Saeid Sanei - EEG Signal Processing and Machine Learning

(4.32)

where k is the discrete frequency index.

At each step, the number of scalar products is divided by two and consequently the signal is smoothed. Using this procedure, the first part of a filter bank is built up. In order to restore the original data, Mallat uses the properties of orthogonal wavelets, but the theory has been generalized to a large class of filters by introducing two other filters and , also called conjugate filters. The restoration is performed with:

(4.33)

where w j + 1(∙) are the wavelet coefficients at the scale j + 1 defined later in this section. For an exact restoration, two conditions have to be satisfied for the conjugate filters:

Anti‐aliasing condition :

(4.34)

Exact restoration:

(4.35)

In the decomposition, the input is successively convolved with the two filters H (low frequencies) and G (high frequencies). Each resulting function is decimated by suppression of one sample out of two. The high frequency signal is left, and we iterate with the low frequency signal (left side of Figure 4.7). In the reconstruction, we restore the sampling by inserting a zero between each sample, then we convolve with the conjugate filters and , we add the resulting outputs and we multiply the result by 2. We iterate up to the smallest scale (right side of Figure 4.7). Orthogonal wavelets correspond to the restricted case where:

(4.36)

Figure 4.7 The filter bank associated with the multiresolution analysis.

(4.37)

(4.38)

and

(4.39)

We can easily see that this set satisfies the two basic relations (4.72)and (4.73). Among various wavelets, Daubechies wavelets are the only compact solutions to satisfy the above conditions. For bi‐orthogonal wavelets we have the relations:

(4.40)

(4.41)

and

(4.42)

The relations (4.34)and (4.35)have to be also satisfied. A large class of compact wavelet functions can be used. Many sets of filters were proposed, especially for coding [22]. It was shown that the choice of these filters must be guided by the regularity of the scaling and the wavelet functions. The complexity is proportional to N . The algorithm provides a pyramid of N elements.

Consider the scalar products c 0( k ) = 〈 f ( t ). φ ( t − k )〉 for continuous wavelets. If φ ( t ) is band limited to half of the sampling frequency, the data are correctly sampled. The data at the resolution j = 1 are:

(4.43)

and we can compute the set c 1( k ) from c 0( k ) with a discrete‐time filter with frequency response :

(4.44)

and for and

(4.45)

Therefore, an estimate of the coefficients is:

(4.46)

The cut‐off frequency is reduced by a factor 2 at each step, allowing a reduction of the number of samples by this factor. The wavelet coefficients at the scale j + 1 are:

(4.47)

and they can be computed directly from C jby:

(4.48)

where G is the following discrete‐time filter:

(4.49)