On the non-uniqueness of the instantaneous frequency

TL;DR

This paper demonstrates that the instantaneous frequency (IF) of oscillatory signals is inherently non-unique, regardless of the method used or prior knowledge of the signal's physical origin, challenging the notion of a definitive IF.

Contribution

It provides a mathematical explanation and quantification of the non-uniqueness of IF, showing that no preferred representation exists without specific assumptions.

Findings

IF is inherently non-unique for oscillatory signals

Different adaptive methods yield different IF representations

Non-uniqueness persists even with known physical models

Abstract

In this article, we investigate the debated Instantaneous Frequency (IF) topic. Here, we show that IF is non-unique inherently. We explain how this non-uniqueness can be quantified and explained from a mathematical perspective. The non-uniqueness of the IF can also be observed if different methods of adaptive signal processing are used. We will also show that even if we know the physical origin of an oscillatory signal, e.g. linear second order ordinary differential equation, the non-uniqueness is still present. All in all, we will end up with the conclusion that, without any a priori assumption about the relationship of the envelope and phase function of an oscillatory signal, there is not any preferred neither best representation of the IF of such oscillatory signal.