Saeid Sanei - EEG Signal Processing and Machine Learning

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Explore cutting edge techniques at the forefront of electroencephalogram research and artificial intelligence from leading voices in the field

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and for EEG Signal Processing and Machine Learning - изображение 205and EEG Signal Processing and Machine Learning - изображение 206:

(4.50) EEG Signal Processing and Machine Learning - изображение 207

The frequency band is also reduced by a factor of two at each step. These relationships are also valid for DWT following Section 4.5.1.4.

4.5.1.6 Reconstruction

The reconstruction of the data from its wavelet coefficients can be performed step‐by‐step, starting from the lowest resolution. At each scale, we compute:

(4.51) 452 we look for C jknowing C j 1 W j 1 h and g Then - фото 208

(4.52) we look for C jknowing C j 1 W j 1 h and g Then is restored by - фото 209

we look for C jknowing C j + 1, W j + 1, h , and g . Then is restored by minimizing 453 using a least minimum squares estimator - фото 210is restored by minimizing:

(4.53) using a least minimum squares estimator and are weight functions whic - фото 211

using a least minimum squares estimator. картинка 212and картинка 213are weight functions which permit a general solution to the restoration of The relationship from of is in the form of 454 where the - фото 214. The relationship from of is in the form of 454 where the conjugate filters have the expressions - фото 215is in the form of:

(4.54) where the conjugate filters have the expressions 455 456 - фото 216

where the conjugate filters have the expressions:

(4.55) 456 It is straightforward to see that these filters satisfy the exact - фото 217

(4.56) It is straightforward to see that these filters satisfy the exact - фото 218

It is straightforward to see that these filters satisfy the exact reconstruction condition given in Eq. (4.35). In fact, Eqs. (4.55)and (4.56)give the general solutions to this equation. In this analysis, the Shannon sampling condition is always respected. No aliasing exists, so that the anti‐aliasing condition (4.76)is not necessary.

The denominator is reduced if we choose:

(4.57) This corresponds to the case where the wavelet is the difference between the - фото 219

This corresponds to the case where the wavelet is the difference between the squares of two resolutions:

(4.58) The reconstruction algorithm then carries out the following steps 1 Compute - фото 220

The reconstruction algorithm then carries out the following steps:

1 Compute the fast Fourier transform (FFT) of the signal at the low resolution.

2 Set j to np (number of WT resolutions); perform the following iteration steps:

3 Compute the FFT of the wavelet coefficients at the scale j.

4 Multiply the wavelet coefficients Wj by .

5 Multiply the signal coefficients at the lower resolution Cj by .

6 The inverse Fourier transform of gives the coefficients Cj‐1.

7 j = j − 1 and return to step 3.

The use of a band‐limited scaling function allows a reduction of sampling at each scale and limits the computation complexity.

The WT has been widely used in EEG signal analysis. Its application to seizure detection, especially for neonates, modelling of the neuron potentials, and the detection of EP and ERPs will be discussed in the corresponding chapters of this book.

4.5.2 Synchro‐Squeezed Wavelet Transform

The synchro‐squeezing wavelet transform (SSWT) has been introduced as a post‐processing technique to enhance the TF spectrum obtained by applying the WT [23]. Assuming that the input f ( t ) is a pure harmonic signal ( f ( t ) = Acos ( ωt )), using Plancherel's theorem, the following equations are derived from (4.20)[23]:

459 One assumption in the above equation Eq 459 is that the selected - фото 221

(4.59) One assumption in the above equation Eq 459 is that the selected mother - фото 222

One assumption in the above equation ( Eq. 4.59) is that the selected mother wavelet is concentrated within the positive energy range, which means картинка 223, for ε < 0. If картинка 224is concentrated around ε = ω 0, then, W ( a , b ) is concentrated around a = ω 0/ ω . This is spread out over a region of the horizontal line (e.g. a = ω 0/ ω ). In the situation where ω is almost but not exactly similar to the actual instantaneous frequency (IF) of the input signal, then, W ( a , b ) has non‐zero energy. By synchro‐squeezing, the idea is to move this energy away from ω . The proposed method in the SSWT aims to reassign the frequency locations which are closer to the actual IF to obtain an enhanced spectrum. To do that, first, the candidate IFs are calculated. For these IFs, W ( a , b ) ≠ 0:

(4.60) Considering the selected pure harmonic signal f t Acos ωt it is - фото 225

Considering the selected pure harmonic signal f ( t ) = Acos ( ωt ), it is simple to observe that ω ( a , b ) = ω . The candidate IFs are exploited to recover the actual frequencies. Therefore, a reallocation technique has been used to map the time domain into TF domain using ( b , a ) ⇒ ( b , ω ( a , b )). Based on this, each value of W ( a , b ) (computed at discrete values of a k) is re‐allocated into T f( ω l, b ) as provided in the following equation ( Eq. 4.61):

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