# 0.9 Calculation of the discrete wavelet transform  (Page 2/5)

 Page 2 / 5

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

$f\left(t\right)\approx \sum _{k=-\infty }^{\infty }⟨f,{\phi }_{{J}_{1},k}⟩{\phi }_{{J}_{1},k}\left(t\right)$

or, in terms of wavelets, [link] becomes

$f\left(t\right)\approx \sum _{k=-\infty }^{\infty }⟨f,{\phi }_{{J}_{0},k}⟩{\phi }_{{J}_{0},k}\left(t\right)+\sum _{k=-\infty }^{\infty }\sum _{j={J}_{0}}^{{J}_{1}-1}⟨f,{\psi }_{j,k}⟩{\psi }_{j,k}\left(t\right).$

This assumes that approximately $f\in {\mathcal{V}}_{{J}_{1}}$ or equivalently, $\parallel f-{P}_{{J}_{1}}f\parallel \approx 0$ , where ${P}_{{J}_{1}}$ denotes the orthogonal projection of $f$ onto ${\mathcal{V}}_{{J}_{1}}$ .

Given $f\left(t\right)\in {\mathcal{V}}_{{J}_{1}}$ so that the expansion in  [link] is exact, one computes the DWT coefficients in two steps.

1. Projection onto finest scale: Compute $⟨f,{\phi }_{{J}_{1},k}⟩$
2. Analysis: Compute $⟨f,{\psi }_{j,k}⟩$ , $j\in \left\{{J}_{0},...,{J}_{1}-1\right\}$ and $⟨f,{\phi }_{{J}_{0},k}⟩$ .

For ${J}_{1}$ large enough, ${\phi }_{{J}_{1},k}\left(t\right)$ can be approximated by a Dirac impulse at its center of mass since $\int \phi \left(t\right)\phantom{\rule{0.166667em}{0ex}}dt=1$ . For large $j$ this gives

${2}^{j}\int f\left(t\right)\phantom{\rule{0.166667em}{0ex}}\phi \left({2}^{j}t\right)\phantom{\rule{0.166667em}{0ex}}dt\approx \int f\left(t\right)\phantom{\rule{0.166667em}{0ex}}\delta \left(t-{2}^{-j}{m}_{0}\right)\phantom{\rule{0.166667em}{0ex}}dt=f\left(t-{2}^{-j}{m}_{0}\right)$

where ${m}_{0}=\int t\phantom{\rule{0.166667em}{0ex}}\phi \left(t\right)\phantom{\rule{0.166667em}{0ex}}dt$ is the first moment of $\phi \left(t\right)$ . Therefore the scaling function coefficients at the $j={J}_{1}$ scale are

${c}_{{J}_{1}}\left(k\right)=⟨f,{\phi }_{{J}_{1},k}⟩={2}^{{J}_{1}/2}\int f\left(t\right)\phantom{\rule{0.166667em}{0ex}}\phi \left({2}^{{J}_{1}}t-k\right)\phantom{\rule{0.166667em}{0ex}}dt={2}^{{J}_{1}/2}\int f\left(t+{2}^{-{J}_{1}}k\right)\phantom{\rule{0.166667em}{0ex}}\phi \left({2}^{{J}_{1}}t\right)\phantom{\rule{0.166667em}{0ex}}dt$

which are approximately

${c}_{{J}_{1}}\left(k\right)=⟨f,{\phi }_{{J}_{1},k}⟩\approx {2}^{-{J}_{1}/2}f\left({2}^{-{J}_{1}}\left({m}_{0}+k\right)\right).$

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 $f\left(t\right)$ is a quadratic polynomial, then

${c}_{{J}_{1}}\left(k\right)=⟨f,{\phi }_{{J}_{1},k}⟩={2}^{-{J}_{1}/2}f\left({2}^{-{J}_{1}}\left({m}_{0}+k\right)\right)\approx {2}^{-{J}_{1}/2}f\left({2}^{-{J}_{1}}k\right).$

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 $f\left(t\right)$ 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 $⟨f,{\phi }_{{J}_{1},k}⟩$ . 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 ${L}^{2}\left[0,P\right]$ .

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 $f\left(t\right)$ or the signal can be made periodic as is done for the DFT.

how can chip be made from sand
are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
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
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
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?
what is a peer
What is meant by 'nano scale'?
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 ?
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!