Linear and synchrosqueezed time-frequency representations revisited. Part I: Overview, standards of use, related issues and algorithms
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paper_title: A friendly guide to wavelets

paper_content:

Preface.- Suggestions to the Reader.- Symbols, Conventions, and Transforms.- Part I: Basic Wavelet Analysis. Preliminaries: Background and Notation.- Windowed Fourier Transforms.- Continuous Wavelet Transforms.- Generalized Frames: Key to Analysis and Synthesis.- Discrete Time-Frequency Analysis and Sampling.- Discrete Time-Scale Analysis.- Multiresolution Analysis.- Daubechies' Orthonormal Wavelet Bases.- Part II: Physical Wavelets.- Introduction to Wavelet Electromagnetics.- Applications to Radar and Scattering.- Wavelet Acoustics.- References.- Index.
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paper_title: Characterization Of Signals By The Ridges Of Their Wavelet Transforms

paper_content:

The characterization and the separation of amplitude and frequency modulated signals is a classical problem of signal analysis and signal processing. We present a couple of new algorithmic procedures for the detection of ridges in the modulus of the (continuous) wavelet transform of one-dimensional (1-D) signals. These detection procedures are shown to be robust to additive white noise. We also derive and test a new reconstruction procedure. The latter uses only information from the restriction of the wavelet transform to a sample of points from the ridge. This provides a very efficient way to code the information contained in the signal.
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paper_title: Foundations of Time-Frequency Analysis

paper_content:

Time-frequency analysis is a modern branch of harmonic analysis. It com- prises all those parts of mathematics and its applications that use the struc- ture of translations and modulations (or time-frequency shifts) for the anal- ysis of functions and operators. Time-frequency analysis is a form of local Fourier analysis that treats time and frequency simultaneously and sym- metrically. My goal is a systematic exposition of the foundations of time-frequency analysis, whence the title of the book. The topics range from the elemen- tary theory of the short-time Fourier transform and classical results about the Wigner distribution via the recent theory of Gabor frames to quantita- tive methods in time-frequency analysis and the theory of pseudodifferential operators. This book is motivated by applications in signal analysis and quantum mechanics, but it is not about these applications. The main ori- entation is toward the detailed mathematical investigation of the rich and elegant structures underlying time-frequency analysis. Time-frequency analysis originates in the early development of quantum mechanics by H. Weyl, E. Wigner, and J. von Neumann around 1930, and in the theoretical foundation of information theory and signal analysis by D.
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paper_title: Detecting couplings between interacting oscillators with time-varying basic frequencies: Instantaneous wavelet bispectrum and information theoretic approach

paper_content:

In the natural world, the properties of interacting oscillatory systems are not constant, but evolve or fluctuating continuously in time. Thus, the basic frequencies of the interacting oscillators are time varying, which makes the system analysis complex. For studying their interactions we propose a complementary approach combining wavelet bispectral analysis and information theory. We show how these methods uncover the interacting properties and reveal the nature, strength, and direction of coupling. Wavelet bispectral analysis is generalized as a technique for detecting instantaneous phase-time dependence for the case of two or more coupled nonlinear oscillators whereas the information theory approach can uncover the directionality of coupling and extract driver-response relationships in complex systems. We generate bivariate time-series numerically to mimic typical situations that occur in real measured data, apply both methods to the same time-series and discuss the results. The approach is applicable quite generally to any system of coupled nonlinear oscillators.
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paper_title: Wavelet Phase Coherence Analysis: Application to Skin Temperature and Blood Flow

paper_content:

The technique of wavelet phase coherence analysis is introduced and used to explore relationships between oscillations on blood flow and temperature in the skin of 10 healthy subjects. Their skin temperature and blood flow were continuously recorded: under basal conditions for 30 min; during local cooling of the skin with an ice-pack for 20 min: and 30 min thereafter. The group mean basal skin temperature of 33.4°C was decreased to 29.2°C during the cooling period, and had recovered to 32.1°C by the end of the recording. The wavelet transform was used to obtain the time–frequency content of the two signals, and their coherence. It is shown that cooling increases coherence to a statistically significant extent in two frequency intervals, around 0.007 and 0.1 Hz, suggesting that these oscillatory components are involved in the regulation of skin temperature when cold is applied as a stress.
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paper_title: Joint Time-Frequency Analysis

paper_content:

It has been well understood that a given signal can be represented in an infinite number of different ways. Different signal representations can be used for different applications. For example, signals obtained from most engineering applications are usually functions of time. But when studying or designing the system, we often like to study signals and systems in the frequency domain. Although the frequency content of the majority of signals in the real world evolves over time, the classical power spectrum does not reveal such important information. In order to overcome this problem, many alternatives, such as the Gabor (1946) expansion, wavelets, and time-dependent spectra, have been developed and widely studied. In contrast to the classical time and frequency analysis, we name these new techniques joint time-frequency analysis. We introduce the basic concepts and well-tested algorithms for joint time-frequency analysis. Analogous to the classical Fourier analysis, we roughly partition this article into two parts: the linear (e.g., short-time Fourier transform, Gabor expansion) and the quadratic transforms (e.g., Wigner-Ville (1932, 1948) distribution). Finally, we introduce the so-called model-based (or parametric) time-frequency analysis method.
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paper_title: Wavelet analysis of oscillations in the peripheral blood circulation measured by laser Doppler technique

paper_content:

The wavelet transform technique, a time-frequency method with logarithmic frequency resolution, was used to analyze oscillations in human peripheral blood flow measured by laser Doppler flowmetry. The oscillations extended over a wide frequency state and their periods varied in time. Within the frequency range studied, 0.0095-1.6 Hz, five characteristic oscillations were revealed, arising from both local and central regulatory mechanisms. After the insertion of endothelium-dependent and endothelium-independent vasodilators the spectra of blood how markedly differed in the frequency interval 0.0095-0.02 Hz. In this way it was demonstrated that endothelial activity is a rhythmic process that contributes to oscillations in blood flow with a characteristic frequency of around 0.01 Hz. The study illustrates the potential of laser Doppler flowmetry combined with dynamical systems analysis for studies of both the micro- and macroscopic mechanisms of blood flow regulation in vivo.
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paper_title: Nonlinear dynamics of cardiovascular ageing

paper_content:

The application of methods drawn from nonlinear and stochastic dynamics to the analysis of cardiovascular time series is reviewed, with particular reference to the identification of changes associated with ageing. The natural variability of the heart rate (HRV) is considered in detail, including the respiratory sinus arrhythmia (RSA) corresponding to modulation of the instantaneous cardiac frequency by the rhythm of respiration. HRV has been intensively studied using traditional spectral analyses, e.g. by Fourier transform or autoregressive methods, and, because of its complexity, has been used as a paradigm for testing several proposed new methods of complexity analysis. These methods are reviewed. The application of time–frequency methods to HRV is considered, including in particular the wavelet transform which can resolve the time-dependent spectral content of HRV. Attention is focused on the cardio-respiratory interaction by introduction of the respiratory frequency variability signal (RFV), which can be acquired simultaneously with HRV by use of a respiratory effort transducer. Current methods for the analysis of interacting oscillators are reviewed and applied to cardio-respiratory data, including those for the quantification of synchronization and direction of coupling. These reveal the effect of ageing on the cardio-respiratory interaction through changes in the mutual modulation of the instantaneous cardiac and respiratory frequencies. Analyses of blood flow signals recorded with laser Doppler flowmetry are reviewed and related to the current understanding of how endothelial-dependent oscillations evolve with age: the inner lining of the vessels (the endothelium) is shown to be of crucial importance to the emerging picture. It is concluded that analyses of the complex and nonlinear dynamics of the cardiovascular system can illuminate the mechanisms of blood circulation, and that the heart, the lungs and the vascular system function as a single entity in dynamical terms. Clear evidence is found for dynamical ageing.
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paper_title: Wavelet analysis for geophysical applications

paper_content:

Wavelet transforms originated in geophysics in the early 1980s for the analysis of seismic signals. Since then, significant mathematical advances in wavelet theory have enabled a suite of applications in diverse fields. In geophysics the power of wavelets for analysis of nonstationary processes that contain multiscale features, detection of singularities, analysis of transient phenom- ena, fractal and multifractal processes, and signal com- pression is now being exploited for the study of several processes including space-time precipitation, remotely sensed hydrologic fluxes, atmospheric turbulence, can- opy cover, land surface topography, seafloor bathymetry, and ocean wind waves. It is anticipated that in the near future, significant further advances in understanding and modeling geophysical processes will result from the use of wavelet analysis. In this paper we review the basic properties of wavelets that make them such an attractive and powerful tool for geophysical applications. We dis- cuss continuous, discrete, orthogonal wavelets and wavelet packets and present applications to geophysical processes.
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paper_title: A Practical Guide to Wavelet Analysis

paper_content:

A practical step-by-step guide to wavelet analysis is given, with examples taken from time series of the El Nino–Southern Oscillation (ENSO). The guide includes a comparison to the windowed Fourier transform, the choice of an appropriate wavelet basis function, edge effects due to finite-length time series, and the relationship between wavelet scale and Fourier frequency. New statistical significance tests for wavelet power spectra are developed by deriving theoretical wavelet spectra for white and red noise processes and using these to establish significance levels and confidence intervals. It is shown that smoothing in time or scale can be used to increase the confidence of the wavelet spectrum. Empirical formulas are given for the effect of smoothing on significance levels and confidence intervals. Extensions to wavelet analysis such as filtering, the power Hovmoller, cross-wavelet spectra, and coherence are described. The statistical significance tests are used to give a quantitative measure of change...
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paper_title: Synchrosqueezing-based Recovery of Instantaneous Frequency from Nonuniform Samples

paper_content:

We propose a new approach for studying the notion of the instantaneous frequency of a signal. We build on ideas from the Synchrosqueezing theory of Daubechies, Lu, and Wu [Appl. Comput. Harmonic Anal., 30 (2010), pp. 243–261] and consider a variant of Synchrosqueezing, based on the short-time Fourier transform, to precisely define the instantaneous frequencies of a multicomponent AM-FM signal. We describe an algorithm to recover these instantaneous frequencies from the uniform or nonuniform samples of the signal and show that our method is robust to noise. We also consider an alternative approach based on the conventional, Hilbert transform-based notion of instantaneous frequency to compare to our new method. We use these methods on several test cases and apply our results to a signal analysis problem in electrocardiography.
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paper_title: Cycles in Politics: Wavelet Analysis of Political Time Series

paper_content:

Spectral analysis and ARMA models have been the most common weapons of choice for the detection of cycles in political time-series. Controversies about cycles, however, tend to revolve about an issue that both techniques are badly equipped to address: the possibility of irregular cycles without fixed periodicity throughout the entire time-series. This has led to two main consequences. On the one hand, proponents of cyclical theories have often dismissed established statistical techniques. On the other hand, proponents of established techniques have dismissed the possibility of cycles without fixed periodicity. Wavelets allow the detection of transient and coexisting cycles and structural breaks in periodicity. In this paper, we present the tools of wavelet analysis and apply them to the study to two lingering puzzles in the political science literature: the existence to cycles in election returns in the United States and in the severity of major power wars.
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paper_title: Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies

paper_content:

