0.5 The scaling function and scaling coefficients, wavelet and  (Page 12/13)

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Because of the iterative form of this algorithm, applying the same process over and over, it is sometimes called the cascade algorithm [link] , [link] .

Iterating the filter bank

An interesting method for calculating the scaling function also uses an iterative procedure which consists of the stages of the filterstructure of Chapter: Filter Banks and the Discrete Wavelet Transform which calculates wavelet expansions coefficients (DWT values) at one scale from those at another. A scalingfunction, wavelet expansion of a scaling function itself would be a single nonzero coefficient at the scale of $j=1$ . Passing this single coefficient through the synthesis filter structure of Figure: Two-Stage Two-Band Synthesis Tree and [link] would result in a fine scale output that for large $j$ would essentially be samples of the scaling function.

Successive approximation in the frequency domain

The Fourier transform of the scaling function defined in [link] is an important tool for studying and developing wavelet theory. It could beapproximately calculated by taking the DFT of the samples of $\phi \left(t\right)$ but a more direct approach is available using the infinite product in [link] . From this formulation we can see how the zeros of $H\left(\omega \right)$ determine the zeros of $\text{Φ}\left(\omega \right)$ . The existence conditions in Theorem 5 require $H\left(\pi \right)=0$ or, more generally, $H\left(\omega \right)=0$ for $\omega =\left(2k+1\right)\pi$ . Equation [link] gives the relation of these zeros of $H\left(\omega \right)$ to the zeros of $\text{Φ}\left(\omega \right)$ . For the index $k=1$ , $H\left(\omega /2\right)=0$ at $\omega =2\left(2k+1\right)\pi$ . For $k=2$ , $H\left(\omega /4\right)=0$ at $\omega =4\left(2k+1\right)\pi$ , $H\left(\omega /8\right)=0$

at $\omega =8\left(2k+1\right)\pi$ , etc. Because [link] is a product of stretched versions of $H\left(\omega \right)$ , these zeros of $H\left(\omega /{2}^{j}\right)$ are the zeros of the Fourier transform of $\phi \left(t\right)$ . Recall from  Theorem 15 that $H\left(\omega \right)$ has no zeros in $-\pi /3<\omega <\pi /3$ . All of this gives a picture of the shape of $\text{Φ}\left(\omega \right)$ and the location of its zeros. From an asymptotic analysis of $\text{Φ}\left(\omega \right)$ as $\omega \to \infty$ , one can study the smoothness of  $\phi \left(t\right)$ .

A Matlab program that calculates $\text{Φ}\left(\omega \right)$ using this frequency domain successive approximations approach suggested by [link] is given in Appendix C . Studying this program gives further insight into the structure of $\text{Φ}\left(\omega \right)$ . Rather than starting the calculations given in [link] for the index $j=1$ , they are started for the largest $j=J$ and worked backwards. If we calculate a length-N DFT consistent with $j=J$ using the FFT, then the samples of $H\left(\omega /{2}^{j}\right)$ for $j=J-1$ are simply every other sample of the case for $j=J$ . The next stage for $j=J-2$ is done likewise and if the original $N$ is chosen a power of two, the process in continued down to $j=1$ without calculating any more FFTs. This results in a very efficient algorithm. The details are in theprogram itself.

This algorithm is so efficient, using it plus an inverse FFT might be a good way to calculate $\phi \left(t\right)$ itself. Examples of the algorithm are illustrated in [link] where the transform is plotted for each step of the iteration.

The dyadic expansion of the scaling function

The next method for evaluating the scaling function uses a completely different approach. It starts by calculating the values of the scalingfunction at integer values of $t$ , which can be done exactly (within our ability to solve simultaneous linear equations). Consider the basicrecursion equation [link] for integer values of $t=k$

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
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
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
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
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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