On a Sturm-Liouville framework for continuous and discrete frequency modulation

It is well known that purely sinusoidal signals satisfy a linear second-order constant coefficient differential equation. It is also well known that a broad class of orthogonal special functions such as the Legendre and Hermite polynomials satisfy the second-order Sturm-Liouville differential equation. Both sinusoidal and AM-FM models have been used for analysis and synthesis of speech signals. In this paper, we present a Sturm-Liouville differential and difference equation approach to both continuous and discrete time frequency modulation. Orthogonal modes of frequency modulation that are not distorted by the Sturm-Liouville operator are described.