IF equation: A feature extractor for high-concentration time–frequency representation and application to mixed signals analysis

High-concentration time-frequency (TF) representation provides a valuable tool for characterizing multi-component non-stationary signals. Our previous work proposed using an instantaneous frequency (IF) equation to sharpen the TF distribution. However, the underlying principle is not comprehensive, and efficient feature extractors are lacking to analyze mixed signals. In this paper, we systematically discuss why the IF equation-based TF analysis methods work and how to use the IF equation to improve TF sharpness. By the analysis of the properties of the IF equation, we prove that a good IF equation can unify the well-known IF and group delay estimators and provides an effective way to characterize the mixture of time-varying and frequency-varying signals. By discussing the post-processing techniques based on the IF equation, we can prove that many popular TF post-processing methods, such as the synchroextracting transform, the multi-synchrosqueezing transform, and the time extracting transform, fall into the IF equation-based category. We also propose a novel approach to combine different IF equations to minimize energy spreading based on local sparsity. Numerical simulations and application to seizure and bearing fault signals are presented to illustrate the performance of the proposed method.