# 0.6 Regularity, moments, and wavelet system design  (Page 9/13)

 Page 9 / 13

An illustration of the effects of these approximations on a signal is shown in [link] where a signal with a very smooth component (a sinusoid) and a discontinuous component (a square wave) is expanded in awavelet series using samples as the high resolution scaling function coefficients. Notice the effects of projecting onto lower and lowerresolution scales.

If we consider a wavelet system where the same number of scaling function and wavelet moments are set zero and this number is as large as possible,then the following is true [link] , [link] :

Theorem 27 If $m\left(\ell \right)={m}_{1}\left(\ell \right)=0$ for $\ell =1,2,\cdots ,L$ and ${m}_{1}\left(0\right)=0$ , then the ${L}^{2}$ error is

${ϵ}_{3}=\parallel f\left(t\right)-{S}^{j}\left\{f\left(t\right)\right\}{\parallel }_{2}\phantom{\rule{0.166667em}{0ex}}\le \phantom{\rule{0.166667em}{0ex}}{C}_{3}\phantom{\rule{0.166667em}{0ex}}{2}^{-j\left(L+1\right)},$

where ${C}_{3}$ is a constant independent of $j$ and $L$ , but dependent on $f\left(t\right)$ and the wavelet system.

Here we see that for this wavelet system called a Coifman wavelet system, that using samples as the inner product expansion coefficients is anexcellent approximation. This justifies that using samples of a signal as input to a filter bank gives a proper wavelet analysis. This approximationis also illustrated in [link] and in [link] .

From the previous approximation theorems, we see that a combination of zero wavelet and zero scaling function moments used with samples of thesignal may give superior results to wavelets with only zero wavelet moments. Not only does forcing zero scaling function moments give abetter approximation of the expansion coefficients by samples, it often causesthe scaling function to be more symmetric. Indeed, that characteristic may be more important than the sample approximation in certainapplications.

Daubechies considered the design of these wavelets which were suggested by Coifman [link] , [link] , [link] . Gopinath [link] , [link] and Wells [link] , [link] show how zero scaling function moments give a better approximation of high-resolution scaling coefficients by samples. Tianand Wells [link] , [link] have also designed biorthogonal systems with mixed zero moments with very interesting properties. Approximations to f ( t ) at a Different Finite Scales

The Coifman wavelet system (Daubechies named the basis functions “coiflets") is an orthonormal multiresolution wavelet system with

$\int {t}^{k}\phi \left(t\right)\phantom{\rule{0.166667em}{0ex}}dt=m\left(k\right)=0,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\text{for}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}k=1,2,\cdots ,L-1$
$\int {t}^{k}\psi \left(t\right)\phantom{\rule{0.166667em}{0ex}}dt={m}_{1}\left(k\right)=0,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\text{for}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}k=1,2,\cdots ,L-1.$

This definition imposes the requirement that there be at least $L-1$ zero scaling function moments and at least $L-1$ wavelet moments in addition to the one zero moment of ${m}_{1}\left(0\right)$ required by orthogonality. This system is said to be of order or degree $L$ and sometime has the additional requirement that the length of the scaling function filter $h\left(n\right)$ , which is denoted $N$ , is minimum [link] , [link] . In the design of these coiflets, one obtains more total zero moments than $N/2-1$ . This was first noted by Beylkin, et al [link] . The length-4 wavelet systemhas only one degree of freedom, so it cannot have both a scaling function moment and wavelet moment of zero (see [link] ). Tian [link] , [link] has derived formulas for four length-6 coiflets. These are:

$h=\left[\frac{-3+\sqrt{7}}{16\sqrt{2}},,,\frac{1-\sqrt{7}}{16\sqrt{2}},,,\frac{7-\sqrt{7}}{8\sqrt{2}},,,\frac{7+\sqrt{7}}{8\sqrt{2}},,,\frac{5+\sqrt{7}}{16\sqrt{2}},,,\frac{1-\sqrt{7}}{16\sqrt{2}}\right],$

or

$h=\left[\frac{-3-\sqrt{7}}{16\sqrt{2}},,,\frac{1+\sqrt{7}}{16\sqrt{2}},,,\frac{7+\sqrt{7}}{8\sqrt{2}},,,\frac{7-\sqrt{7}}{8\sqrt{2}},,,\frac{5-\sqrt{7}}{16\sqrt{2}},,,\frac{1+\sqrt{7}}{16\sqrt{2}}\right],$

or

$h=\left[\frac{-3+\sqrt{15}}{16\sqrt{2}},,,\frac{1-\sqrt{15}}{16\sqrt{2}},,,\frac{3-\sqrt{15}}{8\sqrt{2}},,,\frac{3+\sqrt{15}}{8\sqrt{2}},,,\frac{13+\sqrt{15}}{16\sqrt{2}},,,\frac{9-\sqrt{15}}{16\sqrt{2}}\right],$

or

$h=\left[\frac{-3-\sqrt{15}}{16\sqrt{2}},,,\frac{1+\sqrt{15}}{16\sqrt{2}},,,\frac{3+\sqrt{15}}{8\sqrt{2}},,,\frac{3-\sqrt{15}}{8\sqrt{2}},,,\frac{13-\sqrt{15}}{16\sqrt{2}},,,\frac{9+\sqrt{15}}{16\sqrt{2}}\right],$

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
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
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
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
hi
Loga
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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! By Brooke Delaney By By Brooke Delaney By Anonymous User By Jessica Collett By OpenStax By Sandy Yamane By Vanessa Soledad By David Corey By OpenStax