Unifying Common Signal Analyses with Instantaneous Time-Frequency Atoms

TL;DR

This paper introduces a unified framework for instantaneous time-frequency analysis using atoms, deriving closed-form expressions for various common analyses by viewing them as decompositions into AM-FM components with quadratic chirplets.

Contribution

It unifies multiple time-frequency analysis methods within a single framework using instantaneous time-frequency atoms and provides closed-form expressions for their instantaneous spectra.

Findings

Unified analysis framework for time-frequency methods

Closed-form expressions for instantaneous spectra

Demonstrated with example signals

Abstract

In previous work, we presented a general framework for instantaneous time-frequency analysis but did not provide any specific details of how to compute a particular instantaneous spectrum (IS). In this work, we use instantaneous time-frequency atoms to obtain an IS associated with common signal analyses: time domain analysis, frequency domain analysis, fractional Fourier transform, synchrosqueezed short-time Fourier transform, and synchrosqueezed short-time fractional Fourier transform. By doing so, we demonstrate how the general framework can be used to unify these analyses and we develop closed-form expressions for the corresponding ISs. This is accomplished by viewing these analyses as decompositions into AM--FM components and recognizing that each uses a specialized (or limiting) form of a quadratic chirplet as a template during analysis. With a two-parameter quadratic chirplet, we can organize these ISs into a 2D continuum with points in the plane corresponding to a decomposition related to one of the signal analyses. Finally, using several example signals, we compute in closed-form the ISs for the various analyses.