# 0.6 Regularity, moments, and wavelet system design

 Page 1 / 13

We now look at a particular way to use the remaining $\frac{N}{2}-1$ degrees of freedom to design the $N$ values of $h\left(n\right)$ after satisfying [link] and [link] , which insure the existence and orthogonality (or property of being a tight frame) of the scaling function and wavelets [link] , [link] , [link] .

One of the interesting characteristics of the scaling functions and wavelets is that while satisfying [link] and [link] will guarantee the existence of an integrable scaling function, it may be extraordinarilyirregular, even fractal in nature. This may be an advantage in analyzing rough or fractal signals but it is likely to be a disadvantage for mostsignals and images.

We will see in this section that the number of vanishing moments of ${h}_{1}\left(n\right)$ and $\psi \left(t\right)$ are related to the smoothness or differentiability of $\phi \left(t\right)$ and $\psi \left(t\right)$ . Unfortunately, smoothness is difficult to determine directly because, unlike with differential equations, thedefining recursion [link] does not involve derivatives.

We also see that the representation and approximation of polynomials are related to the number of vanishing or minimized wavelet moments. Sincepolynomials are often a good model for certain signals and images, this property is both interesting and important.

The number of zero scaling function moments is related to the “goodness" of the approximation of high-resolution scaling coefficients by samples of thesignal. They also affect the symmetry and concentration of the scaling function and wavelets.

This section will consider the basic 2-band or multiplier-2 case defined in [link] . The more general M-band or multiplier-M case is discussed in Section: Multiplicity-M (M-Band) Scaling Functions and Wavelets .

## K-regular scaling filters

Here we start by defining a unitary scaling filter to be an FIR filter with coefficients $h\left(n\right)$ from the basic recursive [link] satisfying the admissibility conditions from [link] and orthogonality conditions from [link] as

$\sum _{n}h\left(n\right)=\sqrt{2}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\text{and}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\sum _{k}h\left(k\right)\phantom{\rule{0.166667em}{0ex}}h\left(k+2m\right)=\delta \left(m\right).$

The term “scaling filter" comes from Mallat's algorithm, and the relation to filter banks discussed in Chapter: Filter Banks and the Discrete Wavelet Transform . The term “unitary" comesfrom the orthogonality conditions expressed in filter bank language, which is explained in Chapter: Filter Banks and Transmultiplexers .

A unitary scaling filter is said to be $K$ - regular if its z-transform has $K$ zeros at $z={e}^{i\pi }$ . This looks like

$H\left(z\right)={\left(\frac{1+{z}^{-1}}{2}\right)}^{K}Q\left(z\right)$

where $H\left(z\right)={\sum }_{n}h\left(n\right)\phantom{\rule{0.166667em}{0ex}}{z}^{-n}$ is the z-transform of the scaling coefficients $h\left(n\right)$ and $Q\left(z\right)$ has no poles or zeros at $z={e}^{i\pi }$ . Note that we are presenting a definition of regularity of $h\left(n\right)$ , not of the scaling function $\phi \left(t\right)$ or wavelet $\psi \left(t\right)$ . They are related but not the same. Note also from [link] that any unitary scaling filter is at least $K=1$ regular.

The length of the scaling filter is $N$ which means $H\left(z\right)$ is an $N-1$ degree polynomial. Since the multiple zero at $z=-1$ is order $K$ , the polynomial $Q\left(z\right)$ is degree $N-1-K$ . The existence of $\phi \left(t\right)$ requires the zero th moment be $\sqrt{2}$ which is the result of the linear condition in [link] . Satisfying the conditions for orthogonality requires $N/2$ conditions which are the quadratic equations in [link] . This means the degree of regularity is limited by

Preparation and Applications of Nanomaterial for Drug Delivery
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
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
yes that's correct
Professor
I think
Professor
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
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
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
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
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
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
hi
Loga
Got questions? Join the online conversation and get instant answers! By OpenStax By Richley Crapo By Yasser Ibrahim By Jessica Collett By OpenStax By OpenStax By Mary Cohen By Stephen Voron By Anonymous User By OpenStax