Breaking the Limits -- Redefining the Instantaneous Frequency

TL;DR

This paper redefines the instantaneous frequency to be valid for all signals, nonlinear and nonstationary, improving time-frequency analysis and signal decomposition accuracy using a novel mathematical approach.

Contribution

It introduces a new definition of instantaneous frequency using multivalued inverse tangent, addressing previous limitations and enhancing zero-phase filtering methods.

Findings

Proposed IF is valid for all monocomponent and multicomponent signals.

Enhanced accuracy in time-frequency-energy distribution.

Zero-phase filtering combined with the new IF outperforms existing methods.

Abstract

The Carson and Fry (1937) introduced the concept of variable frequency as a generalization of the constant frequency. The instantaneous frequency (IF) is the time derivative of the instantaneous phase and it is well-defined only when this derivative is positive. If this derivative is negative, the IF creates problem because it does not provide any physical significance. This study proposes a mathematical solution and eliminate this problem by redefining the IF such that it is valid for all monocomponent and multicomponent signals which can be nonlinear and nonstationary in nature. This is achieved by using the property of the multivalued inverse tangent function. The efforts and understanding of all the methods based on the IF would improve significantly by using this proposed definition of the IF. We also demonstrate that the decomposition of a signal, using zero-phase filtering based on the well established Fourier and filter theory, into a set of desired frequency bands with proposed IF produces accurate time-frequency-energy (TFE) distribution that reveals true nature of signal. Simulation results demonstrate the efficacy of the proposed IF that makes zero-phase filter based decomposition most powerful, for the TFE analysis of a signal, as compared to other existing methods in the literature.