A Better Way to Define and Describe Morlet Wavelets for Time-frequency Analysis

Complex 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 spectral 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 (expressed as full-width at half-maximum). This formulation provides clarity on an important data analysis parameter, and can facilitate proper analyses, reporting, and interpretation of results. MATLAB code and sample data are provided.