# 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 $O\left(N\right)$ .

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
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
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