The behavior of the continuous wavelet and Gabor coefficients in the asymptotic limit using stationary phase approximations are investigated. In particular, it is shown how, under some additional assumptions, these coefficients allow the extraction of some characteristics of the analyzed signal, such as frequency and amplitude modulation laws. Applications to spectral line estimations and matched filtering are briefly discussed. >
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paper_title: Time-Frequency Reassignment and Synchrosqueezing: An Overview

paper_content:

This article provides a general overview of time-frequency (T-F) reassignment and synchrosqueezing techniques applied to multicomponent signals, covering the theoretical background and applications. We explain how synchrosqueezing can be viewed as a special case of reassignment enabling mode reconstruction and place emphasis on the interest of using such T-F distributions throughout with illustrative examples.
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paper_title: RECENT ADVANCES IN WAVELET ANALYSES: PART 1. A REVIEW OF CONCEPTS

paper_content:

This contribution provides a review of the most recent wavelet applications in the field of earth sciences and is devoted to introducing and illustrating new wavelet analysis methods in the field of hydrology. Wavelet analysis remains unknown in the field of hydrology even though it clearly overcomes the well-known limits of the classical Fourier analysis. New wavelet-based tools are proposed to hydrologists in order to make wavelet analysis more attractive. First, a multiresolution continuous wavelet analysis method is shown to significantly improve the determination of the temporal-scale structure of a given signal. Second, the concept of wavelet entropy in both continuous and multiresolution frameworks is introduced allowing for an estimation of the temporal evolution of a given hydrological or climatologic signal's complexity. New insights in the scale-dependence of the relationship are exposed by introducing wavelet cross-correlation and wavelet coherence. Continuous wavelet cross-correlation provides a time scale distribution of the correlation between two signals, whereas continuous wavelet coherence provides a qualitative estimator of the temporal evolution of the degree of linearity of the relationship between two signals on a given scale. These methods are applied to four large river runoffs and two global climatic indexes in a companion paper.
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paper_title: Wavelet bispectral analysis for the study of interactions among oscillators whose basic frequencies are significantly time variable

paper_content:

Bispectral analysis, recently introduced as a technique for revealing time-phase relationships, is extended to make use of wavelets rather than Fourier analysis. It is thus able to encompass instantaneous phase-time dependence for the case of two or more coupled nonlinear oscillators. The method is demonstrated and evaluated by use of test signals from a pair of coupled Poincare oscillators. It promises to be useful in a wide range of scientific contexts for studies of interacting oscillators whose basic frequencies are significantly time variable.
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paper_title: Synchrosqueezed wavelet transforms: An empirical mode decomposition-like tool

paper_content:

Abstract The EMD algorithm is a technique that aims to decompose into their building blocks functions that are the superposition of a (reasonably) small number of components, well separated in the time–frequency plane, each of which can be viewed as approximately harmonic locally, with slowly varying amplitudes and frequencies. The EMD has already shown its usefulness in a wide range of applications including meteorology, structural stability analysis, medical studies. On the other hand, the EMD algorithm contains heuristic and ad hoc elements that make it hard to analyze mathematically. In this paper we describe a method that captures the flavor and philosophy of the EMD approach, albeit using a different approach in constructing the components. The proposed method is a combination of wavelet analysis and reallocation method. We introduce a precise mathematical definition for a class of functions that can be viewed as a superposition of a reasonably small number of approximately harmonic components, and we prove that our method does indeed succeed in decomposing arbitrary functions in this class. We provide several examples, for simulated as well as real data.
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paper_title: Time-Frequency Signal Analysis with Applications

paper_content:

Introduction to Fourier Analysis Linear Time-Frequency Representations Quadratic Time-Frequency Distributions Higher Order Time-Frequency Representations Analysis of Non-Stationary Noisy Signals Some Applications of Time-Frequency Analysis.
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paper_title: The Illustrated Wavelet Transform Handbook: Introductory Theory and Applications in Science, Engineering, Medicine and Finance

paper_content:

This book provides an overview of the theory and practice of continuous and discrete wavelet transforms. Divided into seven chapters, the first three chapters of the book are introductory, describing the various forms of the wavelet transform and their computation, while the remaining chapters are devoted to applications in fluids, engineering, medicine and miscellaneous areas. Each chapter is well introduced, with suitable examples to demonstrate key concepts. Illustrations are included where appropriate, thus adding a visual dimension to the text. A noteworthy feature is the inclusion, at the end of each chapter, of a list of further resources from the academic literature which the interested reader can consult. The first chapter is purely an introduction to the text. The treatment of wavelet transforms begins in the second chapter, with the definition of what a wavelet is. The chapter continues by defining the continuous wavelet transform and its inverse and a description of how it may be used to interrogate signals. The continuous wavelet transform is then compared to the short-time Fourier transform. Energy and power spectra with respect to scale are also discussed and linked to their frequency counterparts. Towards the end of the chapter, the two-dimensional continuous wavelet transform is introduced. Examples of how the continuous wavelet transform is computed using the Mexican hat and Morlet wavelets are provided throughout. The third chapter introduces the discrete wavelet transform, with its distinction from the discretized continuous wavelet transform having been made clear at the end of the second chapter. In the first half of the chapter, the logarithmic discretization of the wavelet function is described, leading to a discussion of dyadic grid scaling, frames, orthogonal and orthonormal bases, scaling functions and multiresolution representation. The fast wavelet transform is introduced and its computation is illustrated with an example using the Haar wavelet. The second half of the chapter groups together miscellaneous points about the discrete wavelet transform, including coefficient manipulation for signal denoising and smoothing, a description of Daubechies' wavelets, the properties of translation invariance and biorthogonality, the two-dimensional discrete wavelet transforms and wavelet packets. The fourth chapter is dedicated to wavelet transform methods in the author's own specialty, fluid mechanics. Beginning with a definition of wavelet-based statistical measures for turbulence, the text proceeds to describe wavelet thresholding in the analysis of fluid flows. The remainder of the chapter describes wavelet analysis of engineering flows, in particular jets, wakes, turbulence and coherent structures, and geophysical flows, including atmospheric and oceanic processes. The fifth chapter describes the application of wavelet methods in various branches of engineering, including machining, materials, dynamics and information engineering. Unlike previous chapters, this (and subsequent) chapters are styled more as literature reviews that describe the findings of other authors. The areas addressed in this chapter include: the monitoring of machining processes, the monitoring of rotating machinery, dynamical systems, chaotic systems, non-destructive testing, surface characterization and data compression. The sixth chapter continues in this vein with the attention now turned to wavelets in the analysis of medical signals. Most of the chapter is devoted to the analysis of one-dimensional signals (electrocardiogram, neural waveforms, acoustic signals etc.), although there is a small section on the analysis of two-dimensional medical images. The seventh and final chapter of the book focuses on the application of wavelets in three seemingly unrelated application areas: fractals, finance and geophysics. The treatment on wavelet methods in fractals focuses on stochastic fractals with a short section on multifractals. The treatment on finance touches on the use of wavelets by other authors in studying stock prices, commodity behaviour, market dynamics and foreign exchange rates. The treatment on geophysics covers what was omitted from the fourth chapter, namely, seismology, well logging, topographic feature analysis and the analysis of climatic data. The text concludes with an assortment of other application areas which could only be mentioned in passing. Unlike most other publications in the subject, this book does not treat wavelet transforms in a mathematically rigorous manner but rather aims to explain the mechanics of the wavelet transform in a way that is easy to understand. Consequently, it serves as an excellent overview of the subject rather than as a reference text. Keeping the mathematics to a minimum and omitting cumbersome and detailed proofs from the text, the book is best-suited to those who are new to wavelets or who want an intuitive understanding of the subject. Such an audience may include graduate students in engineering and professionals and researchers in engineering and the applied sciences.
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paper_title: Application of the cross wavelet transform and wavelet coherence to geophysical time series

paper_content:

Many scientists have made use of the wavelet method in analyzing time series, often using popular free software. However, at present there are no similar easy to use wavelet packages for analyzing two time series together. We discuss the cross wavelet transform and wavelet coher- ence for examining relationships in time frequency space be- tween two time series. We demonstrate how phase angle statistics can be used to gain confidence in causal relation- ships and test mechanistic models of physical relationships between the time series. As an example of typical data where such analyses have proven useful, we apply the methods to the Arctic Oscillation index and the Baltic maximum sea ice extent record. Monte Carlo methods are used to assess the statistical significance against red noise backgrounds. A software package has been developed that allows users to perform the cross wavelet transform and wavelet coherence (http://www.pol.ac.uk/home/research/waveletcoherence/). As we are interested in extracting low s/n ratio signals in time series we discuss only CWT in this paper. While CWT is a common tool for analyzing localized intermittent os- cillations in a time series, it is very often desirable to ex- amine two time series together that may be expected to be linked in some way. In particular, to examine whether re- gions in time frequency space with large common power have a consistent phase relationship and therefore are sug- gestive of causality between the time series. Many geophys- ical time series are not Normally distributed and we suggest methods of applying the CWT to such time series. From two CWTs we construct the Cross Wavelet Transform (XWT) which will expose their common power and relative phase in time-frequency space. We will further define a measure of Wavelet Coherence (WTC) between two CWT, which can find significant coherence even though the common power is low, and show how confidence levels against red noise back- grounds are calculated. We will present the basic CWT theory before we move on to XWT and WTC. New developments such as quanti- fying the phase relationship and calculating the WTC sig- nificance level will be treated more fully. When using the methods on time series it is important to have solid mecha- nistic foundations on which to base any relationships found, and we caution against using the methods in a "scatter-gun" approach (particularly if the time series probability density functions are modified). To illustrate how the various meth- ods are used we apply them to two data sets from meteo- rology and glaciology. Finally, we will provide links to a MatLab software package.
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paper_title: Detecting the harmonics of oscillations with time-variable frequencies

paper_content:

A method is introduced for the spectral analysis of complex noisy signals containing several frequency components. It enables components that are independent to be distinguished from the harmonics of nonsinusoidal oscillatory processes of lower frequency. The method is based on mutual information and surrogate testing combined with the wavelet transform, and it is applicable to relatively short time series containing frequencies that are time variable. Where the fundamental frequency and harmonics of a process can be identified, the characteristic shape of the corresponding oscillation can be determined, enabling adaptive filtering to remove other components and nonoscillatory noise from the signal. Thus the total bandwidth of the signal can be correctly partitioned and the power associated with each component then can be quantified more accurately. The method is first demonstrated on numerical examples. It is then used to identify the higher harmonics of oscillations in human skin blood flow, both spontaneous and associated with periodic iontophoresis of a vasodilatory agent. The method should be equally relevant to all situations where signals of comparable complexity are encountered, including applications in astrophysics, engineering, and electrical circuits, as well as in other areas of physiology and biology.
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paper_title: Recent advances in wavelet analyses: Part 2—Amazon, Parana, Orinoco and Congo discharges time scale variability

paper_content:

