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Discrete multiresolution analysis, the discrete-time wavelet transform, and the continuous wavelet transform

Up to this point, we have developed wavelet methods using the series wavelet expansion of continuous-time signals called the discrete wavelettransform (DWT), even though it probably should be called the continuous-time wavelet series. This wavelet expansion is analogous tothe

Local Basis
Local Basis

Fourier series in that both are series expansions that transform continuous-time signals into a discrete sequence of coefficients. However, unlike the Fourier series, the DWT can be made periodic ornonperiodic and, therefore, is more versatile and practically useful.

In this chapter we will develop a wavelet method for expanding discrete-time signals in a series expansion since, in most practicalsituations, the signals are already in the form of discrete samples. Indeed, we have already discussed when it is possible to use samples ofthe signal as scaling function expansion coefficients in order to use the filter bank implementation of Mallat's algorithm. We find there is anintimate connection between the DWT and DTWT, much as there is between the Fourier series and the DFT. One expands signals with the FS but oftenimplements that with the DFT.

To further generalize the DWT, we will also briefly present the continuous wavelet transform which, similar to the Fourier transform, transforms afunction of continuous time to a representation with continuous scale and translation. In order to develop the characteristics of these variouswavelet representations, we will often call on analogies with corresponding Fourier representations. However, it is important tounderstand the differences between Fourier and wavelet methods. Much of that difference is connected to the wavelet being concentrated in bothtime and scale or frequency, to the periodic nature of the Fourier basis, and to the choice of wavelet bases.

Discrete multiresolution analysis and the discrete-time wavelet transform

Parallel to the developments in early chapters on multiresolution analysis, we can define a discrete multiresolution analysis (DMRA) for l 2 , where the basis functions are discrete sequences [link] , [link] , [link] . The expansion of a discrete-time signal in terms of discrete-time basis function is expressed in a form parallel to [link] as

f ( n ) = j , k d j ( k ) ψ ( 2 j n - k )

where ψ ( m ) is the basic expansion function of an integer variable m . If these expansion functions are an orthogonal basis (or form a tight frame), the expansion coefficients (discrete-time wavelet transform) arefound from an inner product by

d j ( k ) = f ( n ) , ψ ( 2 j n - k ) = n f ( n ) ψ ( 2 j n - k )

If the expansion functions are not orthogonal or even independent but do span 2 , a biorthogonal system or a frame can be formed such that a transform and inverse can be defined.

Because there is no underlying continuous-time scaling function or wavelet, many of the questions, properties, and characteristics of the analysisusing the DWT in  Chapter: Introduction to Wavelets , Chapter: A multiresolution formulation of Wavelet Systems , Chapter: Regularity, Moments, and Wavelet System Design , etc. do not arise. In fact, because of the filter bank structure for calculating the DTWT,the design is often done using multirate frequency domain techniques, e.g., the work by Smith and Barnwell and associates [link] . The questions of zero wavelet moments posed by Daubechies,which are related to ideas of convergence for iterations of filter banks, and Coifman's zero scalingfunction moments that were shown to help approximate inner products by samples, seem to have no DTWT interpretation.

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.
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Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
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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
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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
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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
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how did you get the value of 2000N.What calculations are needed to arrive at it
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