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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

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
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
characteristics of micro business
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
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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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