This paper is devoted to illustrating new wavelet analysis methods in the field of hydrology. New wavelet indicators are applied to long-term hydrological and climatologic proxies. They are first applied to four Atlantic large river monthly discharges (Amazon, Parana, Orinoco and Congo) and then applied to two well-known long-term climatologic indexes: the Southern Oscillation and the North Atlantic Oscillation. This approach makes it possible to suggest physical explanations for time scale dependant relationships. Six-month variability is present in all the rivers except in the Parana. In the Congo River this signal is near stationary with strong entropy. The strength of this scale coincides with higher discharge variability before 1920 and over the Orinoco basin with higher discharge values between 1940 and 1955. An annual cycle is present in all the rivers but is much stronger, along with the entropy, in the rivers with a large tropical basin, as in the Amazon and the Orinoco. This annual signal is not permanent in the Congo and much more intermittent in the Parana where the annual recharge appears as very irregular. A quasi biennial variability is apparent in all the equator ward rivers and a nearly permanent 2-year coherence is found between SOI and the Amazon discharge. A 3 6-year oscillation typical of ENSO variability is observed in all the rivers. It is intermittent, observed before 1930 in all the rivers and after 1980 mostly in the Amazon River. These peaks of activity correspond to periods of major SOI variability. An 8-year variability is particularly strong in the Parana River before 1940 and around 1970 and to a lesser extent in the Orinoco River during the low NAO index period, between 1940 and 1970. The Congo River exhibits that time scale around 1970. A 13-year variability common to SOI, NAO and the South Atlantic Ocean circulation is observed in the Parana and Congo discharges. In all the rivers the greatest entropy is observed at bi and multi decadal time scales with the greatest coherence with both NAO and SOI. A near 20-year time scale is observed in most of the series around 1970, although it is strongest in Amazon and Parana discharges. Finally, a 30-year time scale is observed between 1940 and 1970. It is to be noted that most of the time scale dominant activities, in all the rivers, may have been influenced by the interdecadal shifts that took place around 1940 and 1970 in the Pacific and in the Atlantic. Therefore, in this contribution, the application of new wavelet indicators (combined continuous and multi-resolution analysis, wavelet entropy, wavelet coherence, wavelet cross-correlation) leads to several improvements in the analysis of these global hydrological signal fluctuations and of their mutual time varying relationships. It is demonstrated that wavelets should be used more systematically in preference to the classical Fourier analysis, notably in hydrology.
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paper_title: A Wavelet Tour of Signal Processing

paper_content:

Introduction to a Transient World. Fourier Kingdom. Discrete Revolution. Time Meets Frequency. Frames. Wavelet Zoom. Wavelet Bases. Wavelet Packet and Local Cosine Bases. An Approximation Tour. Estimations are Approximations. Transform Coding. Appendix A: Mathematical Complements. Appendix B: Software Toolboxes.
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paper_title: Time-frequency distributions-a review

paper_content:

A review and tutorial of the fundamental ideas and methods of joint time-frequency distributions is presented. The objective of the field is to describe how the spectral content of a signal changes in time and to develop the physical and mathematical ideas needed to understand what a time-varying spectrum is. The basic gal is to devise a distribution that represents the energy or intensity of a signal simultaneously in time and frequency. Although the basic notions have been developing steadily over the last 40 years, there have recently been significant advances. This review is intended to be understandable to the nonspecialist with emphasis on the diversity of concepts and motivations that have gone into the formation of the field. >
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paper_title: The Synchrosqueezing algorithm for time-varying spectral analysis: robustness properties and new paleoclimate applications

paper_content:

We analyze the stability properties of the Synchrosqueezing transform, a time-frequency signal analysis method that can identify and extract oscillatory components with time-varying frequency and amplitude. We show that Synchrosqueezing is robust to bounded perturbations of the signal and to Gaussian white noise. These results justify its applicability to noisy or nonuniformly sampled data that is ubiquitous in engineering and the natural sciences. We also describe a practical implementation of Synchrosqueezing and provide guidance on tuning its main parameters. As a case study in the geosciences, we examine characteristics of a key paleoclimate change in the last 2.5 million years, where Synchrosqueezing provides significantly improved insights. Highlights? We study the stability of the Synchrosqueezing transform for spectral analysis. ? Synchrosqueezing is shown to be robust to bounded errors and Gaussian white noise. ? We describe a numerical implementation and compare it with other techniques. ? We apply Synchrosqueezing to proxies describing the evolution of the global climate.
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paper_title: ON INSTANTANEOUS FREQUENCY

paper_content:

Instantaneous frequency (IF) is necessary for understanding the detailed mechanisms for nonlinear and nonstationary processes. Historically, IF was computed from analytic signal (AS) through the Hilbert transform. This paper offers an overview of the difficulties involved in using AS, and two new methods to overcome the difficulties for computing IF. The first approach is to compute the quadrature (defined here as a simple 90° shift of phase angle) directly. The second approach is designated as the normalized Hilbert transform (NHT), which consists of applying the Hilbert transform to the empirically determined FM signals. Additionally, we have also introduced alternative methods to compute local frequency, the generalized zero-crossing (GZC), and the teager energy operator (TEO) methods. Through careful comparisons, we found that the NHT and direct quadrature gave the best overall performance. While the TEO method is the most localized, it is limited to data from linear processes, the GZC method is the m...
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paper_title: A product theorem for Hilbert transforms

paper_content:

Abstract : A mathematical theorem is presented for finding the Hilbert transform of a product of functions in a simplified fashion, under certain conditions. Engineering applications are discussed. (Author)
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paper_title: On Instantaneous Amplitude and Phase of Signals

paper_content:

In many questions of signal processing, it is important to use the concepts of instantaneous amplitude or phase of signals. This is especially the case in communication systems with amplitude or frequency modulation. These concepts are often introduced empirically. However, it is well known that the correct approach for this purpose is to use the concept of analytic signal. Starting from this point, we show some examples of contradictions appearing when using other definitions of instantaneous amplitude or frequency that are commonly admitted. This introduces the problem of characterizing pure amplitude-modulated or pure phase-modulated signals. It is especially shown that whereas amplitude modulated signals can be characterized by spectral considerations, this is no longer the case for phase-modulated signals. Furthermore, signals with constant amplitude have very specific properties, which are analyzed in detail. Some consequences and extensions to random signals are finally discussed.
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paper_title: On the analytic signal, the Teager-Kaiser energy algorithm, and other methods for defining amplitude and frequency

paper_content:

This paper compares the Teager-Kaiser algorithm (TKA) and other similar local methods to the analytic signal (AS) procedure. The general concepts of the instantaneous amplitude and frequency are discussed. It is shown that only AS meets certain physical conditions for the amplitude, phase, and frequency (APF). The advantage of accuracy and simplicity of the AS is also demonstrated.
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paper_title: Estimating and interpreting the instanteneous frequency of a signal. I. Fundamentals

paper_content:

The frequency of a sinusoidal signal is a well defined quantity. However, often in practice, signals are not truly sinusoidal, or even aggregates of sinusoidal components. Nonstationary signals in particular do not lend themselves well to decomposition into sinusoidal components. For such signals, the notion of frequency loses with effectiveness, and one needs to use a parameter which accounts for the time-varying nature of the process. This need has given rise to the idea of instantaneous frequency. The instantaneous frequency (IF) of a signal is a parameter which is often of significant practical importance
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paper_title: Foundations of Time-Frequency Analysis

paper_content:

Time-frequency analysis is a modern branch of harmonic analysis. It com- prises all those parts of mathematics and its applications that use the struc- ture of translations and modulations (or time-frequency shifts) for the anal- ysis of functions and operators. Time-frequency analysis is a form of local Fourier analysis that treats time and frequency simultaneously and sym- metrically. My goal is a systematic exposition of the foundations of time-frequency analysis, whence the title of the book. The topics range from the elemen- tary theory of the short-time Fourier transform and classical results about the Wigner distribution via the recent theory of Gabor frames to quantita- tive methods in time-frequency analysis and the theory of pseudodifferential operators. This book is motivated by applications in signal analysis and quantum mechanics, but it is not about these applications. The main ori- entation is toward the detailed mathematical investigation of the rich and elegant structures underlying time-frequency analysis. Time-frequency analysis originates in the early development of quantum mechanics by H. Weyl, E. Wigner, and J. von Neumann around 1930, and in the theoretical foundation of information theory and signal analysis by D.
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paper_title: A Wavelet Tour of Signal Processing

paper_content:

Introduction to a Transient World. Fourier Kingdom. Discrete Revolution. Time Meets Frequency. Frames. Wavelet Zoom. Wavelet Bases. Wavelet Packet and Local Cosine Bases. An Approximation Tour. Estimations are Approximations. Transform Coding. Appendix A: Mathematical Complements. Appendix B: Software Toolboxes.
---
paper_title: A Practical Guide to Wavelet Analysis

paper_content:

A practical step-by-step guide to wavelet analysis is given, with examples taken from time series of the El Nino–Southern Oscillation (ENSO). The guide includes a comparison to the windowed Fourier transform, the choice of an appropriate wavelet basis function, edge effects due to finite-length time series, and the relationship between wavelet scale and Fourier frequency. New statistical significance tests for wavelet power spectra are developed by deriving theoretical wavelet spectra for white and red noise processes and using these to establish significance levels and confidence intervals. It is shown that smoothing in time or scale can be used to increase the confidence of the wavelet spectrum. Empirical formulas are given for the effect of smoothing on significance levels and confidence intervals. Extensions to wavelet analysis such as filtering, the power Hovmoller, cross-wavelet spectra, and coherence are described. The statistical significance tests are used to give a quantitative measure of change...
---
paper_title: The Illustrated Wavelet Transform Handbook: Introductory Theory and Applications in Science, Engineering, Medicine and Finance

paper_content:

