# 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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is this allso about nanoscale material
Almas
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yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
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William
currently
William
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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da
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Bhagvanji
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Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
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what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
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Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
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Alexandre
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Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
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Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
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Mahi
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Rafiq
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Anam
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Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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