Generalized MTSFM Signals Expressed as Complex Fourier Series using Generalized Bessel Functions of Anger Type: Representation and Application

The classical multi-tone sinusoidal frequency mod ulation (MTSFM) is a Fourier expansion of the instantaneous frequency or modulation function of a signal, where the Fourier expansion period equals the pulse width. We extended or generalized the MTSFM by introducing Fourier expansions with arbitrary periods and by expanding the phase instead of the instantaneous frequency. This generalization allows the representation of less smooth signals and avoids significant root mean square (RMS) bandwidth (and swept bandwidth) increase. In our (numerical) analysis we computed the signal and its corresponding Fourier transform (FT) and auto-correlation function (ACF) by directly evaluating the MTSFM expansion and calculating the integrals in the FT and ACF by Riemann sums. Instead, an alternative approach in the classical MTSFM has been developed, in which the signal is represented by a complex Fourier expansion with generalized Bessel functions (GBFs) of Anger type as coefficients, briefly called a Jacobi-Anger expansion. In this paper we derive expressions for the Jacobi Anger expansions of a generalized MTSFM signal and its FT and ACF, both without and with window, and present approaches for analytical and numerical evaluation of these expansions. We apply our theory to a classical example MTSFM signal of which we present the numerically evaluated signal, FT, and ACF by Jacobi-Anger expansions, and we comment on the complexity of the expansions in comparison to direct evaluation of the generalized MTSFM signal, its FT, and its ACF, as well as on the additional insight the expansions provide in the behaviour of these quantities. Finally, we describe our future analysis with application of the Jacobi-Anger expansions in optimization.