This book provides an overview of the theory and practice of continuous and discrete wavelet transforms. Divided into seven chapters, the first three chapters of the book are introductory, describing the various forms of the wavelet transform and their computation, while the remaining chapters are devoted to applications in fluids, engineering, medicine and miscellaneous areas. Each chapter is well introduced, with suitable examples to demonstrate key concepts. Illustrations are included where appropriate, thus adding a visual dimension to the text. A noteworthy feature is the inclusion, at the end of each chapter, of a list of further resources from the academic literature which the interested reader can consult. The first chapter is purely an introduction to the text. The treatment of wavelet transforms begins in the second chapter, with the definition of what a wavelet is. The chapter continues by defining the continuous wavelet transform and its inverse and a description of how it may be used to interrogate signals. The continuous wavelet transform is then compared to the short-time Fourier transform. Energy and power spectra with respect to scale are also discussed and linked to their frequency counterparts. Towards the end of the chapter, the two-dimensional continuous wavelet transform is introduced. Examples of how the continuous wavelet transform is computed using the Mexican hat and Morlet wavelets are provided throughout. The third chapter introduces the discrete wavelet transform, with its distinction from the discretized continuous wavelet transform having been made clear at the end of the second chapter. In the first half of the chapter, the logarithmic discretization of the wavelet function is described, leading to a discussion of dyadic grid scaling, frames, orthogonal and orthonormal bases, scaling functions and multiresolution representation. The fast wavelet transform is introduced and its computation is illustrated with an example using the Haar wavelet. The second half of the chapter groups together miscellaneous points about the discrete wavelet transform, including coefficient manipulation for signal denoising and smoothing, a description of Daubechies' wavelets, the properties of translation invariance and biorthogonality, the two-dimensional discrete wavelet transforms and wavelet packets. The fourth chapter is dedicated to wavelet transform methods in the author's own specialty, fluid mechanics. Beginning with a definition of wavelet-based statistical measures for turbulence, the text proceeds to describe wavelet thresholding in the analysis of fluid flows. The remainder of the chapter describes wavelet analysis of engineering flows, in particular jets, wakes, turbulence and coherent structures, and geophysical flows, including atmospheric and oceanic processes. The fifth chapter describes the application of wavelet methods in various branches of engineering, including machining, materials, dynamics and information engineering. Unlike previous chapters, this (and subsequent) chapters are styled more as literature reviews that describe the findings of other authors. The areas addressed in this chapter include: the monitoring of machining processes, the monitoring of rotating machinery, dynamical systems, chaotic systems, non-destructive testing, surface characterization and data compression. The sixth chapter continues in this vein with the attention now turned to wavelets in the analysis of medical signals. Most of the chapter is devoted to the analysis of one-dimensional signals (electrocardiogram, neural waveforms, acoustic signals etc.), although there is a small section on the analysis of two-dimensional medical images. The seventh and final chapter of the book focuses on the application of wavelets in three seemingly unrelated application areas: fractals, finance and geophysics. The treatment on wavelet methods in fractals focuses on stochastic fractals with a short section on multifractals. The treatment on finance touches on the use of wavelets by other authors in studying stock prices, commodity behaviour, market dynamics and foreign exchange rates. The treatment on geophysics covers what was omitted from the fourth chapter, namely, seismology, well logging, topographic feature analysis and the analysis of climatic data. The text concludes with an assortment of other application areas which could only be mentioned in passing. Unlike most other publications in the subject, this book does not treat wavelet transforms in a mathematically rigorous manner but rather aims to explain the mechanics of the wavelet transform in a way that is easy to understand. Consequently, it serves as an excellent overview of the subject rather than as a reference text. Keeping the mathematics to a minimum and omitting cumbersome and detailed proofs from the text, the book is best-suited to those who are new to wavelets or who want an intuitive understanding of the subject. Such an audience may include graduate students in engineering and professionals and researchers in engineering and the applied sciences.
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paper_title: Sampling theory and wave propagation

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In the seismic reflection method, the seismic signal does vary in amplitude, shape, frequency and phase versus the propagation time.
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paper_title: Synchrosqueezing-based Recovery of Instantaneous Frequency from Nonuniform Samples

paper_content:

We propose a new approach for studying the notion of the instantaneous frequency of a signal. We build on ideas from the Synchrosqueezing theory of Daubechies, Lu, and Wu [Appl. Comput. Harmonic Anal., 30 (2010), pp. 243–261] and consider a variant of Synchrosqueezing, based on the short-time Fourier transform, to precisely define the instantaneous frequencies of a multicomponent AM-FM signal. We describe an algorithm to recover these instantaneous frequencies from the uniform or nonuniform samples of the signal and show that our method is robust to noise. We also consider an alternative approach based on the conventional, Hilbert transform-based notion of instantaneous frequency to compare to our new method. We use these methods on several test cases and apply our results to a signal analysis problem in electrocardiography.
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paper_title: Time-Frequency Reassignment and Synchrosqueezing: An Overview

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This article provides a general overview of time-frequency (T-F) reassignment and synchrosqueezing techniques applied to multicomponent signals, covering the theoretical background and applications. We explain how synchrosqueezing can be viewed as a special case of reassignment enabling mode reconstruction and place emphasis on the interest of using such T-F distributions throughout with illustrative examples.
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paper_title: Synchrosqueezed wavelet transforms: An empirical mode decomposition-like tool

paper_content:

Abstract The EMD algorithm is a technique that aims to decompose into their building blocks functions that are the superposition of a (reasonably) small number of components, well separated in the time–frequency plane, each of which can be viewed as approximately harmonic locally, with slowly varying amplitudes and frequencies. The EMD has already shown its usefulness in a wide range of applications including meteorology, structural stability analysis, medical studies. On the other hand, the EMD algorithm contains heuristic and ad hoc elements that make it hard to analyze mathematically. In this paper we describe a method that captures the flavor and philosophy of the EMD approach, albeit using a different approach in constructing the components. The proposed method is a combination of wavelet analysis and reallocation method. We introduce a precise mathematical definition for a class of functions that can be viewed as a superposition of a reasonably small number of approximately harmonic components, and we prove that our method does indeed succeed in decomposing arbitrary functions in this class. We provide several examples, for simulated as well as real data.
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paper_title: Recent advances in wavelet analyses: Part 2—Amazon, Parana, Orinoco and Congo discharges time scale variability

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This paper is devoted to illustrating new wavelet analysis methods in the field of hydrology. New wavelet indicators are applied to long-term hydrological and climatologic proxies. They are first applied to four Atlantic large river monthly discharges (Amazon, Parana, Orinoco and Congo) and then applied to two well-known long-term climatologic indexes: the Southern Oscillation and the North Atlantic Oscillation. This approach makes it possible to suggest physical explanations for time scale dependant relationships. Six-month variability is present in all the rivers except in the Parana. In the Congo River this signal is near stationary with strong entropy. The strength of this scale coincides with higher discharge variability before 1920 and over the Orinoco basin with higher discharge values between 1940 and 1955. An annual cycle is present in all the rivers but is much stronger, along with the entropy, in the rivers with a large tropical basin, as in the Amazon and the Orinoco. This annual signal is not permanent in the Congo and much more intermittent in the Parana where the annual recharge appears as very irregular. A quasi biennial variability is apparent in all the equator ward rivers and a nearly permanent 2-year coherence is found between SOI and the Amazon discharge. A 3 6-year oscillation typical of ENSO variability is observed in all the rivers. It is intermittent, observed before 1930 in all the rivers and after 1980 mostly in the Amazon River. These peaks of activity correspond to periods of major SOI variability. An 8-year variability is particularly strong in the Parana River before 1940 and around 1970 and to a lesser extent in the Orinoco River during the low NAO index period, between 1940 and 1970. The Congo River exhibits that time scale around 1970. A 13-year variability common to SOI, NAO and the South Atlantic Ocean circulation is observed in the Parana and Congo discharges. In all the rivers the greatest entropy is observed at bi and multi decadal time scales with the greatest coherence with both NAO and SOI. A near 20-year time scale is observed in most of the series around 1970, although it is strongest in Amazon and Parana discharges. Finally, a 30-year time scale is observed between 1940 and 1970. It is to be noted that most of the time scale dominant activities, in all the rivers, may have been influenced by the interdecadal shifts that took place around 1940 and 1970 in the Pacific and in the Atlantic. Therefore, in this contribution, the application of new wavelet indicators (combined continuous and multi-resolution analysis, wavelet entropy, wavelet coherence, wavelet cross-correlation) leads to several improvements in the analysis of these global hydrological signal fluctuations and of their mutual time varying relationships. It is demonstrated that wavelets should be used more systematically in preference to the classical Fourier analysis, notably in hydrology.
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paper_title: The Synchrosqueezing algorithm for time-varying spectral analysis: robustness properties and new paleoclimate applications

paper_content:

We analyze the stability properties of the Synchrosqueezing transform, a time-frequency signal analysis method that can identify and extract oscillatory components with time-varying frequency and amplitude. We show that Synchrosqueezing is robust to bounded perturbations of the signal and to Gaussian white noise. These results justify its applicability to noisy or nonuniformly sampled data that is ubiquitous in engineering and the natural sciences. We also describe a practical implementation of Synchrosqueezing and provide guidance on tuning its main parameters. As a case study in the geosciences, we examine characteristics of a key paleoclimate change in the last 2.5 million years, where Synchrosqueezing provides significantly improved insights. Highlights? We study the stability of the Synchrosqueezing transform for spectral analysis. ? Synchrosqueezing is shown to be robust to bounded errors and Gaussian white noise. ? We describe a numerical implementation and compare it with other techniques. ? We apply Synchrosqueezing to proxies describing the evolution of the global climate.
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paper_title: Wavelet analysis of oscillations in the peripheral blood circulation measured by laser Doppler technique

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The wavelet transform technique, a time-frequency method with logarithmic frequency resolution, was used to analyze oscillations in human peripheral blood flow measured by laser Doppler flowmetry. The oscillations extended over a wide frequency state and their periods varied in time. Within the frequency range studied, 0.0095-1.6 Hz, five characteristic oscillations were revealed, arising from both local and central regulatory mechanisms. After the insertion of endothelium-dependent and endothelium-independent vasodilators the spectra of blood how markedly differed in the frequency interval 0.0095-0.02 Hz. In this way it was demonstrated that endothelial activity is a rhythmic process that contributes to oscillations in blood flow with a characteristic frequency of around 0.01 Hz. The study illustrates the potential of laser Doppler flowmetry combined with dynamical systems analysis for studies of both the micro- and macroscopic mechanisms of blood flow regulation in vivo.
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paper_title: Nonlinear dynamics of cardiovascular ageing

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The application of methods drawn from nonlinear and stochastic dynamics to the analysis of cardiovascular time series is reviewed, with particular reference to the identification of changes associated with ageing. The natural variability of the heart rate (HRV) is considered in detail, including the respiratory sinus arrhythmia (RSA) corresponding to modulation of the instantaneous cardiac frequency by the rhythm of respiration. HRV has been intensively studied using traditional spectral analyses, e.g. by Fourier transform or autoregressive methods, and, because of its complexity, has been used as a paradigm for testing several proposed new methods of complexity analysis. These methods are reviewed. The application of time–frequency methods to HRV is considered, including in particular the wavelet transform which can resolve the time-dependent spectral content of HRV. Attention is focused on the cardio-respiratory interaction by introduction of the respiratory frequency variability signal (RFV), which can be acquired simultaneously with HRV by use of a respiratory effort transducer. Current methods for the analysis of interacting oscillators are reviewed and applied to cardio-respiratory data, including those for the quantification of synchronization and direction of coupling. These reveal the effect of ageing on the cardio-respiratory interaction through changes in the mutual modulation of the instantaneous cardiac and respiratory frequencies. Analyses of blood flow signals recorded with laser Doppler flowmetry are reviewed and related to the current understanding of how endothelial-dependent oscillations evolve with age: the inner lining of the vessels (the endothelium) is shown to be of crucial importance to the emerging picture. It is concluded that analyses of the complex and nonlinear dynamics of the cardiovascular system can illuminate the mechanisms of blood circulation, and that the heart, the lungs and the vascular system function as a single entity in dynamical terms. Clear evidence is found for dynamical ageing.
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paper_title: Instantaneous frequency and wave shape functions (I)

paper_content:

