I am trying to understand frequency modulation (applied to sound spectrum synthesis, not radio transmission), and all explanations of side bands I have found make a huge leap. For reference, frequency modulation (in its purest form mathematically) is changing the frequency of a “carrier” sine wave by a “modulator” sine wave, so that when the modulator is at its peak “amplitude” the resultant wave is at its highest frequency and when the modulator is at its lowest “amplitude” the resultant wave is at its lowest frequency. This generates “side bands,” which are mathematically defined and acoustically perceptible but not immediately apparent in the waveform (the same way that when you add two sine waves of different frequencies, neither is apparent in the resulting wave). I know how to determine the frequencies of the side bands (by adding and subtracting the modulating frequency to the carrier an infinite number of times) and to determine their respective amplitudes (using Bessel functions of the first kind evaluated at the modulation index, which is the peak frequency deviation related to the frequency of the modulator — with a low enough index value, most higher order side bands will end up with amplitudes close enough to 0 to be negligible, but I would guess that mathematical proofs would not consider them negligible). This is as far as the most specific explanations of sidebands get.
All of the explanations I’ve found skip from a function defining the resultant (modulated) wave intuitively: $y=A\sin((f_ct)+I\sin(f_mt))$, where $A$ is the peak amplitude of the carrier wave, $f_c$ is the frequency of the carrier wave, $I$ is the modulation index (the peak frequency deviation divided by the frequency of the modulator wave), and $f_m$ is the frequency of the modulator wave
To a function defining the resultant (modulated) wave as a sum of its frequency (sine wave) components: $y=A \sum\limits_{n=-\infty}^\infty J_n(I)\sin((f_c+nf_m)t)$, where $J_n(I)$ is a Bessel function of the first kind of order $n$ evaluated at $x=I$.
I haven’t been able to find any proof of how the second one is equivalent to the first. I want to find some kind of proof, or even some indication as to how a proof might work, because I want to have some intuitive understanding of why side bands are produced at frequencies defined by the frequency of the modulator rather than its amplitude (the peak frequency deviation of the resultant wave). I’ve been able to work all of this out for amplitude modulation, but so far the mathematics behind frequency modulation have eluded me. Because Bessel functions are involved (and I don’t understand Bessel functions that well either), I’m guessing some kind of calculus involved, either differential equations or integrals, but I can’t see right now where or why they would get involved. And I’ve never seen a trig identity that results in an infinite sum. I also considered using a Taylor series, but I wouldn’t know how to resolve it into the Bessel functions or how to differentiate the original equation. I don’t suppose anyone could help me, or point me in the right direction? Any advice would be appreciated.
edit I completely forgot about the existence of Fourier series and Fourier transforms, and I think they’ve provided the answer, though I still need to work out the math to double check. I’m going to leave the question up anyway though, in case anyone else finds it relevant, and because I have next to no experience with Bessel functions, and I’m almost positive the transform is gonna resolve to Bessel functions.
By definition $$J_n(x)= \frac{1}{2\pi} \int_{-\pi}^\pi e^{ix \sin \tau} e^{-in\tau}d\tau \qquad \implies e^{ix \sin \tau}= \sum_{n=-\infty}^\infty e^{i n \tau} J_n(x)$$