results of a Fourier transform are given below.

Used in this manner, a FFT can be used to construct complete tables, and graphs, of Bessel functions. The FFT will also give a phase of either 0 degrees, or 180 degrees, where 180 degrees indicates a negative amplitude.