# 0.8 Filter banks and transmultiplexers  (Page 21/23)

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$\mathbf{d}=\left[\begin{array}{cc}\mathbf{I}& 0\\ 0& \mathbf{U}\\ 0& \mathbf{H}\end{array}\right]\mathbf{x}.$

Next consider switching from an $M$ -channel filter bank to a one-channel filter bank. Until $n=-1$ , the $M$ -channel filter bank is operational. From $n=0$ onwards the inputs leaks to the output. In this case, there are exit filterscorresponding to flushing the states in the first filter bank implementation at $n=0$ .

$\mathbf{d}=\left[\begin{array}{cc}\mathbf{H}& 0\\ \mathbf{W}& 0\\ 0& \mathbf{I}\end{array}\right]\mathbf{x}.$

Finally, switching from an ${M}_{1}$ -band filter bank to an ${M}_{2}$ -band filter bank can be accomplished as follows:

$\mathbf{d}=\left[\begin{array}{cc}{\mathbf{H}}_{\mathbf{1}}& 0\\ {\mathbf{W}}_{\mathbf{1}}& 0\\ 0& {\mathbf{U}}_{\mathbf{2}}\\ 0& {\mathbf{H}}_{\mathbf{2}}\end{array}\right]\mathbf{x}.$

The transition region is given by the exit filters of the first filter bank and the entry filters of the second. Clearly the transition filters areabrupt (they do not overlap). One can obtain overlapping transition filters as follows: replace them by any orthogonal basis for the row space ofthe matrix $\left[\begin{array}{cc}{\mathbf{W}}_{\mathbf{1}}& 0\\ 0& {\mathbf{U}}_{\mathbf{2}}\end{array}\right]$ . For example, consider switching between two-channel filter banks with length-4and length-6 Daubechies' filters. In this case, there is one exit filter ( ${\mathbf{W}}_{\mathbf{1}}$ ) and two entry filters ( ${\mathbf{U}}_{\mathbf{2}}$ ).

## Growing a filter bank tree

Consider growing a filter bank tree at $n=0$ by replacing a certain output channel in the tree (point of tree growth) by an $M$ channel filter bank. This is equivalent to switching from a one-channel to an $M$ -channel filter bank at the point of tree growth. The transition filters associated with this change are related to the entry filters of the $M$ -channel filter bank. In fact, every transition filter is the net effect of an entry filterat the point of tree growth seen from the perspective of the input rather than the output point at which the treeis grown. Let the mapping from the input to the output “growth” channel be as shown in [link] . The transition filters are given by the system in [link] , which is driven by the entry filters of the newly added filter bank. Every transition filter is obtained byrunning the corresponding time-reversed entry filter through the synthesis bank of the corresponding branch of the extant tree.

## Pruning a filter bank tree

In the more general case of tree pruning, if the map from the input to the point of pruning is given as in [link] , then the transition filters are given by [link] .

## Wavelet bases for the interval

By taking the effective input/output map of an arbitrary unitary time-varying filter bank tree, one readily obtains time-varying discrete-timewavelet packet bases. Clearly we have such bases for one-sided and finite signals also. These bases are orthonormal because they are built from unitary building blocks.We now describe the construction of continuous-time time-varying wavelet bases. What follows is the most economical (in terms of number of entry/exit functions)continuous-time time-varying wavelet bases.

## Wavelet bases for ${L}^{2}\left(\left[0,\infty \right)\right)$

Recall that an $M$ channel unitary filter bank (with synthesis filters $\left\{{h}_{i}\right\}$ ) such that ${\sum }_{n}{h}_{0}\left(n\right)=\sqrt{M}$ gives rise to an $M$ -band wavelet tight frame for ${L}^{2}\left(\text{ℝ}\right)$ . If

${W}_{i,j}=Span\left\{{\psi }_{i,j,k}\right\}\stackrel{\mathrm{def}}{=}\left\{{M}^{j/2},{\psi }_{i},\left({M}^{j}t-k\right)\right\}\phantom{\rule{1.25em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}k\in \mathbf{Z},$

then ${W}_{0,j}$ forms a multiresolution analysis of ${L}^{2}\left(\text{ℝ}\right)$ with

${W}_{0,j}={W}_{0,j-1}\oplus {W}_{1,j-1}...\oplus {W}_{M-1,j-1}\phantom{\rule{1.25em}{0ex}}\forall j\in \mathbf{Z}.$

In [link] , Daubechies outlines an approach due to Meyer to construct a wavelet basis for ${L}^{2}\left(\left[0,\infty \right)\right)$ . One projects ${W}_{0,j}$ onto ${W}_{0,j}^{half}$ which is the space spanned by the restrictions of ${\psi }_{0,j,k}\left(t\right)$ to $t>0$ . We give a different construction based on the following idea. For $k\in \mathrm{I}\phantom{\rule{-1.99997pt}{0ex}}\mathrm{N}$ , support of ${\psi }_{i,j,k}\left(t\right)$ is in $\left[0,\infty \right)$ . With this restriction (in [link] ) define the spaces ${W}_{i,j}^{+}$ . As $j\to \infty$ (since ${W}_{0,j}\to {L}^{2}\left(\text{ℝ}\right)$ ) ${W}_{0,j}^{+}\to {L}^{2}\left(\left[0,\infty \right)\right)$ . Hence it suffices to have a multiresolution

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