EEG Signal Processing and Machine Learning. Saeid Sanei

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      (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.

      4.5.1.5 Wavelet Transform Using Fourier Transform

      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_j by:

      (4.48)

      where G is the following discrete‐time filter:

      (4.49)

      and for and :

      (4.50)

      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)

      (4.52)

      we look for C_j knowing C_{j + 1}, W_{j + 1}, h, and g. Then is restored by minimizing:

      (4.53)

      using a least minimum squares estimator. and are weight functions which permit a general solution to the restoration of . The relationship from of is in the form of:

      (4.54)

      where the conjugate filters have the expressions:

(4.55)

(4.56)

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)

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