Citing from that page > In other words, $ J_k(\beta)$ is the amplitude of the $ k$ th harmonic in the Fourier-series expansion of the periodic signal $ x_m(t)

Citing from that page > In other words, $ J_k(\beta)$ is the amplitude of the $ k$ th harmonic in the Fourier-series expansion of the periodic signal $ x_m(t)$ . Or from the parent page about FM > It is well known that sinusoidal frequency-modulation of a sinusoid creates sinusoidal components that are uniformly spaced in frequency by multiples of the modulation frequency, with amplitudes given by the Bessel functions of the first kind [15]. As a special case, frequency-modulation of a sinusoid by itself generates a harmonic spectrum in which the $ k$ th harmonic amplitude is proportional to $ J_k(\beta)$ , where $ k$ is the order of the Bessel function and $ \beta $ is the FM index. In other words, Bessel functions come up in Fourier transforms of frequency modulation signals, because integrals. The calculations have been figured out centuries ago. Regarding phase and arbitrary frequencies, it's easier to understand in the discrete time case: a vector of N complex numbers (time domain samples) goes in, it's multiplied by a matrix, and a vector of N complex numbers (Fourier coefficients) comes out. Regardless of the input signal, such a transform can be inverted (up to rounding errors and the like) by multiplication with the inverse of the forward matrix: the difference between signals of different frequencies is that if periods are a whole fraction of N samples the projections fall on one Fourier coefficient instead of spreading around all of them. Aligning the signal's period with the window length M allows reproducing the whole infinite duration signal from a N coefficient slice because all slices are identical.