# 0.7 Generalizations of the basic multiresolution wavelet system  (Page 6/28)

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We next define the ${k}^{th}$ moments of ${\psi }_{\ell }\left(t\right)$ as

${m}_{\ell }\left(k\right)=\int {t}^{k}\phantom{\rule{0.166667em}{0ex}}{\psi }_{\ell }\left(t\right)\phantom{\rule{0.166667em}{0ex}}dt$

and the ${k}^{th}$ discrete moments of ${h}_{\ell }\left(n\right)$ as

${\mu }_{\ell }\left(k\right)=\sum _{n}{n}^{k}\phantom{\rule{0.166667em}{0ex}}{h}_{\ell }\left(n\right).$

Theorem 36 (Equivalent Characterizations of K-Regular M-Band Filters) A unitary scaling filter is K-regular if and only if the following equivalentstatements are true:

1. All moments of the wavelet filters are zero, ${\mu }_{\ell }\left(k\right)=0$ , for $k=0,1,\cdots ,\left(K-1\right)$ and for $\ell =1,2,\cdots ,\left(M-1\right)$
2. All moments of the wavelets are zero, ${m}_{\ell }\left(k\right)=0$ , for $k=0,1,\cdots ,\left(K-1\right)$ and for $\ell =1,2,\cdots ,\left(M-1\right)$
3. The partial moments of the scaling filter are equal for $k=0,1,\cdots ,\left(K-1\right)$
4. The frequency response of the scaling filter has zeros of order $K$ at the ${M}^{th}$ roots of unity, $\text{ω}=2\pi \phantom{\rule{0.166667em}{0ex}}\ell /M$ for $\ell =1,2,\cdots ,M-1$ .
5. The magnitude-squared frequency response of the scaling filter is flat to order 2K at $\text{ω}=0$ . This follows from [link] .
6. All polynomial sequences up to degree $\left(K-1\right)$ can be expressed as a linear combination of integer-shifted scaling filters.
7. All polynomials of degree up to $\left(K-1\right)$ can be expressed as a linear combination of integer-shifted scaling functions for all $j$ .

This powerful result [link] , [link] is similar to the $M=2$ case presented in Chapter: Regularity, Moments, and Wavelet System Design . It not only ties the number of zero moments to the regularity but also to the degree of polynomials that canbe exactly represented by a sum of weighted and shifted scaling functions. Note the location of the zeros of $H\left(z\right)$ are equally spaced around the unit circle, resulting in a narrower frequency response than for thehalf-band filters if $M=2$ . This is consistent with the requirements given in [link] and illustrated in [link] .

Sketches of some of the derivations in this section are given in the Appendix or are simple extensions of the $M=2$ case. More details are given in [link] , [link] , [link] .

## M-band scaling function design

Calculating values of $\phi \left(n\right)$ can be done by the same methods given in Section: Calculating the Basic Scaling Function and Wavelet . However, the design of the scaling coefficients $h\left(n\right)$ parallels that for the two-band case but is somewhat more difficult [link] .

One special set of cases turns out to be a simple extension of the two-band system. If the multiplier $M={2}^{m}$ , then the scaling function is simply a scaled version of the $M=2$ case and a particular set of corresponding wavelets are those obtained by iterating the waveletbranches of the Mallat algorithm tree as is done for wavelet packets described in [link] . For other values of $M$ , especially odd values, the situation is more complex.

## M-band wavelet design and cosine modulated methods

For $M>2$ the wavelet coefficients ${h}_{\ell }\left(n\right)$ are not uniquely determined by the scaling coefficients, as was the case for $M=2$ . This is both a blessing and a curse. It gives us more flexibility in designing specific systems,but it complicates the design considerably. For small $N$ and $M$ , the designs can be done directly, but for longer lengths and/or for large $M$ , direct design becomes impossible and something like the cosine modulateddesign of the wavelets from the scaling function as described in Chapter: Filter Banks and Transmultiplexers , is probably the bestapproach [link] , [link] , [link] , [link] , [link] [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] .

## Wavelet packets

The classical $M=2$ wavelet system results in a logarithmic frequency resolution. The low frequencies have narrow bandwidths and the highfrequencies have wide bandwidths, as illustrated in Figure: Frequency Bands for the Analysis Tree . This is called “constant-Q" filtering and is appropriate for someapplications but not all. The wavelet packet system was proposed by Ronald Coifman [link] , [link] to allow a finer and adjustable resolution of frequencies at high frequencies. It also gives a rich structure thatallows adaptation to particular signals or signal classes. The cost of this richer structureis a computational complexity of $O\left(Nlog\left(N\right)\right)$ , similar to the FFT, in contrast to the classical wavelet transform which is $O\left(N\right)$ .

#### Questions & Answers

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
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Alexandre
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Rafiq
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Damian
How we are making nano material?
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What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
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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?
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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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