Abstract Although one can formulate an intuitive notion of instantaneous frequency, generalizing “frequency” as we understand it in e.g. the Fourier transform, a rigorous mathematical definition is lacking. In this paper, we consider a class of functions composed of waveforms that repeat nearly periodically, and for which the instantaneous frequency can be given a rigorous meaning. We show that Synchrosqueezing can be used to determine the instantaneous frequency of functions in this class, even if the waveform is not harmonic, thus generalizing earlier results for cosine wave functions. We also provide real-life examples and discuss the advantages, for these examples, of considering such non-harmonic waveforms.
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paper_title: A Wavelet Tour of Signal Processing

paper_content:

Introduction to a Transient World. Fourier Kingdom. Discrete Revolution. Time Meets Frequency. Frames. Wavelet Zoom. Wavelet Bases. Wavelet Packet and Local Cosine Bases. An Approximation Tour. Estimations are Approximations. Transform Coding. Appendix A: Mathematical Complements. Appendix B: Software Toolboxes.
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paper_title: Atomic Decomposition by Basis Pursuit

paper_content:

The time-frequency and time-scale communities have recently developed a large number of overcomplete waveform dictionaries --- stationary wavelets, wavelet packets, cosine packets, chirplets, and warplets, to name a few. Decomposition into overcomplete systems is not unique, and several methods for decomposition have been proposed, including the method of frames (MOF), Matching pursuit (MP), and, for special dictionaries, the best orthogonal basis (BOB). ::: Basis Pursuit (BP) is a principle for decomposing a signal into an "optimal" superposition of dictionary elements, where optimal means having the smallest l1 norm of coefficients among all such decompositions. We give examples exhibiting several advantages over MOF, MP, and BOB, including better sparsity and superresolution. BP has interesting relations to ideas in areas as diverse as ill-posed problems, in abstract harmonic analysis, total variation denoising, and multiscale edge denoising. ::: BP in highly overcomplete dictionaries leads to large-scale optimization problems. With signals of length 8192 and a wavelet packet dictionary, one gets an equivalent linear program of size 8192 by 212,992. Such problems can be attacked successfully only because of recent advances in linear programming by interior-point methods. We obtain reasonable success with a primal-dual logarithmic barrier method and conjugate-gradient solver.
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paper_title: The Chirplet Transform: Physical Considerations

paper_content:

We consider a multidimensional parameter space formed by inner products of a parameterizable family of chirp functions with a signal under analysis. We propose the use of quadratic chirp functions (which we will call q-chirps for short), giving rise to a parameter space that includes both the time-frequency plane and the time-scale plane as 2-D subspaces. The parameter space contains a "time-frequency-scale volume" and thus encompasses both the short-time Fourier transform (as a slice along the time and frequency axes) and the wavelet transform (as a slice along the time and scale axes). In addition to time, frequency, and scale, there are two other coordinate axes within this transform space: shear in time (obtained through convolution with a q-chirp) and shear in frequency (obtained through multiplication by a q-chirp). Signals in this multidimensional space can be obtained by a new transform, which we call the "q-chirplet transform" or simply the "chirplet transform". The proposed chirplets are generalizations of wavelets related to each other by 2-D affine coordinate transformations (translations, dilations, rotations, and shears) in the time-frequency plane, as opposed to wavelets, which are related to each other by 1-D affine coordinate transformations (translations and dilations) in the time domain only.
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paper_title: Short-time fractional Fourier methods for the time-frequency representation of chirp signals

paper_content:

The fractional Fourier transform (FrFT) provides a valuable tool for the analysis of linear chirp signals. This paper develops two short-time FrFT variants which are suited to the analysis of multicomponent and nonlinear chirp signals. Outputs have similar properties to the short-time Fourier transform (STFT) but show improved time-frequency resolution. The FrFT is a parameterized transform with parameter, α, related to chirp rate. The two short-time implementations differ in how the value of α is chosen. In the first, a global optimization procedure selects one value of α with reference to the entire signal. In the second, α values are selected independently for each windowed section. Comparative variance measures based on the Gaussian function are given and are shown to be consistent with the uncertainty principle in fractional domains. For appropriately chosen FrFT orders, the derived fractional domain uncertainty relationship is minimized for Gaussian windowed linear chirp signals. The two short-time FrFT algorithms have complementary strengths demonstrated by time-frequency representations for a multicomponent bat chirp, a highly nonlinear quadratic chirp, and an output pulse from a finite-difference sonar model with dispersive change. These representations illustrate the improvements obtained in using FrFT based algorithms compared to the STFT.The fractional Fourier transform (FrFT) provides a valuable tool for the analysis of linear chirp signals. This paper develops two short-time FrFT variants which are suited to the analysis of multicomponent and nonlinear chirp signals. Outputs have similar properties to the short-time Fourier transform (STFT) but show improved time-frequency resolution. The FrFT is a parameterized transform with parameter, α, related to chirp rate. The two short-time implementations differ in how the value of α is chosen. In the first, a global optimization procedure selects one value of α with reference to the entire signal. In the second, α values are selected independently for each windowed section. Comparative variance measures based on the Gaussian function are given and are shown to be consistent with the uncertainty principle in fractional domains. For appropriately chosen FrFT orders, the derived fractional domain uncertainty relationship is minimized for Gaussian windowed linear chirp signals. The two short-time ...
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paper_title: A real-time time-frequency based instantaneous frequency estimator

paper_content:

Commonly used time-frequency representations, like the short-time Fourier transform, the Wigner distribution and the higher order polynomial distributions, estimate the instantaneous frequency at the middle of the time-interval used in the analysis. For real time applications, like for example in radar signal processing, where the target parameters are estimated in the same way as the instantaneous frequency, the delay of a half of the considered interval may be unacceptable. Here, we propose a distribution that inherently estimates the instantaneous frequency at the end of the considered time interval. With a presented procedure for on-line implementation it can outperform other time-frequency representations for real time instantaneous frequency estimation.
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paper_title: A new form of the Fourier transform for time-varying frequency estimation

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The local polynomial Fourier transform (LPFT) and the local polynomial periodogram (LPP) are proposed in order to estimate the rapidly time-varying instantaneous frequency (IF) /spl Omega/(t) of a harmonic signal. The LPFT gives the time-frequency power distribution over the t-(/spl Omega/(t), d/spl Omega/(t)/dt,...,d/sup m-1//spl Omega/(t)/dt/sup m-1/) space, where m is a degree of the LPFT. The LPFT enables one to estimate both the time-varying frequency and its derivatives. The technique is based on fitting the local polynomial approximation of the frequency which implements a high-order nonparametric regression. The a priori information about bounds for the frequency and its derivatives can be incorporated to improve the accuracy of the estimation. The asymptotic mean square errors, bias and covariance, of the estimators of d/sup s//spl Omega/(t)/dt/sup s/, s=0, 1,2,...,m/sup -1/, are obtained. The considered estimators are high-order generalization of the short-time Fourier transform. The comparative study of the asymptotic variance and bias of the estimates is presented.
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paper_title: Time-frequency signal analysis based on the windowed fractional Fourier transform

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A new signal-adaptive joint time-frequency distribution for the analysis of nonstationary signals is proposed. It is based on a fractional-Fourier-domain realization of the weighted Wigner distribution producing auto-terms close to the ones in the Wigner distribution itself, but with reduced cross-terms. Improvement over the standard time-frequency representations is achieved when the principal axes of a signal (defined as mutually orthogonal directions in the time-frequency plane for which the width of the signal's fractional power spectrum is minimum or maximum) do not correspond to time and frequency. The computational cost of this fractional-domain realization is the same as the computational cost of the realizations in the time or the frequency domain, since the windowed Fourier transform of the fractional Fourier transform of a signal corresponds to the short-time Fourier transform of the signal itself, with the window being the fractional Fourier transform of the initial one. The appropriate fractional domain is found from the knowledge of three second-order fractional Fourier transform moments. Numerical simulations confirm a qualitative advantage in the time-frequency representation, when the calculation is done in the optimal fractional domain. The approach can be generalized to the time-frequency distributions from the Cohen class.
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paper_title: Improving the readability of time-frequency and time-scale representations by the reassignment method

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In this paper, the use of the reassignment method, first applied by Kodera, Gendrin, and de Villedary (1976) to the spectrogram, is generalized to any bilinear time-frequency or time-scale distribution. This method creates a modified version of a representation by moving its values away from where they are computed, so as to produce a better localization of the signal components. We first propose a new formulation of this method, followed by a thorough theoretical study of its characteristics. Its practical use for a large variety of known time-frequency and time-scale distributions is then addressed. Finally, some experimental results are reported to demonstrate the performance of this method. >
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paper_title: Matching pursuits with time-frequency dictionaries

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The authors introduce an algorithm, called matching pursuit, that decomposes any signal into a linear expansion of waveforms that are selected from a redundant dictionary of functions. These waveforms are chosen in order to best match the signal structures. Matching pursuits are general procedures to compute adaptive signal representations. With a dictionary of Gabor functions a matching pursuit defines an adaptive time-frequency transform. They derive a signal energy distribution in the time-frequency plane, which does not include interference terms, unlike Wigner and Cohen class distributions. A matching pursuit isolates the signal structures that are coherent with respect to a given dictionary. An application to pattern extraction from noisy signals is described. They compare a matching pursuit decomposition with a signal expansion over an optimized wavepacket orthonormal basis, selected with the algorithm of Coifman and Wickerhauser see (IEEE Trans. Informat. Theory, vol. 38, Mar. 1992). >
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paper_title: A method for time-frequency analysis

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A method for time-frequency signal analysis is presented. The proposed method belongs to the general class of smoothed pseudo Wigner distributions. It is derived from the analysis of the Wigner distribution defined in the frequency domain. This method provides some substantial advantages over the Wigner distribution. The well-known cross term effects are reduced or completely removed. The oversampling of signal is not necessary. In addition, the computation time can be significantly shorter. The results are demonstrated on two numerical examples with frequency modulated signals. >
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paper_title: Short-Time Fractional Fourier Transform and Its Applications

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The fractional Fourier transform (FRFT) is a potent tool to analyze the chirp signal. However, it fails in locating the fractional Fourier domain (FRFD)-frequency contents which is required in some applications. The short-time fractional Fourier transform (STFRFT) is proposed to solve this problem. It displays the time and FRFD-frequency information jointly in the short-time fractional Fourier domain (STFRFD). Two aspects of its performance are considered: the 2-D resolution and the STFRFD support. The time-FRFD-bandwidth product (TFBP) is defined to measure the resolvable area and the STFRFD support. The optimal STFRFT is obtained with the criteria that maximize the 2-D resolution and minimize the STFRFD support. Its inverse transform, properties and computational complexity are presented. Two applications are discussed: the estimations of the time-of-arrival (TOA) and pulsewidth (PW) of chirp signals, and the STFRFD filtering. Simulations verify the validity of the proposed algorithms.
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paper_title: The fractional Fourier transform and time-frequency representations

paper_content:

