On the extraction of instantaneous frequencies from ridges in time-frequency representations of signals

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 figure

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

Glassy states and superrelaxation in populations of coupled phase oscillators

Large networks of coupled oscillators appear in many branches of science, so that the kinds of phenomena they exhibit are not only of intrinsic interest but also of very wide importance. In 1975, Kuramoto proposed an analytically tractable model to describe such systems, which has since been successfully applied in many contexts and remains a subject of intensive research. Some related problems, however, remain unclarified for decades, such as the properties of the oscillator glass state discovered by Daido in 1992. Here we present a detailed analysis of a very general form of the Kuramoto model. In particular, we find the conditions when it can exhibit glassy behavior, which represents a kind of synchronous disorder in the present case. Furthermore, we discover a new and intriguing phenomenon that we refer to as {\it superrelaxation} where, for a class of parameter distributions, the oscillators feel no interaction at all during relaxation to incoherence, a phenomenon reminiscent of superfluidity or superconductivity. Our findings offer the possibility of creating glassy states and observing superrelaxation in real systems, thus paving the way to a cascade of applications and further research in the field.Comment: 9 pages, 5 figure

Dynamics of cardiovascular ageing

We gain wrinkles and lose hair, as we age, but our bodies also change in less obvious but much more important ways. This project studied the age-related alterations that occur in the cardiovascular system – the heart, lungs and network of arteries and veins that carry oxygenated blood and nutrients to every cell of the body and remove the waste products of metabolism. It was already known that the phase of breathing affects the rate at which the heart beats, but that this effect decreases as we age. The research has associated this reduction in heart-lung interaction with changes in the endothelium, the inner lining of all the blood vessels. It involved making non-invasive measurements of blood flow in the skin of 200 healthy subjects of all ages. The analysis focused on very low frequency oscillations in blood flow that can give a measure of the state of the endothelium. The main conclusions are, first, that to age healthily, you should look after your endothelium and, secondly, that it should be feasible to design an instrument for assessing endothelial health – an endotheliometer

Monitoring the ageing of the cardiovascular system

This research developed a way of measuring blood flow through the capillaries and thereby monitoring the health of the endothelium, the inner lining of the blood vessels

Nonlinear Mode Decomposition: a new noise-robust, adaptive decomposition method

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 figure

Mean-field and mean-ensemble frequencies of a system of coupled oscillators

We investigate interacting phase oscillators whose mean field is at a different frequency from the mean or mode of their natural frequencies. The associated asymmetries lead to a macroscopic traveling wave. We show that the mean-ensemble frequency of such systems differs from their entrainment frequency. In some scenarios these frequencies take values that, counterintuitively, lie beyond the limits of the natural frequencies. The results indicate that a clear distinction should be drawn between the two variables describing the macroscopic dynamics of cooperative systems. This has important implications for real systems where a nontrivial distribution of parameters is common