Linear and synchrosqueezed time–frequency representations revisited: Overview, standards of use, resolution, reconstruction, concentration, and algorithms

Time–frequency representations (TFRs) of signals, such as the windowed Fourier transform (WFT), wavelet transform (WT) and their synchrosqueezed versions (SWFT, SWT), provide powerful analysis tools. Here we present a thorough review of these TFRs, summarizing all practically relevant aspects of their use, reconsidering some conventions and introducing new concepts and procedures to advance their applicability and value. Furthermore, a detailed numerical and theoretical study of three specific questions is provided, relevant to the application of these methods, namely: the effects of the window/wavelet parameters on the resultant TFR; the relative performance of different approaches for estimating parameters of the components present in the signal from its TFR; and the advantages/drawbacks of synchrosqueezing. In particular, we show that the higher concentration of the synchrosqueezed transforms does not seem to imply better resolution properties, so that the SWFT and SWT do not appear to provide any significant advantages over the original WFT and WT apart from a more visually appealing pictures. The algorithms and Matlab codes used in this work, e.g. those for calculating (S)WFT and (S)WT, are freely available for download.