0.9 Calculation of the discrete wavelet transform  (Page 3/5)

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Periodic or cyclic discrete wavelet transform

If $f\left(t\right)$ has finite support, create a periodic version of it by

$\stackrel{˜}{f}\left(t\right)=\sum _{n}f\left(t+Pn\right)$

where the period $P$ is an integer. In this case, $⟨f,{\phi }_{j,k}⟩$ and $⟨f,{\psi }_{j,k}⟩$ are periodic sequences in $k$ with period $P\phantom{\rule{0.166667em}{0ex}}{2}^{j}$ (if $j\ge 0$ and 1 if $j<0$ ) and

$d\left(j,k\right)={2}^{j/2}\int \stackrel{˜}{f}\left(t\right)\psi \left({2}^{j}t-k\right)\phantom{\rule{0.166667em}{0ex}}dt$
$d\left(j,k\right)={2}^{j/2}\int \stackrel{˜}{f}\left(t+{2}^{-j}k\right)\psi \left({2}^{j}t\right)\phantom{\rule{0.166667em}{0ex}}dt={2}^{j/2}\int \stackrel{˜}{f}\left(t+{2}^{-j}\ell \right)\psi \left({2}^{j}t\right)\phantom{\rule{0.166667em}{0ex}}dt$

where $\ell =}_{P\phantom{\rule{0.166667em}{0ex}}{2}^{j}}$ ( $k$ modulo $P\phantom{\rule{0.166667em}{0ex}}{2}^{j}$ ) and $l\in \left\{0,1,...,P\phantom{\rule{0.166667em}{0ex}}{2}^{j}-1\right\}$ . An obvious choice for ${J}_{0}$ is 1. Notice that in this case given $L={2}^{{J}_{1}}$ samples of the signal, $⟨f,{\phi }_{{J}_{1},k}⟩$ , the wavelet transform has exactly $1+1+2+{2}^{2}+\cdots +{2}^{{J}_{1}-1}={2}^{{J}_{1}}=L$ terms. Indeed, this gives a linear, invertible discrete transform which can be considered apart fromany underlying continuous process similar the discrete Fourier transform existing apart from the Fourier transform or series.

There are at least three ways to calculate this cyclic DWT and they are based on the equations [link] , [link] , and [link] later in this chapter. The first method simply convolves the scalingcoefficients at one scale with the time-reversed coefficients $h\left(-n\right)$ to give an $L+N-1$ length sequence. This is aliased or wrapped as indicated in [link] and programmed in dwt5.m in Appendix 3. The second method creates a periodic ${\stackrel{˜}{c}}_{j}\left(k\right)$ by concatenating an appropriate number of ${c}_{j}\left(k\right)$ sections together then convolving $h\left(n\right)$ with it. That is illustrated in [link] and in dwt.m in Appendix 3. The third approach constructs a periodic $\stackrel{˜}{h}\left(n\right)$ and convolves it with ${c}_{j}\left(k\right)$ to implement [link] . The Matlab programs should be studied to understand how these ideas are actually implemented.

Because the DWT is not shift-invariant, different implementations of the DWT may appear to give different results because of shifts of the signaland/or basis functions. It is interesting to take a test signal and compare the DWT of it with different circular shifts of the signal.

Making $f\left(t\right)$ periodic can introduce discontinuities at 0 and $P$ . To avoid this, there are several alternative constructions of orthonormalbases for ${L}^{2}\left[0,P\right]$ [link] , [link] , [link] , [link] . All of these constructions use (directly or indirectly) the concept of time-varyingfilter banks. The basic idea in all these constructions is to retain basis functions with support in $\left[0,P\right]$ , remove ones with support outside $\left[0,P\right]$ and replace the basis functions that overlap across the endpoints with special entry/exit functions that ensure completeness . These boundary functions are chosen so that the constructed basis isorthonormal. This is discussed in Section: Time-Varying Filter Bank Trees . Another way to deal with edges or boundaries uses “lifting" as mentioned in Section: Lattices and Lifting .

Filter bank structures for calculation of the dwt and complexity

Given that the wavelet analysis of a signal has been posed in terms of the finite expansion of [link] , the discrete wavelet transform (expansion coefficients) can be calculated using Mallat's algorithm implemented by afilter bank as described in Chapter: Filter Banks and the Discrete Wavelet Transform and expanded upon in   Chapter: Filter Banks and Transmultiplexers . Using the direct calculations described by the one-sided tree structure of filters and down-samplers in Figure: Three-Stage Two-Band Analysis Tree allows a simple determination of the computational complexity.

where we get a research paper on Nano chemistry....?
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
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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
what is variations in raman spectra for nanomaterials
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
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Alexandre
what is the stm
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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?
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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
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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