The functional Fourier transform (FRFT), which is a generalization of the classical Fourier transform, was introduced a number of years ago in the mathematics literature but appears to have remained largely unknown to the signal processing community, to which it may, however, be potentially useful. The FRFT depends on a parameter /spl alpha/ and can be interpreted as a rotation by an angle /spl alpha/ in the time-frequency plane. An FRFT with /spl alpha/=/spl pi//2 corresponds to the classical Fourier transform, and an FRFT with /spl alpha/=0 corresponds to the identity operator. On the other hand, the angles of successively performed FRFTs simply add up, as do the angles of successive rotations. The FRFT of a signal can also be interpreted as a decomposition of the signal in terms of chirps. The authors briefly introduce the FRFT and a number of its properties and then present some new results: the interpretation as a rotation in the time-frequency plane, and the FRFT's relationships with time-frequency representations such as the Wigner distribution, the ambiguity function, the short-time Fourier transform and the spectrogram. These relationships have a very simple and natural form and support the FRFT's interpretation as a rotation operator. Examples of FRFTs of some simple signals are given. An example of the application of the FRFT is also given. >
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paper_title: A high-resolution quadratic time-frequency distribution for multicomponent signals analysis

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The paper introduces a new kernel for the design of a high resolution time-frequency distribution (TFD). We show that this distribution can solve problems that the Wigner-Ville distribution (WVD) or the spectrogram cannot. In particular, the proposed distribution can resolve two close signals in the time-frequency domain that the two other distributions cannot. Moreover, we show that the proposed distribution is more accurate than the WVD and the spectrogram in the estimation of the instantaneous frequency of a stepped FM signal embedded in additive Gaussian noise. Synthetic and real data collected from real-world applications are shown to validate the proposed distribution.
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paper_title: Time-Frequency Reassignment and Synchrosqueezing: An Overview

paper_content:

This article provides a general overview of time-frequency (T-F) reassignment and synchrosqueezing techniques applied to multicomponent signals, covering the theoretical background and applications. We explain how synchrosqueezing can be viewed as a special case of reassignment enabling mode reconstruction and place emphasis on the interest of using such T-F distributions throughout with illustrative examples.
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paper_title: Time-Frequency Signal Analysis with Applications

paper_content:

Introduction to Fourier Analysis Linear Time-Frequency Representations Quadratic Time-Frequency Distributions Higher Order Time-Frequency Representations Analysis of Non-Stationary Noisy Signals Some Applications of Time-Frequency Analysis.
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paper_title: Compressed sensing

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Suppose x is an unknown vector in Ropfm (a digital image or signal); we plan to measure n general linear functionals of x and then reconstruct. If x is known to be compressible by transform coding with a known transform, and we reconstruct via the nonlinear procedure defined here, the number of measurements n can be dramatically smaller than the size m. Thus, certain natural classes of images with m pixels need only n=O(m1/4log5/2(m)) nonadaptive nonpixel samples for faithful recovery, as opposed to the usual m pixel samples. More specifically, suppose x has a sparse representation in some orthonormal basis (e.g., wavelet, Fourier) or tight frame (e.g., curvelet, Gabor)-so the coefficients belong to an lscrp ball for 0<ples1. The N most important coefficients in that expansion allow reconstruction with lscr2 error O(N1/2-1p/). It is possible to design n=O(Nlog(m)) nonadaptive measurements allowing reconstruction with accuracy comparable to that attainable with direct knowledge of the N most important coefficients. Moreover, a good approximation to those N important coefficients is extracted from the n measurements by solving a linear program-Basis Pursuit in signal processing. The nonadaptive measurements have the character of "random" linear combinations of basis/frame elements. Our results use the notions of optimal recovery, of n-widths, and information-based complexity. We estimate the Gel'fand n-widths of lscrp balls in high-dimensional Euclidean space in the case 0<ples1, and give a criterion identifying near- optimal subspaces for Gel'fand n-widths. We show that "most" subspaces are near-optimal, and show that convex optimization (Basis Pursuit) is a near-optimal way to extract information derived from these near-optimal subspaces
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paper_title: A Wavelet Tour of Signal Processing

paper_content:

Introduction to a Transient World. Fourier Kingdom. Discrete Revolution. Time Meets Frequency. Frames. Wavelet Zoom. Wavelet Bases. Wavelet Packet and Local Cosine Bases. An Approximation Tour. Estimations are Approximations. Transform Coding. Appendix A: Mathematical Complements. Appendix B: Software Toolboxes.
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paper_title: Time-frequency localization operators: A geometric phase space approach

paper_content:

The author defines a set of operators which localize in both time and frequency. These operators are similar to but different from the low-pass time-limiting operator, the singular functions of which are the prolate spheroidal wave functions. The author's construction differs from the usual approach in that she treats the time-frequency plane as one geometric whole (phase space) rather than as two separate spaces. For disk-shaped or ellipse-shaped domains in time-frequency plane, the associated localization operators are remarkably simple. Their eigenfunctions are Hermite functions, and the corresponding eigenvalues are given by simple explicit formulas involving the incomplete gamma functions. >
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paper_title: On the Quantum Correction for Thermodynamic Equilibrium

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The probability of a configuration is given in classical theory by the Boltzmann formula exp [— V/hT] where V is the potential energy of this configuration. For high temperatures this of course also holds in quantum theory. For lower temperatures, however, a correction term has to be introduced, which can be developed into a power series of h. The formula is developed for this correction by means of a probability function and the result discussed.
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paper_title: Restructuring speech representations using a pitch-adaptive time-frequency smoothing and an instantaneous-frequency-based F0 extraction: possible role of a repetitive structure in sounds

paper_content:

Abstract A set of simple new procedures has been developed to enable the real-time manipulation of speech parameters. The proposed method uses pitch-adaptive spectral analysis combined with a surface reconstruction method in the time–frequency region. The method also consists of a fundamental frequency (F0) extraction using instantaneous frequency calculation based on a new concept called `fundamentalness'. The proposed procedures preserve the details of time–frequency surfaces while almost perfectly removing fine structures due to signal periodicity. This close-to-perfect elimination of interferences and smooth F0 trajectory allow for over 600% manipulation of such speech parameters as pitch, vocal tract length, and speaking rate, while maintaining high reproductive quality.
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paper_title: Signal energy distribution in time and frequency

paper_content:

The Fourier representation of signals and its relation to the signal structure in time and frequency, and more generally the inherent properties of phase-modulated signals, have received considerable attention in the past. These topics have led to such seemingly unrelated studies as the representation of a signal in a combined time-frequency plane, "instantaneous power spectra," and the ambiguity function and its transform relations. It is shown in this paper that the studies can be unified by the introduction of the concept of the complex energy density function of a signal. The function is an extension and combination of the one-dimensional energy density functions in time and frequency, the energy density spectrum |\Psi(f)|^{2} , and energy density waveform |\psi (t)|^{2} . On the basis of the complex energy density function, the significance of complicated-appearing transform relations is readily understood. The new concept also conveys a good insight into the internal structure of phase-modulated signals.
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paper_title: A friendly guide to wavelets

paper_content:

Preface.- Suggestions to the Reader.- Symbols, Conventions, and Transforms.- Part I: Basic Wavelet Analysis. Preliminaries: Background and Notation.- Windowed Fourier Transforms.- Continuous Wavelet Transforms.- Generalized Frames: Key to Analysis and Synthesis.- Discrete Time-Frequency Analysis and Sampling.- Discrete Time-Scale Analysis.- Multiresolution Analysis.- Daubechies' Orthonormal Wavelet Bases.- Part II: Physical Wavelets.- Introduction to Wavelet Electromagnetics.- Applications to Radar and Scattering.- Wavelet Acoustics.- References.- Index.
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paper_title: A Wavelet Tour of Signal Processing

paper_content:

Introduction to a Transient World. Fourier Kingdom. Discrete Revolution. Time Meets Frequency. Frames. Wavelet Zoom. Wavelet Bases. Wavelet Packet and Local Cosine Bases. An Approximation Tour. Estimations are Approximations. Transform Coding. Appendix A: Mathematical Complements. Appendix B: Software Toolboxes.
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paper_title: A friendly guide to wavelets

paper_content:

Preface.- Suggestions to the Reader.- Symbols, Conventions, and Transforms.- Part I: Basic Wavelet Analysis. Preliminaries: Background and Notation.- Windowed Fourier Transforms.- Continuous Wavelet Transforms.- Generalized Frames: Key to Analysis and Synthesis.- Discrete Time-Frequency Analysis and Sampling.- Discrete Time-Scale Analysis.- Multiresolution Analysis.- Daubechies' Orthonormal Wavelet Bases.- Part II: Physical Wavelets.- Introduction to Wavelet Electromagnetics.- Applications to Radar and Scattering.- Wavelet Acoustics.- References.- Index.
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paper_title: The Illustrated Wavelet Transform Handbook: Introductory Theory and Applications in Science, Engineering, Medicine and Finance

paper_content:

This book provides an overview of the theory and practice of continuous and discrete wavelet transforms. Divided into seven chapters, the first three chapters of the book are introductory, describing the various forms of the wavelet transform and their computation, while the remaining chapters are devoted to applications in fluids, engineering, medicine and miscellaneous areas. Each chapter is well introduced, with suitable examples to demonstrate key concepts. Illustrations are included where appropriate, thus adding a visual dimension to the text. A noteworthy feature is the inclusion, at the end of each chapter, of a list of further resources from the academic literature which the interested reader can consult. The first chapter is purely an introduction to the text. The treatment of wavelet transforms begins in the second chapter, with the definition of what a wavelet is. The chapter continues by defining the continuous wavelet transform and its inverse and a description of how it may be used to interrogate signals. The continuous wavelet transform is then compared to the short-time Fourier transform. Energy and power spectra with respect to scale are also discussed and linked to their frequency counterparts. Towards the end of the chapter, the two-dimensional continuous wavelet transform is introduced. Examples of how the continuous wavelet transform is computed using the Mexican hat and Morlet wavelets are provided throughout. The third chapter introduces the discrete wavelet transform, with its distinction from the discretized continuous wavelet transform having been made clear at the end of the second chapter. In the first half of the chapter, the logarithmic discretization of the wavelet function is described, leading to a discussion of dyadic grid scaling, frames, orthogonal and orthonormal bases, scaling functions and multiresolution representation. The fast wavelet transform is introduced and its computation is illustrated with an example using the Haar wavelet. The second half of the chapter groups together miscellaneous points about the discrete wavelet transform, including coefficient manipulation for signal denoising and smoothing, a description of Daubechies' wavelets, the properties of translation invariance and biorthogonality, the two-dimensional discrete wavelet transforms and wavelet packets. The fourth chapter is dedicated to wavelet transform methods in the author's own specialty, fluid mechanics. Beginning with a definition of wavelet-based statistical measures for turbulence, the text proceeds to describe wavelet thresholding in the analysis of fluid flows. The remainder of the chapter describes wavelet analysis of engineering flows, in particular jets, wakes, turbulence and coherent structures, and geophysical flows, including atmospheric and oceanic processes. The fifth chapter describes the application of wavelet methods in various branches of engineering, including machining, materials, dynamics and information engineering. Unlike previous chapters, this (and subsequent) chapters are styled more as literature reviews that describe the findings of other authors. The areas addressed in this chapter include: the monitoring of machining processes, the monitoring of rotating machinery, dynamical systems, chaotic systems, non-destructive testing, surface characterization and data compression. The sixth chapter continues in this vein with the attention now turned to wavelets in the analysis of medical signals. Most of the chapter is devoted to the analysis of one-dimensional signals (electrocardiogram, neural waveforms, acoustic signals etc.), although there is a small section on the analysis of two-dimensional medical images. The seventh and final chapter of the book focuses on the application of wavelets in three seemingly unrelated application areas: fractals, finance and geophysics. The treatment on wavelet methods in fractals focuses on stochastic fractals with a short section on multifractals. The treatment on finance touches on the use of wavelets by other authors in studying stock prices, commodity behaviour, market dynamics and foreign exchange rates. The treatment on geophysics covers what was omitted from the fourth chapter, namely, seismology, well logging, topographic feature analysis and the analysis of climatic data. The text concludes with an assortment of other application areas which could only be mentioned in passing. Unlike most other publications in the subject, this book does not treat wavelet transforms in a mathematically rigorous manner but rather aims to explain the mechanics of the wavelet transform in a way that is easy to understand. Consequently, it serves as an excellent overview of the subject rather than as a reference text. Keeping the mathematics to a minimum and omitting cumbersome and detailed proofs from the text, the book is best-suited to those who are new to wavelets or who want an intuitive understanding of the subject. Such an audience may include graduate students in engineering and professionals and researchers in engineering and the applied sciences.
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paper_title: A Wavelet Tour of Signal Processing

