Quantitative analysis of EEG signals: Time-frequency methods and ...

4.2.3 Multiresolution Analysis
Contracted versions of the wavelet function will match high frequency components of
the original signal and on the other hand, dilated versions will match low frequency
oscillations. By correlating the original signal with wavelet functions of dierent sizes
we can obtain the details of the signal in dierent scale levels. Then, the information
given by the dyadic Wavelet Transform can be organized according to a hierarchical
scheme called multiresolution analysis (Mallat, 1989 Chui, 1992).
If we denote by W j the subspaces of L 2 generated by the wavelets jk for each level
j, the space L 2 can be decomposed as a direct sum of the subspaces W j ,
L 2 = X j2Z
W j (30)
Let us dene the closed subspaces
V j = W j+1 W j+2 ::: j 2Z (31)
The subspaces V j are a multiresolution approximation of L 2 and they are generated by
scalings and translations of a single function jk called the scaling function. Then, for
the subspaces V j we have the orthogonal complementary subspaces W j , namely:
V j;1 = V j W j j 2Z (32)
Let us suppose we have a discrete signal X(n), which we will denote as x 0 , with nite
energy and without loss of generality, let us suppose that the sampling rate is t =1.
Then, we can successively decompose it with the following recursive scheme
x j;1 ( n ) = x j ( n ) r j ( n ) (33)
where the terms x j (n) 2 V j give the coarser representation of the signal and r j (n) 2 W j
give the details for each scale j =0 1 N. Then, for any resolution level N > 0, we
can write the decomposition of the signal as
x N (k) ( 2 ;N n ; k ) +
NX
X
C j (k) jk (n) (34)
X(n) = X k
j=1
k
where () is the scaling function, C j (k) are the wavelet coecients, and the sequence
fx N (k)g represents the coarser signal at the resolution level N. The second term is
the wavelet expansion. The wavelet coecients C j (k) can be interpreted as the local
residual errors between successive signal approximations at scales j ; 1 and j, and
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