A better way to define and describe Morlet wavelets for time-frequency analysis

TLDR: Alternative formulations of Morlet wavelets in time and in frequency are presented that allow parameterizing the wavelets directly in terms of the desired temporal and spectral smoothing (expressed as full-width at half-maximum).

About: This article is published in NeuroImage.The article was published on 2019-10-01 and is currently open access. It has received 160 citations till now. The article focuses on the topics: Wavelet & Smoothing.

Figures

Figure 3 . The empirical FWHM of the Gaussian (y-axis) used to define complex Morlet wavelets over a range of frequencies (x-axis), defined using equations 1-2 vs. equation 3.

Figure 1 . In the presence of signal non-stationarities (panel A), the “static” power spectrum from the Fourier transform can be difficult to interpret (panel B). In these cases, a time-frequency analysis (panel C) is often insightful.

Figure 2 . Three Morlet wavelets in the time domain (upper row) and in the frequency domain (lower row) with two frequencies and different time-frequency trade-off parameters. The argument of this paper is that reporting this parameter in terms of full-width at half-maximum (FWHM) in the time and/or frequency domains is more informative than number of cycles.

Figure 4 . An example of specifying the wavelet shape in the frequency domain (top panel) and computing its inverse Fourier transform to obtain a time-domain Morlet wavelet (bottom panel). The temporal FWHM is defined using the envelope.

Frequently asked questions

Q1. What are the contributions mentioned in the paper "A better way to define and describe morlet wavelets for time-frequency analysis" ?

The purpose of this paper is to present alternative formulations of Morlet wavelets in time and in frequency that allow parameterizing the wavelets directly in terms of the desired temporal and spectral smoothing ( as full-width at half-maximum ). This formulation provides clarity on an important data analysis parameter, and should facilitate proper analyses, reporting, and interpretation of results. CC-BY-NC-ND 4. 0 International license available under a not certified by peer review ) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. An example of a “ static ” and “ dynamic ” spectral representation of a non-stationary signal is presented in Figure 1. 4. 0 International license available under a not certified by peer review ) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. The key argument of this paper is that this crucial parameter is created and reported in a way that obscures both the signal-processing and the theoretical assumptions that are imposed on the data, and that shape the results. The key upshot of this paper is two alternative methods for creating and reporting Morlet wavelets in a way that makes this key parameter transparent and easily interpretable. This parameter is typically defined as the “ number of cycles, ” but the purpose of this paper is to argue that it would be better to define the Gaussian width as the full-width at half-maximum ( FWHM ), which is the distance in time between 50 % gain before the peak to 50 % gain after the peak. CC-BY-NC-ND 4. 0 International license available under a not certified by peer review ) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. The argument of this paper is that reporting this parameter in terms of full-width at half-maximum ( FWHM ) in the time and/or frequency domains is more informative than number of cycles. 4. 0 International license available under a not certified by peer review ) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. The key goal of this paper is to present two different methods of formulating and describing Morlet wavelets in a way that makes this assumption transparent and easily interpretable. Furthermore, the resulting smoothness is often advantageous for averaging data across stimulus repetitions and individuals ; thus, arbitrarily good precision can be detrimental in applied data analysis of biological systems. The absence of sharp edges minimizes ripple effects that can be misinterpreted as oscillations ( this is a potential danger associated with plateau-shaped filters ). 

Q2. What is the primary motivation for time-frequency analyses?

Signal non-stationarities are the primary motivation for time-frequency analyses, in which the power spectrum is computed over short windows of time. 

Q3. What is the FWHM of a wavelet?

Note that a FWHM corresponding to one cycle actually means that more than one cycle will contribute to the wavelet, because there is still non-zero energy beyond the 50% gain boundaries used to define FWHM. 

Q4. What is the FWHM of the Gaussian that is used to create the Morlet?

Interpreting the results of Morlet wavelet convolution relies on the assumption that the signal is stationary within the time window that the wavelet has non-zero energy. 

Q5. What is the key parameter in time-frequency analysis?

A key parameter in time-frequency analysis is the one that governs the trade-off between temporal precision and spectral precision; it is not possible to have simultaneously arbitrarily good precision in both time and in frequency. 

Q6. What is the funding for the research?

Funding: MXC is funded by an ERC-StG 638589 Abstract Morlet wavelets are frequently used for time-frequency analysis of non-stationary time series data,such as neuroelectrical signals recorded from the brain. 

Q7. What is the inverse Fourier transform of the Gaussian?

The Gaussian in equation 4 creates the amplitude shape in the frequency domain; the time-domain wavelet can be obtained by the inverse Fourier transform of this envelope. 

Q8. what is the definition of a complex morlet wavelet?

Typical definition of Morlet wavelets A complex Morlet wavelet w can be defined as the product of a complex sine wave and a Gaussian window:(1)where i is the imaginary operator ( ), f is frequency in Hz, and t is time in seconds. 

Q9. What are the two considerations for selecting the width of a wavelet for time-frequency?

As mentioned earlier, there are two considerations for selecting the width of a wavelet for time-frequency analysis: signal processing and system quasi-stationary. 

Q10. What is the power spectrum of the Fourier transform?

the power spectrum resulting from the Fourier transform is easily visually interpretable only for stationary signals; non-stationarities are encoded in the phase spectrum, which is typically impossible to interpret visually. 

Q11. What is the main advantage of a Morlet wavelet for time-frequency analysis?

Advantages and assumptions of Morlet wavelets for time-frequency analysis A Morlet wavelet is defined as a sine wave tapered by a Gaussian (Figure 2, top row). 

Q12. what is the % of the wavelet freq2?

bioRxiv preprintMATLAB code to reproduce figure 2 % simulation parameters freq1 = 7; % of the wavelet freq2 = 12; srate = 1000; time = -2:1/srate:2; pnts = length(time); % define number of cycles numcycles = [ 3 8 ]; % create the sine wave and two Gaussian windows sinwave1 = cos(2*pi*freq1*time); sinwave2 = cos(2*pi*freq2*time); gauswin1 = exp( -time. 

Q13. Who is the author/funder of the preprint?

CC-BY-NC-ND 4.0 International licenseavailable under a not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity.