<< Chapter < Page Chapter >> Page >

Vanishing scaling function moments

While the moments of the wavelets give information about flatness of H ( ω ) and smoothness of ψ ( t ) , the moments of φ ( t ) and h ( n ) are measures of the “localization" and symmetry characteristics of the scaling function and, therefore, the wavelet transform. We know from [link] that n h ( n ) = 2 and, after normalization, that φ ( t ) d t = 1 . Using [link] , one can show [link] that for K 2 , we have

m ( 2 ) = m 2 ( 1 ) .

This can be seen in [link] . A generalization of this result has been developed by Johnson [link] and is given in [link] through [link] .

A more general picture of the effects of zero moments can be seen by next considering two approximations. Indeed, this analysis gives a veryimportant insight into the effects of zero moments. The mixture of zero scaling function moments with other specifications is addressed laterin [link] .

Approximation of signals by scaling function projection

The orthogonal projection of a signal f ( t ) on the scaling function subspace V j is given and denoted by

P j { f ( t ) } = k f ( t ) , φ j , k ( t ) φ j , k ( t )

which gives the component of f ( t ) which is in V j and which is the best least squares approximation to f ( t ) in V j .

As given in [link] , the t h moment of ψ ( t ) is defined as

m 1 ( ) = t ψ ( t ) d t .

We can now state an important relation of the projection [link] as an approximation to f ( t ) in terms of the number of zero wavelet moments and the scale.

Theorem 25 If m 1 ( ) = 0 for = 0 , 1 , , L then the L 2 error is

ϵ 1 = f ( t ) - P j { f ( t ) } 2 C 2 - j ( L + 1 ) ,

where C is a constant independent of j and L but dependent on f ( t ) and the wavelet system [link] , [link] .

This states that at any given scale, the projection of the signal on the subspace at that scale approaches the function itself as the numberof zero wavelet moments (and the length of the scaling filter) goes to infinity. It also states that for any given length, the projection goesto the function as the scale goes to infinity. These approximations converge exponentially fast. This projection is illustrated in [link] .

Approximation of scaling coefficients by samples of the signal

A second approximation involves using the samples of f ( t ) as the inner product coefficients in the wavelet expansion of f ( t ) in [link] . We denote this sampling approximation by

S j { f ( t ) } = k 2 - j / 2 f ( k / 2 j ) φ j , k ( t )

and the scaling function moment by

m ( ) = t φ ( t ) d t

and can state [link] the following

Theorem 26 If m ( ) = 0 for = 1 , 2 , , L then the L 2 error is

ϵ 2 = S j { f ( t ) } - P j { f ( t ) } 2 C 2 2 - j ( L + 1 ) ,

where C 2 is a constant independent of j and L but dependent on f ( t ) and the wavelet system.

This is a similar approximation or convergence result to the previous theorem but relates the projection of f ( t ) on a j -scale subspace to the sampling approximation in that same subspace. These approximationsare illustrated in [link] .

Approximation and Projection of f(t) at a Finite Scale
Approximation and Projection of f ( t ) at a Finite Scale

This “vector space" illustration shows the nature and relationships of the two types of approximations. The use of samples as inner productsis an approximation within the expansion subspace V j . The use of a finite expansion to represent a signal f ( t ) is an approximation from L 2 onto the subspace V j . Theorems  [link] and [link] show the nature of those approximations, which, for wavelets, is very good.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Wavelets and wavelet transforms. OpenStax CNX. Aug 06, 2015 Download for free at https://legacy.cnx.org/content/col11454/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Wavelets and wavelet transforms' conversation and receive update notifications?