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Although when using the wavelet expansion as a tool in an abstract mathematical analysis, the infinite sum and the continuous description of t R are appropriate, as a practical signal processing or numerical analysis tool, the function or signal f ( t ) in [link] is available only in terms of its samples, perhaps with additionalinformation such as its being band-limited. In this chapter, we examine the practical problem of numerically calculating the discrete wavelettransform.

Finite wavelet expansions and transforms

The wavelet expansion of a signal f ( t ) as first formulated in [link] is repeated here by

f ( t ) = k = - j = - f , ψ j , k ψ j , k ( t )

where the { ψ j , k ( t ) } form a basis or tight frame for the signal space of interest (e.g., L 2 ). At first glance, this infinite series expansion seems to have the same practical problems in calculation that aninfinite Fourier series or the Shannon sampling formula has. In a practical situation, this wavelet expansion, where the coefficients arecalled the discrete wavelet transform (DWT), is often more easily calculated. Both the time summation over the index k and the scale summation over the index j can be made finite with little or no error.

The Shannon sampling expansion [link] , [link] of a signal with infinite support in terms of sinc ( t ) = sin ( t ) t expansion functions

f ( t ) = n = - f ( T n ) sinc ( π T t - π n )

requires an infinite sum to evaluate f ( t ) at one point because the sinc basis functions have infinite support. This is not necessarily true for awavelet expansion where it is possible for the wavelet basis functions tohave finite support and, therefore, only require a finite summation over k in [link] to evaluate f ( t ) at any point.

The lower limit on scale j in [link] can be made finite by adding the scaling function to the basis set as was done in [link] . By using the scaling function, the expansion in [link] becomes

f ( t ) = k = - f , φ J 0 , k φ J 0 , k ( t ) + k = - j = J 0 f , ψ j , k ψ j , k ( t ) .

where j = J 0 is the coarsest scale that is separately represented. The level of resolution or coarseness to start the expansion with is arbitrary,as was shown in  Chapter: A multiresolution formulation of Wavelet Systems in [link] , [link] , and [link] . The space spanned by the scaling function contains all the spaces spanned by the lower resolution wavelets from j = - up to the arbitrary starting point j = J 0 . This means V J 0 = W - W J 0 - 1 . In a practical case, this would be the scale where separating detail becomes important. For asignal with finite support (or one with very concentrated energy), the scaling function might be chosen so that the support of the scalingfunction and the size of the features of interest in the signal being analyzed were approximately the same.

This choice is similar to the choice of period for the basis sinusoids in a Fourier series expansion. If the period of the basis functions ischosen much larger than the signal, much of the transform is used to describe the zero extensions of the signal or the edge effects.

The choice of a finite upper limit for the scale j in [link] is more complicated and usually involves some approximation. Indeed, forsamples of f ( t ) to be an accurate description of the signal, the signal should be essentially bandlimited and the samples taken at least at theNyquist rate (two times the highest frequency in the signal's Fourier transform).

Questions & Answers

what are the products of Nano chemistry?
Maira Reply
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
learn
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
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
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?
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
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
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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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