14.2 Discrete Wavelet Transform

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 $ k$th element of vector $ w$ can be represented as the inner product of the data $ y$ and the $ k$th eigenfunction $ e_k$: $ w_k =\langle y, e_k\rangle$. For a better understanding of the wavelet transform you can look at the plot of the eigenvector $ e_k,\ (k=1,\dots,N)$.


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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 $ 1,\dots,N$. 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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