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We now look at a particular way to use the remaining N 2 - 1 degrees of freedom to design the N values of h ( n ) 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 ( n ) and ψ ( t ) are related to the smoothness or differentiability of φ ( t ) and ψ ( t ) . 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 ( n ) from the basic recursive [link] satisfying the admissibility conditions from [link] and orthogonality conditions from [link] as

n h ( n ) = 2 and k h ( k ) h ( k + 2 m ) = δ ( m ) .

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 π . This looks like

H ( z ) = 1 + z - 1 2 K Q ( z )

where H ( z ) = n h ( n ) z - n is the z-transform of the scaling coefficients h ( n ) and Q ( z ) has no poles or zeros at z = e i π . Note that we are presenting a definition of regularity of h ( n ) , not of the scaling function φ ( t ) or wavelet ψ ( t ) . 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 ( z ) is an N - 1 degree polynomial. Since the multiple zero at z = - 1 is order K , the polynomial Q ( z ) is degree N - 1 - K . The existence of φ ( t ) requires the zero th moment be 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

Questions & Answers

what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
Brian Reply
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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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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
what is Nano technology ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
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?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
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?
Damian Reply
absolutely yes
Daniel
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Source:  OpenStax, Wavelets and wavelet transforms. OpenStax CNX. Aug 06, 2015 Download for free at https://legacy.cnx.org/content/col11454/1.6
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