The Role of Bessel Functions in Frequency Modulation Theory

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Bessel functions play a crucial role in frequency modulation (FM) and phase modulation (PM) theory by helping to describe the output of these systems when subjected to sinusoidal input. The output function is essentially a sinusoid with the input function influencing its phase, particularly in FM where the phase is the time integral of the input function. When analyzing the Fourier series of this output, one encounters integrals that lead to the Bessel functions of the first kind. These functions are integral in determining the coefficients of the Fourier series for sinusoidal excitation. Understanding this relationship is essential for grasping the mathematical foundations of FM and PM systems.
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What role do Bessel functions play in frequency modulation theory?
 
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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.
 
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