paper_content:

Introduction to a Transient World. Fourier Kingdom. Discrete Revolution. Time Meets Frequency. Frames. Wavelet Zoom. Wavelet Bases. Wavelet Packet and Local Cosine Bases. An Approximation Tour. Estimations are Approximations. Transform Coding. Appendix A: Mathematical Complements. Appendix B: Software Toolboxes.
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paper_title: A Practical Guide to Wavelet Analysis

paper_content:

A practical step-by-step guide to wavelet analysis is given, with examples taken from time series of the El Nino–Southern Oscillation (ENSO). The guide includes a comparison to the windowed Fourier transform, the choice of an appropriate wavelet basis function, edge effects due to finite-length time series, and the relationship between wavelet scale and Fourier frequency. New statistical significance tests for wavelet power spectra are developed by deriving theoretical wavelet spectra for white and red noise processes and using these to establish significance levels and confidence intervals. It is shown that smoothing in time or scale can be used to increase the confidence of the wavelet spectrum. Empirical formulas are given for the effect of smoothing on significance levels and confidence intervals. Extensions to wavelet analysis such as filtering, the power Hovmoller, cross-wavelet spectra, and coherence are described. The statistical significance tests are used to give a quantitative measure of change...
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paper_title: ENSEMBLE EMPIRICAL MODE DECOMPOSITION: A NOISE-ASSISTED DATA ANALYSIS METHOD

paper_content:

A new Ensemble Empirical Mode Decomposition (EEMD) is presented. This new approach consists of sifting an ensemble of white noise-added signal (data) and treats the mean as the final true result. Finite, not infinitesimal, amplitude white noise is necessary to force the ensemble to exhaust all possible solutions in the sifting process, thus making the different scale signals to collate in the proper intrinsic mode functions (IMF) dictated by the dyadic filter banks. As EEMD is a time–space analysis method, the added white noise is averaged out with sufficient number of trials; the only persistent part that survives the averaging process is the component of the signal (original data), which is then treated as the true and more physical meaningful answer. The effect of the added white noise is to provide a uniform reference frame in the time–frequency space; therefore, the added noise collates the portion of the signal of comparable scale in one IMF. With this ensemble mean, one can separate scales naturall...
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paper_title: The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis

paper_content:

A new method for analysing nonlinear and non-stationary data has been developed. The key part of the method is the empirical mode decomposition method with which any complicated data set can be dec...
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paper_title: One or Two Frequencies? The Empirical Mode Decomposition Answers

paper_content:

This paper investigates how the empirical mode decomposition (EMD), a fully data-driven technique recently introduced for decomposing any oscillatory waveform into zero-mean components, behaves in the case of a composite two-tones signal. Essentially two regimes are shown to exist, depending on whether the amplitude ratio of the tones is greater or smaller than unity, and the corresponding resolution properties of the EMD turn out to be in good agreement with intuition and physical interpretation. A refined analysis is provided for quantifying the observed behaviors and theoretical claims are supported by numerical experiments. The analysis is then extended to a nonlinear model where the same two regimes are shown to exist and the resolution properties of the EMD are assessed.
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paper_title: Analytical basics of the EMD: Two harmonics decomposition

paper_content:

The aim of the present work is to theoretically and simply explain how the empirical mode decomposition (EMD) operates for harmonic functions and why it selects out the highest frequency oscillation, leaving the lower frequency oscillation in the signal. Also derived are the obtainable frequency resolutions of the method and the existence of a critical frequency limit that allows separation of the closest harmonics.
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paper_title: Characterization Of Signals By The Ridges Of Their Wavelet Transforms

paper_content:

The characterization and the separation of amplitude and frequency modulated signals is a classical problem of signal analysis and signal processing. We present a couple of new algorithmic procedures for the detection of ridges in the modulus of the (continuous) wavelet transform of one-dimensional (1-D) signals. These detection procedures are shown to be robust to additive white noise. We also derive and test a new reconstruction procedure. The latter uses only information from the restriction of the wavelet transform to a sample of points from the ridge. This provides a very efficient way to code the information contained in the signal.
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paper_title: On the Analytic Wavelet Transform

paper_content:

An exact and general expression for the analytic wavelet transform of a real-valued signal is constructed, resolving the time-dependent effects of nonnegligible amplitude and frequency modulation. The analytic signal is first locally represented as a modulated oscillation, demodulated by its own instantaneous frequency, and then Taylor-expanded at each point in time. The terms in this expansion, called the instantaneous modulation functions, are time-varying functions which quantify, at increasingly higher orders, the local departures of the signal from a uniform sinusoidal oscillation. Closed-form expressions for these functions are found in terms of Bell polynomials and derivatives of the signal's instantaneous frequency and bandwidth. The analytic wavelet transform is shown to depend upon the interaction between the signal's instantaneous modulation functions and frequency-domain derivatives of the wavelet, inducing a hierarchy of departures of the transform away from a perfect representation of the signal. The form of these deviation terms suggests a set of conditions for matching the wavelet properties to suit the variability of the signal, in which case our expressions simplify considerably. One may then quantify the time-varying bias associated with signal estimation via wavelet ridge analysis, and choose wavelets to minimize this bias.
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paper_title: Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies

paper_content:

The behavior of the continuous wavelet and Gabor coefficients in the asymptotic limit using stationary phase approximations are investigated. In particular, it is shown how, under some additional assumptions, these coefficients allow the extraction of some characteristics of the analyzed signal, such as frequency and amplitude modulation laws. Applications to spectral line estimations and matched filtering are briefly discussed. >
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paper_title: Time-Frequency Reassignment and Synchrosqueezing: An Overview

paper_content:

This article provides a general overview of time-frequency (T-F) reassignment and synchrosqueezing techniques applied to multicomponent signals, covering the theoretical background and applications. We explain how synchrosqueezing can be viewed as a special case of reassignment enabling mode reconstruction and place emphasis on the interest of using such T-F distributions throughout with illustrative examples.
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paper_title: On the extraction of instantaneous frequencies from ridges in time-frequency representations of signals

paper_content:

In signal processing applications, it is often necessary to extract oscillatory components and their properties from time-frequency representations, e.g. the windowed Fourier transform or wavelet transform. 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 and providing a measure of its instantaneous frequency. 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 analyzed. 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. HighlightsIdentifying and following instantaneous (ridge) frequencies in complex signals.Adaptive, universal, extraction of ridge frequencies from time-frequency representations.Comparison of the performance of different approaches for ridge curve extraction.Relative performance of the synchrosqueezed transforms in terms of curve extraction.Freely available codes implementing the new method.
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paper_title: The Synchrosqueezing algorithm for time-varying spectral analysis: robustness properties and new paleoclimate applications

paper_content:

We analyze the stability properties of the Synchrosqueezing transform, a time-frequency signal analysis method that can identify and extract oscillatory components with time-varying frequency and amplitude. We show that Synchrosqueezing is robust to bounded perturbations of the signal and to Gaussian white noise. These results justify its applicability to noisy or nonuniformly sampled data that is ubiquitous in engineering and the natural sciences. We also describe a practical implementation of Synchrosqueezing and provide guidance on tuning its main parameters. As a case study in the geosciences, we examine characteristics of a key paleoclimate change in the last 2.5 million years, where Synchrosqueezing provides significantly improved insights. Highlights? We study the stability of the Synchrosqueezing transform for spectral analysis. ? Synchrosqueezing is shown to be robust to bounded errors and Gaussian white noise. ? We describe a numerical implementation and compare it with other techniques. ? We apply Synchrosqueezing to proxies describing the evolution of the global climate.
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paper_title: On the Analytic Wavelet Transform

paper_content:

An exact and general expression for the analytic wavelet transform of a real-valued signal is constructed, resolving the time-dependent effects of nonnegligible amplitude and frequency modulation. The analytic signal is first locally represented as a modulated oscillation, demodulated by its own instantaneous frequency, and then Taylor-expanded at each point in time. The terms in this expansion, called the instantaneous modulation functions, are time-varying functions which quantify, at increasingly higher orders, the local departures of the signal from a uniform sinusoidal oscillation. Closed-form expressions for these functions are found in terms of Bell polynomials and derivatives of the signal's instantaneous frequency and bandwidth. The analytic wavelet transform is shown to depend upon the interaction between the signal's instantaneous modulation functions and frequency-domain derivatives of the wavelet, inducing a hierarchy of departures of the transform away from a perfect representation of the signal. The form of these deviation terms suggests a set of conditions for matching the wavelet properties to suit the variability of the signal, in which case our expressions simplify considerably. One may then quantify the time-varying bias associated with signal estimation via wavelet ridge analysis, and choose wavelets to minimize this bias.
---
paper_title: Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies

paper_content:

The behavior of the continuous wavelet and Gabor coefficients in the asymptotic limit using stationary phase approximations are investigated. In particular, it is shown how, under some additional assumptions, these coefficients allow the extraction of some characteristics of the analyzed signal, such as frequency and amplitude modulation laws. Applications to spectral line estimations and matched filtering are briefly discussed. >
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paper_title: Synchrosqueezed wavelet transform for frequency and damping identification from noisy signals

paper_content:

