Amplitudes of mono-components and representation by generalized sampling functions

TL;DR

This paper investigates how the amplitude of mono-components in signals can be exactly reconstructed from phase information using generalized sampling functions, offering new insights into band-limited functions.

Contribution

It introduces a novel sampling formula for reconstructing amplitudes of mono-components via generalized sampling functions and characterizes their regularity and relation to band-limited functions.

Findings

Amplitude can be perfectly reconstructed from phase using generalized sampling functions.

The amplitude's regularity is at least continuous.

Provides a new characterization of band-limited 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.