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

TL;DR: 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).

Abstract: Morlet wavelets are frequently used for time-frequency analysis of non-stationary time series data, such as neuroelectrical signals recorded from the brain. The crucial parameter of Morlet wavelets is the width of the Gaussian that tapers the sine wave. This width parameter controls the trade-off between temporal precision and frequency precision. It is typically defined as the "number of cycles," but this parameter is opaque, and often leads to uncertainty and suboptimal analysis choices, as well as being difficult to interpret and evaluate. 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. MATLAB code is provided.

## Summary (3 min read)

### MX Cohen Morlet wavelet definition 2 of 14

- Motivation for time-frequency analysis Many biological and physical systems exhibit rhythmic processes.
- Rhythmic temporal structure embedded in time series data can be extracted and quantified using the Fourier transform or other spectral analysis methods.
- The Fourier transform has a “soft assumption” of signal stationarity, which means that the spectral and other features of the signal remain constant over time.
- The primary assumption here is that the signal is roughly stationary over some shorter time window.

### MX Cohen Morlet wavelet definition 3 of 14

- Overview of methods for 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.
- 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.
- There are several advantages of Morlet wavelets for time-frequency analysis.
- 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.

### MX Cohen Morlet wavelet definition 4 of 14

- There are two key assumptions of Morlet wavelet-based time-frequency analysis.
- The first is that the signal is roughly sinusoidal.
- On the other hand, the overwhelming success and ubiquity of time-frequency analysis methods indicates that minor violations of this assumption are not deleterious for practical data analysis, hypothesis-testing, data mining, and feature discovery.
- 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.
- The copyright holder for this preprint (which wasthis version posted August 21, 2018.

### MX Cohen Morlet wavelet definition 5 of 14

- Selected and reported in a way that obscures the assumption underlying the data analysis.
- 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.
- Other applications may require a different number of cycles.
- The relationship between the number of cycles and the temporal-spectral smoothing is unclear, partly because most people do not think about time in terms of “number of cycles,” and partly because the effect of n on the temporal smoothing depends on the frequency of the wavelet .
- Many researchers and students struggle with how to specify and interpret this parameter, which leads to confusion and possibly suboptimal data analyses.

### MX Cohen Morlet wavelet definition 6 of 14

- The empirical FWHM can be obtained by subtracting the time point where the post-peak Gaussian is closest to 50% gain (if the Gaussian is normalized to a peak amplitude of 1, then this is the value closest to .5) from the time point where the pre-peak Gaussian is closest to 50% gain.
- (This algorithm is presented in the MATLAB code at the end of this document.).
- Figure 3 shows the relationship between the empirical FWHM derived from equations 1-2 vs. equation 3.
- It is recommended to set the FWHM no lower than one cycle at the frequency of the sine wave used to create the wavelet.
- This corresponds to ~2⅔ cycles (the n parameter in equation 2).

### MX Cohen Morlet wavelet definition 7 of 14

- Have wider spectral energy windows (Cohen 2014) , so creating wavelets using the minimum bound is not necessarily optimal.
- The discussion above concerns defining the wavelet in the time domain.
- The empirical FWHM in the time domain Morlet wavelet can be obtained by applying the FWHM estimation procedure to the magnitude of the complex time series.
- 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.
- The copyright holder for this preprint (which wasthis version posted August 21, 2018.

### MX Cohen Morlet wavelet definition 8 of 14

- It is sensible and useful to report the FWHM in both the time and the frequency domains.
- This will facilitate interpretation of the results and replication of analysis methods.
- Here is a suggestion for describing the wavelets:.
- The authors implemented time-frequency analysis by convolving the signal with a set of complex Morlet wavelets, defined as complex sine waves tapered by a Gaussian.
- The full-width at half-maximum (FWHM) ranged from XXX to XXX ms with increasing wavelet peak frequency.

### MX Cohen Morlet wavelet definition 9 of 14

- 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.
- The copyright holder for this preprint (which wasthis version posted August 21, 2018.

### MX Cohen Morlet wavelet definition 11 of 14

- The copyright holder for this preprint (which wasthis version posted August 21, 2018.

### MX Cohen Morlet wavelet definition 13 of 14

- The copyright holder for this preprint (which wasthis version posted August 21, 2018.

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### Cites background from "A better way to define and describe..."

...A Morlet wavelet is a complex sine wave tapered by a Gaussian function that minimizes edge effects and retains the smoothness crucial to the analysis of electrophysiological data (Cohen, 2018)....

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9 citations

### Cites methods from "A better way to define and describe..."

...We selected these parameters such that the half width full mass of the low and high frequencies were more closely matched (Cohen, 2019)....

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##### References

1,089 citations

### "A better way to define and describe..." refers background or result in this paper

...There are several time-frequency analysis methods, most of which produce qualitatively or quantitatively similar results (Bruns 2004; Cohen 2014)....

[...]

...One should also keep in mind that wavelets with narrower Gaussians (smaller FWHM) have wider spectral energy windows (Cohen 2014), so creating wavelets using the minimum bound is not necessarily optimal....

[...]

386 citations

### "A better way to define and describe..." refers result in this paper

...There are several time-frequency analysis methods, most of which produce qualitatively or quantitatively similar results (Bruns 2004; Cohen 2014)....

[...]

280 citations

### "A better way to define and describe..." refers background in this paper

...Some researchers have noted that brain oscillations are probably not shaped like pure sine waves (Cole and Voytek 2017; Jones 2016)....

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147 citations

### "A better way to define and describe..." refers background in this paper

...Some researchers have noted that brain oscillations are probably not shaped like pure sine waves (Cole and Voytek 2017; Jones 2016)....

[...]