# Time frequency analysis of the EEG using the wavelet transform

Several methods have been proposed to reveal frequency-specific, time-locked event-related modulations of the amplitude of ongoing EEG oscillations (ERD and ERS). All these methods rely on estimating within each single EEG epoch, the amplitude of the signal as a function time and frequency. Indeed, averaging oscillation amplitude regardless of phase allows enhancing both ‘phase-locked’ and ‘non phase-locked’ changes in signal amplitude.

An often proposed method consists in band-pass filtering EEG epochs within a predefined frequency band and squaring amplitude values (Pfurtscheller and Lopes da Silva 1999). This easily implemented method has an important disadvantage: the range of explored frequencies must be arbitrarily defined prior to performing the analysis. Therefore, this method does not allow the simultaneous exploration of the whole range of the EEG frequency spectrum. This limitation may have important consequences as studies have shown, for example, that subtle differences in topographical distribution and temporal course could be found within different frequencies of the alpha-band. Furthermore, studies have also shown that the peak frequency of these EEG rhythms may significantly vary across subjects.

The Fourier transform can also be used to express the signals oscillation amplitude across the entire frequency spectrum. However, the Fourier transform contains no temporal information. To circumvent this problem, the Fourier transform can be performed on successive EEG segments defined by a ‘windowing’ function. This procedure allows the extraction of both time and frequency domain information. The windowed Fourier transform and the wavelet transform are such Fourier derived methods. In analogy to Heisenberg’s uncertainty principle, the width of the ‘windowing function’ limits both the time and frequency resolution of the analysis. Windowed Fourier transform uses a fixed and arbitrarily defined window width resulting in a fixed time-frequency resolution ratio.

By adapting the window width in function of the frequency, time-frequency decomposition of EEG epochs using the Wavelet transform offers an optimal compromise for time-frequency resolution. A time-frequency representation based on the continuous Morlet wavelet transform of EEG epochs may thus be used to optimally identify stimulus-induced amplitude modulations of oscillatory activities.

# Continuous Morlet wavelet transform of EEG epochs

To enhance EEG changes that are time-locked but not necessarily phase-locked to stimulus onset (i.e. ERD and ERS but also ERPs), the Continuous Morlet Transform is applied to each individual trial. Resulting amplitude maps are then averaged across trials. These maps express the average oscillation amplitude as a function of latency and frequency.

According to the additive noise model, time-averaging will only enhance stimulus-related EEG changes whose waveform is identical through trials. Stimulus related oscillations satisfy this condition only if phase-locked to the stimulus onset. The TF transform of time-averaged EEG epochs (i.e. the standard-average) can be computed to assess whether amplitude enhancements are phase-locked or non phase-locked to the stimulus onset. Indeed, activities present in both the ‘TF-average’ and the ‘TF-single’ transforms may be considered as phase-locked to stimulus onset (ERPs) while activities present only in the ‘TF-single’ transform may be considered as non phase-locked to stimulus onset (ERD and ERS).