Fourier-, Hilbert- and wavelet-based signal analysis: are they really different approaches?

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Abstract

Spectral signal analysis constitutes one of the most important and most commonly used analytical tools for the evaluation of neurophysiological signals. It is not only the spectral parameters per se (amplitude and phase) which are of interest, but there is also a variety of measures derived from them, including important coupling measures like coherence or phase synchrony. After reviewing some of these measures in order to underline the widespread relevance of spectral analysis, this report compares the three classical spectral analysis approaches: Fourier, Hilbert and wavelet transform. Recently, there seems to be increasing acceptance of the notion that Hilbert- or wavelet-based analyses be in some way superior to Fourier-based analyses. The present article counters such views by demonstrating that the three techniques are in fact formally (i.e. mathematically) equivalent when using the class of wavelets that is typically applied in spectral analyses. Moreover, spectral amplitude serves as an example to show that Fourier, Hilbert and wavelet analysis also yield equivalent results in practical applications to neuronal signals.

Introduction

Many studies in neurosciences include the measurement and evaluation of dynamic electromagnetic physiological parameters (measured, e.g., with electro-/magnetoencephalography, electrocorticography, electroretinography, electrooculography, electromyography or electrocardiography or microelectrodes). While the analysis of such neurophysiological signals increasingly often incorporates model-based (e.g., autoregressive) or information-theoretic approaches, extensive use is also still made of traditional spectral analysis techniques in which amplitude and phase values are directly estimated from a time–frequency transform of the measured signal. It has long been known, for example, that the spectral composition of electrical brain signals depends on the current state of the brain (Adrian and Matthews, 1934, Berger, 1929). Motivated by this fact, classical electroencephalography has been using an empirical subdivision of the frequency axis into separate bands or ranges (δ, ϑ, α, β, γ). If one wants to characterize not only different general states, but also the dynamics of neuronal processes, this requires a time-resolved (e.g., event-related) spectral analysis. Currently, three different classical approaches to time–frequency analysis—Fourier, Hilbert and wavelet approach—are very prevalent in the neuroscience community. Any of these methods yields quantities which can serve as a basis for further analyses. In particular, some of the most common coupling or synchrony measures (like coherence or phase synchrony) can be derived from the results of these spectral analysis techniques.

Certainly, the Fourier transform can be said to constitute the most widely used operation to obtain a spectral representation of a given signal. And if the Fourier approach is used in combination with a sliding time window (short-time Fourier transform), a spectro-temporal representation of the signal is obtained, which allows tracking of the temporal evolution of spectral values. During the last one or two decades, however, two other approaches have increasingly often been applied in neurophysiological signal analysis. Firstly, the idea of the wavelet transform is to convolve the signal to be analyzed with several oscillatory filter kernels representing different frequency bands, respectively. Secondly, the Hilbert transform can be used to determine the instantaneous, so-called analytic amplitude and phase of a given bandpass signal.

In several fields, the notion appears to have become more and more established that spectral analyses based on Hilbert or wavelet transforms yield better time–frequency resolutions or are less susceptible to non-stationarity than Fourier-based analyses. But this is actually not the case. The present paper is meant as a tutorial report in order to demonstrate that Fourier, Hilbert and wavelet transforms and their derived parameters are formally equivalent and that results are essentially the same as long as the relevant analysis parameters are matched with each other.

Section snippets

Spectral parameters and derived measures

In the following, let s(t) be the neurophysiological signal of interest. A time-resolved spectral analysis of this signal will yield a time-dependent spectrum S(f,t), i.e., a two-dimensional representation of the signal in time–frequency product space. Such a spectro-temporal representation is complex-valued, i.e., at each point, it consists of an amplitude value |S(f,t)| and a phase value ϕ(f,t), such thatS(f,t)=|S(f,t)|·eiϕ(f,t).In other words, in order to estimate at a certain time t and

Getting the spectro-temporal representation

There are several different possibilities for determining a signal’s spectro-temporal representation. The most fundamental distinction is between those approaches which are model-based and those which are not. Model-based (e.g., autoregressive) techniques are not considered in the present report. Instead, the focus will be on the three most commonly used techniques which estimate amplitude and phase as direct signal parameters. These techniques are the Fourier, Hilbert, and wavelet approach,

Empirical comparison of the approaches

Section 2 has been devoted to the most common measures which can be derived from a signal’s complex-valued spectro-temporal representation, regardless of the applied analysis approach. In Section 3, the formal equivalence of Fourier, Hilbert and wavelet spectral analyses has been demonstrated. This section now is meant to show, by means of a quantitative comparison, that the three approaches also yield largely comparable results for typical parameter settings used in practice. For this purpose,

Discussion

The concern of this article was to demonstrate that short-time Fourier transform, Hilbert transform and wavelet transform constitute formally and effectively equivalent spectral analysis approaches. This equivalence refers to phase as well as amplitude and thus also to all quantities derived from these two parameters of the complex spectro-temporal representation. The reason why the techniques are often regarded as being completely different lies in the way they are typically used.

Firstly, in

Acknowledgements

I thank Professor Reinhard Eckhorn (Philipps University, Marburg, Germany) and Dr. Alexander Gail (Caltech, Pasadena, CA, USA) for their help in improving the manuscript. This work was supported by the Deutsche Forschungsgemeinschaft (German Research Foundation; grant no. EC 53/9-3 given to Reinhard Eckhorn).

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