# FM theory

1. Nov 30, 2003

### bosonics

What role do Bessel functions play in frequency modulation theory?

2. Dec 1, 2003

### uart

In FM and PM (phase mdulation) the output function is a sinusoid with the input function as the argument (or phase) of the sinusoid. (Actually it's the time integral of the input function for the phase in the case of FM, but in the context of your question which relates to the case of a sinusoidal input function then the distinction is not too important).

Ok, I'm a little rusty on the exact details, but essentually when you have a sinusoidal input to an FM or a PM system then your output is a sinusoid of another sinusoid (like a nested sinusoid). Now when you try to find the Fourier series of this function (appropriately normalized) then you come up against the following integral.

$$J_n (\beta) = \frac{1}{2\pi} \int_{\theta=-\pi}^{\pi} \cos ( n \theta - \beta \sin ( \theta ) ) \ d\theta$$

So that's where the Bessel function (of the fisrt kind) creeps in. In a nut shell, it's when you calculate the coefficients of the Fourier series for the case of sinusoidal excitation.

Last edited: Dec 1, 2003