Orthogonal Modes of Frequency Modulation and the Sturm–Liouville Frequency Modulation Model

Sinusoidal signals and complex exponentials play a critical role in LTI system theory in that they are eigenfunctions of the LTI convolution operator. While processing frequency-modulated (FM) waveforms using LTI systems, restrictive assumptions must be placed on the system so that a quasi-eigenfunction approximation holds. Upon deviation from these assumptions, FM waveforms incur significant distortion. In this paper, a Sturm-Liouville (S-L) model for frequency modulation introduced by the author, is extended to a) study orthogonal modes of continuous and discrete frequency modulation and b) to develop system theoretical underpinnings for FM waveforms. These FM modes have the same special connection with respect to the FM S-L system operator, that complex exponentials have with LTI systems and the convolution operator. The finite S-L-FM spectrum or transform that measures the strength of the orthogonal FM modes present in a FM signal, analogous to the discrete Fourier spectrum for sinusoids, is introduced. Finally, similarities between the orthogonal S-L-FM modes and angular Mathieu functions are exposed, and a conjecture connecting the two is put forth.