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The question of how one can calculate the Fourier series coefficients of a continuous signal from the discrete Fourier transform of samples of thesignal is similar to asking how one calculates the discrete wavelet transform from samples of the signal. And the answer is similar. Thesamples must be “dense" enough. For the Fourier series, if a frequency can be found above which there is very little energy in the signal (abovewhich the Fourier coefficients are very small), that determines the Nyquist frequency and the necessary sampling rate. For the waveletexpansion, a scale must be found above which there is negligible detail or energy. If this scale is j = J 1 , the signal can be written

f ( t ) k = - f , φ J 1 , k φ J 1 , k ( t )

or, in terms of wavelets, [link] becomes

f ( t ) k = - f , φ J 0 , k φ J 0 , k ( t ) + k = - j = J 0 J 1 - 1 f , ψ j , k ψ j , k ( t ) .

This assumes that approximately f V J 1 or equivalently, f - P J 1 f 0 , where P J 1 denotes the orthogonal projection of f onto V J 1 .

Given f ( t ) V J 1 so that the expansion in  [link] is exact, one computes the DWT coefficients in two steps.

  1. Projection onto finest scale: Compute f , φ J 1 , k
  2. Analysis: Compute f , ψ j , k , j { J 0 , ... , J 1 - 1 } and f , φ J 0 , k .

For J 1 large enough, φ J 1 , k ( t ) can be approximated by a Dirac impulse at its center of mass since φ ( t ) d t = 1 . For large j this gives

2 j f ( t ) φ ( 2 j t ) d t f ( t ) δ ( t - 2 - j m 0 ) d t = f ( t - 2 - j m 0 )

where m 0 = t φ ( t ) d t is the first moment of φ ( t ) . Therefore the scaling function coefficients at the j = J 1 scale are

c J 1 ( k ) = f , φ J 1 , k = 2 J 1 / 2 f ( t ) φ ( 2 J 1 t - k ) d t = 2 J 1 / 2 f ( t + 2 - J 1 k ) φ ( 2 J 1 t ) d t

which are approximately

c J 1 ( k ) = f , φ J 1 , k 2 - J 1 / 2 f ( 2 - J 1 ( m 0 + k ) ) .

For all 2-regular wavelets (i.e., wavelets with two vanishing moments, regular wavelets other than the Haar wavelets—even in the M -band case where one replaces 2 by M in the above equations, m 0 = 0 ), one can show that the samples of the functions themselves form a third-orderapproximation to the scaling function coefficients of the signal [link] . That is, if f ( t ) is a quadratic polynomial, then

c J 1 ( k ) = f , φ J 1 , k = 2 - J 1 / 2 f ( 2 - J 1 ( m 0 + k ) ) 2 - J 1 / 2 f ( 2 - J 1 k ) .

Thus, in practice, the finest scale J 1 is determined by the sampling rate. By rescaling the function and amplifying it appropriately, onecan assume the samples of f ( t ) are equal to the scaling function coefficients. These approximations are made better by setting some ofthe scaling function moments to zero as in the coiflets. These are discussed in Section: Approximation of Scaling Coefficients by Samples of the Signal .

Finally there is one other aspect to consider. If the signal has finite support and L samples are given, then we have L nonzero coefficients f , φ J 1 , k . However, the DWT will typically have more than L coefficients since the wavelet and scaling functions are obtained by convolution and downsampling. In other words,the DWT of a L -point signal will have more than L points. Considered as a finite discrete transform of one vector into another, this situationis undesirable. The reason this “expansion" in dimension occurs is that one is using a basis for L 2 to represent a signal that is of finite duration, say in L 2 [ 0 , P ] .

When calculating the DWT of a long signal, J 0 is usually chosen to give the wavelet description of the slowly changing or longer duration featuresof the signal. When the signal has finite support or is periodic, J 0 is generally chosen so there is a single scaling coefficient for the entire signal or for one period of the signal. To reconcile thedifference in length of the samples of a finite support signal and the number of DWT coefficients, zeros can be appended to the samples of f ( t ) or the signal can be made periodic as is done for the DFT.

Questions & Answers

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
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
hi
Loga
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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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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