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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 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
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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
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
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
Devang Reply
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
Abhijith Reply
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?
s. Reply
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 ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
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Cied
how did you get the value of 2000N.What calculations are needed to arrive at it
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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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