Fractional geometric modulators of sinusoidal baseband signals

Abstract

A sinusoidal-like signal modulation technique is presented, incorporating concepts from differential geometry of surfaces and fractional calculus, specifically involving geodesic flows and Caputo's fractional derivative. We start by introducing geometric objects called funnels, which are revolution surfaces with boundaries. Over these surfaces, we define a geodesic flow based on a carrier signal. We demonstrate that if we define the output signal point-to-point as the normal component of the velocity vector of a geodesic, then, under the first-order derivative, both the carrier signal and the output signal share the same frequency. However, we establish that when Caputo's fractional derivative with a time-dependent fractional order is considered, especially when the variable order depends on a baseband signal, both the carrier signal and the output signal generally exhibit distinct parameters. As a result, an applicable modulation method becomes available where the output signal depends not only on the baseband signal but also on the geometry of the surface. This characteristic not only enables signal modulation but also facilitates the encryption of the information contained in the baseband signal.