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Having reviewed the aspects of the linear and synchrosqueezed time-frequency representations (TFRs) needed for their understanding and correct use in Part I of this review, we now consider three more subtle issues that are nonetheless of crucial importance for effective application of these methods. (i) What effect do the window/wavelet parameters have on the resultant TFR, and how can they most appropriately be chosen? (ii) What are the errors inherent in the two reconstruction methods (direct and ridge) and which of them is the better? (iii) What are the advantages and drawbacks associated with synchrosqueezing? To answer these questions, we perform a detailed numerical and theoretical study of the TFRs under consideration. We consider the relevant estimates in the presence of the complications that arise in practical applications including interference between components, amplitude modulation, frequency modulation, and noise. Taken together, the results provide an in-depth understanding of the issues in question., Comment: 39 pages, 28 figures

Time-frequency representations (TFRs) of signals, such as the windowed Fourier transform (WFT), wavelet transform (WT) and their synchrosqueezed variants (SWFT, SWT), provide powerful analysis tools. However, there are many important issues related to the practical use of TFRs that need to be clarified. Here we present a thorough review of these TFRs, summarizing all theoretical, practical and numerical aspects of their use, reconsidering some conventions and introducing new concepts and procedures. The purposes of this work are: (i) to provide a consistent overview of the computation, properties, and use of the (S)WFT/(S)WT methods; (ii) to establish general standards related to their use, both theoretical and practical; and (iii) to provide clean and optimized algorithms and MatLab codes, appropriate for any window or wavelet., Comment: 45 pages, 14 figures

The extraction of oscillatory components and their properties from different time-frequency representations, such as windowed Fourier transform and wavelet transform, is an important topic in signal processing. The first step in this procedure is to find an appropriate ridge curve: a sequence of amplitude peak positions (ridge points), corresponding to the component of interest. This is not a trivial issue, and the optimal method for extraction is still not settled or agreed. We discuss and develop procedures that can be used for this task and compare their performance on both simulated and real data. In particular, we propose a method which, in contrast to many other approaches, is highly adaptive so that it does not need any parameter adjustment for the signal to be analysed. Being based on dynamic path optimization and fixed point iteration, the method is very fast, and its superior accuracy is also demonstrated. In addition, we investigate the advantages and drawbacks that synchrosqueezing offers in relation to curve extraction. The codes used in this work are freely available for download., Comment: 13 pages, 7 figures, plus 4 supplementary figures

We introduce a new adaptive decomposition tool, which we refer to as Nonlinear Mode Decomposition (NMD). It decomposes a given signal into a set of physically meaningful oscillations for any waveform, simultaneously removing the noise. NMD is based on the powerful combination of time-frequency analysis techniques - which together with the adaptive choice of their parameters make it extremely noise-robust - and surrogate data tests, used to identify interdependent oscillations and to distinguish deterministic from random activity. We illustrate the application of NMD to both simulated and real signals, and demonstrate its qualitative and quantitative superiority over the other existing approaches, such as (ensemble) empirical mode decomposition, Karhunen-Loeve expansion and independent component analysis. We point out that NMD is likely to be applicable and useful in many different areas of research, such as geophysics, finance, and the life sciences. The necessary MATLAB codes for running NMD are freely available at http://www.physics.lancs.ac.uk/research/nbmphysics/diats/nmd/., Comment: 38 pages, 13 figures