Amplitudes of mono-components and representation by generalized sampling functions

Abstract: A mono-component is a real-valued signal of finite energy that has non-negative instantaneous frequencies, which may be defined as the derivative of the phase function of the given real-valued signal through the approach of canonical amplitude-phase modulation. We study in this article how the amplitude is determined by its phase in a canonical amplitude-phase modulation. Our finding is that such an amplitude can be perfectly reconstructed by a sampling formula using the so-called generalized sampling functions and their Hilbert transforms. The regularity of such an amplitude is identified to be at least continuous. Meanwhile, we also make a very interesting and new characterization of the band-limited functions.