One of the objective of signal processing is to extracts keys information embedded within a given signal, these features can be interpreted in the frequency domain [Intech]. Fast Fourier transform (FFT) is the classical and most widely method uses to establish a relationship between the time and frequency domain of a signal. FFT is best suited for so called stationary signal (.i.e. that does not change over time), however in many practical applications most signals show a huge time variation and non-stationarity behaviour [Intech], hence applying FFT on these types of signals will only at best theirs frequencies spectrum with no time information regarding theirs occurrences. In order to know the locations of these frequencies components, a time-frequency description must be implemented. The short time Fourier transform (STFT) develop by Gabor in 1946 is one of time-frequency analysis method available for signal processing. The underline idea behind STFT is to segment the time varying signal of interest in fixed window length then apply Fourier transform within each windows while assuming that the signal is will be stationary inside the window [Intech]. The fundamental question as well as the main limitation of STFT is: how small should the window length be? Knowing that by making the length of the window function too narrow, there might be no frequency component (number of cycle per second) of the signal present. In the other hand, increasing its length violate the stationarity assumption on which STFT relies.

A complete and details analysis of the short time Fourier transform showing how it was applied on the recorded EEG signals along with a background theory can be found in the final year project work entitled “Brain-Computer Interface” submitted by Cillian Brewitt to the School of Engineering, University College Cork in partial fulfilment of the requirements for the award of the degree of Bachelor of Engineering in Electrical and Electronic Engineering.

The main conclusion drawn from this section was that the STFT was not suited for analysing the recorded EEG signal as they present lots of random abrupt changes over time; this has then lead the author’s to search for a more adequate time-frequency signal processing methods known as Wavelet Transform (WT). A description of WT along with its implementation in this project and the advantage over STFT for EEG will be presented in the next section

Wavelet Tansform Algorithm

In the recent years, Wavelet analysis has gained popularity amount scientists to process non-stationary signal. The WT is modern signal processing tool which allows to localize in both time and frequency domains buried patterns within a varying signal [Intowa]. Put it another way, WT analysis permits to isolate specific hidden traits in a non-stationary signal in an analogous way someone will identify a tree in the forest or an individual in a crowded concert.

The Continuous Wavelet Transform (CWT)

The term the Wavelet refers to “small oscillation” so Wavelet analysis focus on examining signal with short duration finite energy [intw]. In order words, they transform a continuous time fluctuating signal into a more useful representation. This transformed signal is called continuous wavelet transform (CWT) and is defined as follow [wtuto]:

'〖'CWT'〗'_X^ψ (τ,s)='⟨'X,ψ_(τ,s) '⟩'=∫_(-∞)^∞'▒'〖'X(t) (ψ_(τ,s) ) ̅(t)dt'〗

=∫_(-∞)^∞'▒'〖'X(t) ψ ̅((t-τ)/s)dt'〗'

(3.8)

Where '⟨' ,'⟩' denotes the scalar product; X (t) is the time varying signal of interest; τ and s are respectively the translation and scaling parameters. And finally ψ (t) represent the mother wavelet function.

Thus the basic idea of CWT is to express a given signal as a linear combinations of mathematical functions which are obtained by scaling, dilating and shifting a base function called mother wavelet figure 8[tutow].

Figure 8: Continuous Wavelet Transform. The signal is decomposed into wavelet by shifting and scaling the mother function from [ ]

Mother wavelets (ψ): family of wavelets which can be assimilated to bandpass filters banks each having its own frequency range such when their compressed and stretched version are convolved over the EEG signal the resulting wavelet is bounded to the range of frequencies contained inside the wavelet [inside]. There are a huge range of wavelet family types, the choice of a particular family mainly depends on the nature of the signal which being analysed. In this project, the Daubechies family (db8) shown in figure 9 was used. The db8 mother wavelet have an optimal frequencies localization properties and also the waveform shape is similar to those present in normal EEG figure 9 [paper].

Figure 9: Daubechies mother wavelet and scaling function. These plot were generated in MATLAB

Scaling and Translation parameters: the mother wavelet is manipulated as follow, first it is moved or convolved over the time varying signal by changing its central location; then different version are obtained by scaling the mother wavelet [Inside]. As shown in figure 10, the scaling parameter allows a global and details view of the signal.

Figure 10: Scaled and time shifted version of the mother wavelet [inside]

The amount of correlation between the wavelet waveform and the shape of the signal (EEG) at a specific scale and location is given by the transform value CWT, which is commonly referred to as wavelet coefficients. They are very useful in analysing EEG signal as it will be shown in chapter3 due to the fact they provide a measure of how much similar is the scaled wavelet similar to the signal at a defined time location ((t-τ)/s)

**...(download the rest of the essay above)**