Newton time-extracting wavelet transform: An effective tool for characterizing frequency-varying signals with weakly-separated components and theoretical analysis

Abstract

In this paper, we propose a high resolution time-frequency (TF) analysis method called Newton time-extracting wavelet transform (NTEWT), which is designed to analyze frequency-varying signals with fast varying group delay (GD). Firstly, we discuss the relationship among the fixed points of time reassignment operator, the ridge points of wavelet transform and GD of the signal. Combining the above relations and Newton algorithm, we propose a Newton GD estimator. By only retaining the TF information most related to frequency-varying features of the signal and removing the weakly-related TF coefficients, we further introduce the NTEWT, which can not only achieve a more concentrated TF representation, but also enable signal reconstruction. Meanwhile, we develop a theoretical analysis of NTEWT under the mathematical framework. Firstly, we introduce a precise mathematical definition of a class of weakly-separated frequency-varying chirp-like components, and we prove that Newton GD estimator can accurately estimate GD of arbitrary function in this class, and NTEWT does indeed succeed in decomposing these functions. Finally, we use numerical experiments to evaluate the performance of the proposed NTEWT in terms of TF concentration, GD estimation and signal reconstruction.