Identification of vibration parameters from the analysis of the dynamic response of a structure plays a key role in current health monitoring systems. This study evaluates the capabilities of the recently developed Synchrosqueezed Wavelet Transform (SWT) to extract instant frequencies and damping values from the simulated noise-contaminated response of a structure. Two approaches to estimate the modal damping ratio from the results of the SWT are presented. The results obtained are compared to other signal processing methods based on Continuous Wavelet (CWT) and Hilbert-Huang (HHT) transforms. It was found that the time-frequency representation obtained via SWT is sharped than the obtained using just the CWT and it allows a more robust extraction of the individual modal responses than using the HHT. However, the identification of damping ratios is more stable when the CWT coefficients are employed.
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paper_title: On the extraction of instantaneous frequencies from ridges in time-frequency representations of signals

paper_content:

In signal processing applications, it is often necessary to extract oscillatory components and their properties from time-frequency representations, e.g. the windowed Fourier transform or wavelet transform. 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 and providing a measure of its instantaneous frequency. 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 analyzed. 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. HighlightsIdentifying and following instantaneous (ridge) frequencies in complex signals.Adaptive, universal, extraction of ridge frequencies from time-frequency representations.Comparison of the performance of different approaches for ridge curve extraction.Relative performance of the synchrosqueezed transforms in terms of curve extraction.Freely available codes implementing the new method.
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paper_title: A Wavelet Tour of Signal Processing

paper_content:

Introduction to a Transient World. Fourier Kingdom. Discrete Revolution. Time Meets Frequency. Frames. Wavelet Zoom. Wavelet Bases. Wavelet Packet and Local Cosine Bases. An Approximation Tour. Estimations are Approximations. Transform Coding. Appendix A: Mathematical Complements. Appendix B: Software Toolboxes.
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paper_title: Nonlinear dynamics of cardiovascular ageing

paper_content:

The application of methods drawn from nonlinear and stochastic dynamics to the analysis of cardiovascular time series is reviewed, with particular reference to the identification of changes associated with ageing. The natural variability of the heart rate (HRV) is considered in detail, including the respiratory sinus arrhythmia (RSA) corresponding to modulation of the instantaneous cardiac frequency by the rhythm of respiration. HRV has been intensively studied using traditional spectral analyses, e.g. by Fourier transform or autoregressive methods, and, because of its complexity, has been used as a paradigm for testing several proposed new methods of complexity analysis. These methods are reviewed. The application of time–frequency methods to HRV is considered, including in particular the wavelet transform which can resolve the time-dependent spectral content of HRV. Attention is focused on the cardio-respiratory interaction by introduction of the respiratory frequency variability signal (RFV), which can be acquired simultaneously with HRV by use of a respiratory effort transducer. Current methods for the analysis of interacting oscillators are reviewed and applied to cardio-respiratory data, including those for the quantification of synchronization and direction of coupling. These reveal the effect of ageing on the cardio-respiratory interaction through changes in the mutual modulation of the instantaneous cardiac and respiratory frequencies. Analyses of blood flow signals recorded with laser Doppler flowmetry are reviewed and related to the current understanding of how endothelial-dependent oscillations evolve with age: the inner lining of the vessels (the endothelium) is shown to be of crucial importance to the emerging picture. It is concluded that analyses of the complex and nonlinear dynamics of the cardiovascular system can illuminate the mechanisms of blood circulation, and that the heart, the lungs and the vascular system function as a single entity in dynamical terms. Clear evidence is found for dynamical ageing.
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paper_title: Selective discrete Fourier transform algorithm for time-frequency analysis: method and application on simulated and cardiovascular signals

paper_content:

The Selective Discrete Fourier transform (DFT) Algorithm [SDA] method for the calculation and display of time-frequency distribution has been developed and validated. For each time and frequency, the algorithm selects the shortest required trace length and calculates the corresponding spectral component by means of DFT. This approach can be extended to any cardiovascular related signal and provides time-dependent power spectra which are intuitively easy to consider, due to their close relation to the classical spectral analysis approach. The optimal parameters of the SDA for cardiovascular-like signals were chosen. The SDA perform standard spectral analysis on stationary simulated signals as well as reliably detect abrupt changes in the frequency content of nonstationary signals. The SDA applied during a stimulated respiration experiment, accurately; detected the changes in the frequency location and amplitude of the respiratory peak in the heart rate (HR) spectrum. It also detected and quantified the expected increase in vagal tone during vagal stimuli. Furthermore, the HR time-dependent power spectrum displayed the increase in sympathetic activity and the vagal withdrawal on standing. Such transient changes in HR control would have been smeared out by standard heart rate variability (HRV), which requires consideration of long trace lengths. The SDA provides a reliable tool for the evaluation and quantification of the control exerted by the Central Nervous System, during clinical and experimental procedures resulting in nonstationary signals.
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paper_title: A new look at the statistical model identification

paper_content:

The history of the development of statistical hypothesis testing in time series analysis is reviewed briefly and it is pointed out that the hypothesis testing procedure is not adequately defined as the procedure for statistical model identification. The classical maximum likelihood estimation procedure is reviewed and a new estimate minimum information theoretical criterion (AIC) estimate (MAICE) which is designed for the purpose of statistical identification is introduced. When there are several competing models the MAICE is defined by the model and the maximum likelihood estimates of the parameters which give the minimum of AIC defined by AIC = (-2)log-(maximum likelihood) + 2(number of independently adjusted parameters within the model). MAICE provides a versatile procedure for statistical model identification which is free from the ambiguities inherent in the application of conventional hypothesis testing procedure. The practical utility of MAICE in time series analysis is demonstrated with some numerical examples.
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paper_title: Model selection and multimodel inference : a practical information-theoretic approach

paper_content:

The second edition of this book is unique in that it focuses on methods for making formal statistical inference from all the models in an a priori set (Multi-Model Inference). A philosophy is presented for model-based data analysis and a general strategy outlined for the analysis of empirical data. The book invites increased attention on a priori science hypotheses and modeling. Kullback-Leibler Information represents a fundamental quantity in science and is Hirotugu Akaike's basis for model selection. The maximized log-likelihood function can be bias-corrected as an estimator of expected, relative Kullback-Leibler information. This leads to Akaike's Information Criterion (AIC) and various extensions. These methods are relatively simple and easy to use in practice, but based on deep statistical theory. The information theoretic approaches provide a unified and rigorous theory, an extension of likelihood theory, an important application of information theory, and are objective and practical to employ across a very wide class of empirical problems. The book presents several new ways to incorporate model selection uncertainty into parameter estimates and estimates of precision. An array of challenging examples is given to illustrate various technical issues. This is an applied book written primarily for biologists and statisticians wanting to make inferences from multiple models and is suitable as a graduate text or as a reference for professional analysts.
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paper_title: Regression and time series model selection in small samples

paper_content:

SUMMARY A bias correction to the Akaike information criterion, AIC, is derived for regression and autoregressive time series models. The correction is of particular use when the sample size is small, or when the number of fitted parameters is a moderate to large fraction of the sample size. The corrected method, called AICC, is asymptotically efficient if the true model is infinite dimensional. Furthermore, when the true model is of finite dimension, AICC is found to provide better model order choices than any other asymptotically efficient method. Applications to nonstationary autoregressive and mixed autoregressive moving average time series models are also discussed.
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Title: Linear and synchrosqueezed time-frequency representations revisited. Part I: Overview, standards of use, related issues and algorithms

Section 1: Introduction
Description 1: Introduce the topic of time-frequency analysis, its usefulness, and the objectives of the paper.

Section 2: Analytic signal
Description 2: Discuss the concept of the analytic signal, components of AM/FM signals, and the analytic approximation.

Section 3: Time-frequency representations (TFRs)
Description 3: Explain the general concept of TFR, the types of TFRs and their importance in analyzing signals.

Section 4: Windowed Fourier Transform (WFT)
Description 4: Explore the definition, calculation, and properties of the Windowed Fourier Transform.

Section 5: Gaussian window
Description 5: Discuss the Gaussian window function specific to the WFT and its properties.

Section 6: Wavelet transform (WT)
Description 6: Delve into the definition, calculation, and properties of the Wavelet Transform.

Section 7: Morlet wavelet
Description 7: Explain the specifics of the Morlet wavelet used in the WT and its resolution properties.

Section 8: Lognormal wavelet
Description 8: Discuss the lognormal wavelet, its advantages, and properties.

Section 9: Synchrosqueezed WFT (SWFT) and WT (SWT)
Description 9: Introduce the concept of synchrosqueezing and how it applies to WFT and WT to create SWFT and SWT.

Section 10: Difference between the (S)WFT and the (S)WT
Description 10: Compare and contrast the properties and applications of the (S)WFT and the (S)WT.

Section 11: Other time-frequency representations
Description 11: Explore other types of TFRs beyond (S)WFT and (S)WT such as quadratic representations and their properties.

Section 12: Time-, frequency-, and time-frequency resolution
Description 12: Discuss the concepts and definitions of time, frequency, and time-frequency resolution for TFRs. 

Section 13: General formulation
Description 13: Provide a detailed formulation of how to define and measure resolutions within the TFR framework.

Section 14: Classical definitions and their flaws
Description 14: Review traditional definitions of time and frequency resolutions and point out their limitations.

Section 15: Notion of the window/wavelet-support
Description 15: Introduce and define the concepts of window and wavelet support in time and frequency domains.

Section 16: Reconsidered definitions
Description 16: Present improved and more universal definitions for time and frequency resolution.

Section 17: Extraction of components from the TFR
Description 17: Discuss the methods for identifying and extracting components from TFRs and their advantages.

Section 18: Identification of the components
Description 18: Explain the steps and criteria for identifying components within the TFR.

Section 19: Estimation of the component's parameters
Description 19: Detail methods for estimating component parameters such as amplitude, phase, and frequency from TFR.

Section 20: Direct reconstruction
Description 20: Describe the direct reconstruction method to estimate signal parameters.

Section 21: Ridge reconstruction
Description 21: Explore the ridge reconstruction method and its application in parameter estimation.

Section 22: Practical issues
Description 22: Address practical considerations such as signal preprocessing, frequency discretization, and dealing with boundary effects when using TFRs.

Section 23: Signal preprocessing
Description 23: Discuss methods for preprocessing signals such as removing trends and bandpass filtering.

Section 24: Frequency discretization
Description 24: Provide guidance on the choice of frequency discretization parameters and their impact on TFRs.

Section 25: Boundary effects and padding
Description 25: Explain the impact of boundary effects and different schemes for padding signals to mitigate these effects.

Section 26: Padding and its schemes
Description 26: Describe various padding schemes and their effects on TFRs.

Section 27: Error estimates and the cone-of-influence
Description 27: Present methods for estimating boundary errors and defining the cone-of-influence for reliable TFR.

Section 28: TFR frequency range
Description 28: Discuss the criteria for selecting the frequency range over which to calculate TFRs. 

Section 29: Summary: step-by-step algorithms and codes
Description 29: Summarize the step-by-step algorithms and provide details on the available MatLab codes for calculating different kinds of TFRs.