Time-Extracting Wavelet Transform for Characterizing Impulsive-Like Signals and Theoretical Analysis

In this paper, a high-resolution time–frequency (TF) analysis method, called time-extracting wavelet transform (TEWT), is introduced to analyze impulsive-like signals whose TF ridge curves are nearly perpendicular to the time axis. For impulsive-like signals, the instantaneous frequency with almost infinite rate of change is difficult to estimate, but the group delay (GD) with nearly zero rate of change is easier to estimate. Since the GD is the key feature of frequency-domain signals, it indicates that one can try to understand impulsive-like signals from the perspective of frequency-domain signals. In this regard, for an impulsive signal and its Fourier transform (i.e., the frequency-domain harmonic signal), we propose the TEWT that achieves highly concentrated TF representations while allowing signal reconstruction, only by retaining the TF coefficients closely related to TF features of signals, while removing weakly related TF information. The two contributions of this paper are the proposal of TEWT and the theoretical analysis of TEWT for frequency-domain signals. On the other hand, we provide a rigorous theoretical analysis of TEWT under a mathematical framework for frequency-domain signals. Specifically, we define a function class as a set of all superposition of well-separated frequency-domain harmonic-like functions, where each function can be locally regarded as a sum of a finite number of harmonic signals in the frequency domain, and establish error bounds for WT approximate expression, GD estimation, and component recovery. Finally, we verify the effectiveness of TEWT in terms of the energy concentration, robustness, and invertibility through numerical experiments with synthetic and real signals.