Читать книгу EEG Signal Processing and Machine Learning - Saeid Sanei - Страница 72

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.