# 0.7 Generalizations of the basic multiresolution wavelet system  (Page 9/28)

 Page 9 / 28

## Two channel biorthogonal filter banks

In previous chapters for orthogonal wavelets, the analysis filters and synthesis filters are time reversal of each other; i.e., $\stackrel{˜}{h}\left(n\right)=h\left(-n\right)$ , $\stackrel{˜}{g}\left(n\right)=g\left(-n\right)$ . Here, for the biorthogonal case, we relax these restrictions. However, in order to perfectlyreconstruct the input, these four filters still have to satisfy a set of relations.

Let ${c}_{1}\left(n\right),n\in \mathbf{Z}$ be the input to the filter banks in [link] , then the outputs of the analysis filter banks are

${c}_{0}\left(k\right)=\sum _{n}\stackrel{˜}{h}\left(2k-n\right){c}_{1}\left(n\right),\phantom{\rule{2.em}{0ex}}{d}_{0}\left(k\right)=\sum _{n}\stackrel{˜}{g}\left(2k-n\right){c}_{1}\left(n\right).$

The output of the synthesis filter bank is

${\stackrel{˜}{c}}_{1}\left(m\right)=\sum _{k}\left[h,\left(2k-m\right),{c}_{0},\left(k\right),+,g,\left(2k-m\right),{d}_{0},\left(k\right)\right].$

${\stackrel{˜}{c}}_{1}\left(m\right)=\sum _{n}\sum _{k}\left[h,\left(2k-m\right),\stackrel{˜}{h},\left(2k-n\right),+,g,\left(2k-m\right),\stackrel{˜}{g},\left(2k-n\right)\right]{c}_{1}\left(n\right).$

For perfect reconstruction, i.e., ${\stackrel{˜}{c}}_{1}\left(m\right)={c}_{1}\left(m\right),\forall m\in \mathbf{Z}$ , we need

$\sum _{k}\left[h,\left(2k-m\right),\stackrel{˜}{h},\left(2k-n\right),+,g,\left(2k-m\right),\stackrel{˜}{g},\left(2k-n\right)\right]=\delta \left(m-n\right).$

Fortunately, this condition can be greatly simplified. In order for it to hold, the four filters have to be related as [link]

$\stackrel{˜}{g}\left(n\right)={\left(-1\right)}^{n}h\left(1-n\right),\phantom{\rule{2.em}{0ex}}g\left(n\right)={\left(-1\right)}^{n}\stackrel{˜}{h}\left(1-n\right),$

up to some constant factors. Thus they are cross-related by time reversal and flipping signs of every other element. Clearly, when $\stackrel{˜}{h}=h$ , we get the familiar relations between the scaling coefficients and the wavelet coefficients for orthogonal wavelets, $g\left(n\right)={\left(-1\right)}^{n}h\left(1-n\right)$ . Substituting [link] back to [link] , we get

$\sum _{n}\stackrel{˜}{h}\left(n\right)h\left(n+2k\right)=\delta \left(k\right).$

In the orthogonal case, we have ${\sum }_{n}h\left(n\right)h\left(n+2k\right)=\delta \left(k\right)$ ; i.e., $h\left(n\right)$ is orthogonal to even translations of itself. Here $\stackrel{˜}{h}$ is orthogonal to $h$ , thus the name biorthogonal .

Equation  [link] is the key to the understanding of the biorthogonal filter banks. Let's assume $\stackrel{˜}{h}\left(n\right)$ is nonzero when ${\stackrel{˜}{N}}_{1}\le n\le {\stackrel{˜}{N}}_{2}$ , and $h\left(n\right)$ is nonzero when ${N}_{1}\le n\le {N}_{2}$ . Equation  [link] implies that [link]

${N}_{2}-{\stackrel{˜}{N}}_{1}=2k+1,\phantom{\rule{1.em}{0ex}}{\stackrel{˜}{N}}_{2}-{N}_{1}=2\stackrel{˜}{k}+1,\phantom{\rule{2.em}{0ex}}k,\stackrel{˜}{k}\in \mathbf{Z}.$

In the orthogonal case, this reduces to the well-known fact that the length of $h$ has to be even.  [link] also imply that the difference between the lengths of $\stackrel{˜}{h}$ and $h$ must be even. Thus their lengths must be both even or both odd.

## Biorthogonal wavelets

We now look at the scaling function and wavelet to see how removing orthogonality and introducing a dual basis changes their characteristics.We start again with the basic multiresolution definition of the scaling function and add to that a similar definition of a dual scaling function.

$\Phi \left(t\right)=\sum _{n}h\left(n\right)\sqrt{2}\Phi \left(2t-n\right),$
$\stackrel{˜}{\Phi }\left(t\right)=\sum _{n}\stackrel{˜}{h}\left(n\right)\sqrt{2}\stackrel{˜}{\Phi }\left(2t-n\right).$

From Theorem  [link] in  Chapter: The Scaling Function and Scaling Coefficients, Wavelet and Wavelet Coefficients , we know that for $\phi$ and $\stackrel{˜}{\phi }$ to exist,

$\sum _{n}h\left(n\right)\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}\sum _{n}\stackrel{˜}{h}\left(n\right)\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}\sqrt{2}.$

Continuing to parallel the construction of the orthogonal wavelets, we also define the wavelet and the dual wavelet as

$\psi \left(t\right)=\sum _{n}g\left(n\right)\sqrt{2}\Phi \left(2t-n\right)=\sum _{n}{\left(-1\right)}^{n}\stackrel{˜}{h}\left(1-n\right)\sqrt{2}\Phi \left(2t-n\right),$
$\stackrel{˜}{\psi }\left(t\right)=\sum _{n}\stackrel{˜}{g}\left(n\right)\sqrt{2}\stackrel{˜}{\Phi }\left(2t-n\right)=\sum _{n}{\left(-1\right)}^{n}h\left(1-n\right)\sqrt{2}\stackrel{˜}{\Phi }\left(2t-n\right).$

Now that we have the scaling and wavelet functions and their duals, the question becomes whether we can expand and reconstruct arbitrary functionsusing them. The following theorem [link] answers this important question.

Theorem 37 For $\stackrel{˜}{h}$ and $h$ satisfying [link] , suppose that for some C, $ϵ>0$ ,

$|\text{Φ}\left(\text{ω}\right)|\le C{\left(1+\text{ω}\right)}^{-1/2-ϵ},\phantom{\rule{2.em}{0ex}}|\stackrel{˜}{\text{Φ}}\left(\text{ω}\right)|\le C{\left(1+\text{ω}\right)}^{-1/2-ϵ}.$

If $\Phi$ and $\stackrel{˜}{\Phi }$ defined above have sufficient decay in the frequency domain, then ${\psi }_{j,k}\stackrel{\mathrm{def}}{=}{2}^{j/2}\psi \left({2}^{j}x-k\right),\phantom{\rule{0.277778em}{0ex}}j,k\in \mathbf{Z}$ constitute a frame in ${L}^{2}\left(\mathbf{R}\right)$ . Their dual frame is given by ${\stackrel{˜}{\psi }}_{j,k}\stackrel{\mathrm{def}}{=}{2}^{j/2}\stackrel{˜}{\psi }\left({2}^{j}x-k\right),\phantom{\rule{0.277778em}{0ex}}j,k\in \mathbf{Z}$ ; for any $f\in {L}^{2}\left(\mathbf{R}\right)$ ,

What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Got questions? Join the online conversation and get instant answers!