Amplitudes of mono-component signals and the generalized sampling functions

Author-Highlights
• The amplitude of a mono-component is determined by its phase.
• The amplitude is represented by the so-called generalized sampling functions that are related to the phase.
• The amplitude and its Hilbert transform must be continuous.
• A new characterization of the band-limited functions in terms of its phase is given.

There is a recent trend to use mono-components to represent nonlinear and non-stationary signals rather than the usual Fourier basis with linear phase, such as the intrinsic mode functions used in Norden Huang's empirical mode decomposition [12]. 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 paper how the amplitude is determined by its phase for a class of signals, of which the instantaneous frequency is periodic and described by the Poisson kernel. Our finding is that such an amplitude can be perfectly represented by a sampling formula using the so-called generalized sampling functions that are related to the phase. The regularity of such an amplitude is identified to be at least continuous. Such characterization of mono-components provides the theory to adaptively decompose non-stationary signals. Meanwhile, we also make a very interesting and new characterization of the band-limited functions.