In this section we describe a periodic version of the discrete wavelet transform.
The display shows a plot of an eigenvector with an index chosen
by Change index in the interactive menu.
Suppose that we have a vector y of the length N,
where N is a power of 2. Then the vector w of wavelet coefficients
of y is defined by w = W y. The matrix W is orthogonal: all its N
eigenvalues are equal to 1. Therefore the kth element of vector w
can be represented as the inner product of the data y and the kth
eigenfunction ek:
.
For a better understanding
of the wavelet transform you can look at the plot of the
eigenvector
.
Changing the index k by using the
item Change index you will see the plots of the eigenvectors
on the display. We advise you to change the index k
successively starting from 1. If k is less than the number chosen by
Change level then you will see the eigenvectors associated
to the father wavelet. Otherwise the plot shows the vectors
associated to the mother wavelet. Note that the eigenvector
approximates a father or mother wavelet of the continuous wavelet
transform if the ``support'' of the
eigenvector is strictly embedded in .
It is very important for the speed of the algorithm that
the multiplication W y is
implemented not by a matrix multiplication, but by a sequence of
special filtering steps which result in O(N) operations.
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