<< Chapter < Page Chapter >> Page >

If the redundancy measure A in [link] and [link] is one, the matrices must be square and the system has an orthonormal basis.

Frames are over-complete versions of non-orthogonal bases and tight frames are over-complete versions of orthonormal bases. Tight frames areimportant in wavelet analysis because the restrictions on the scaling function coefficients discussed in  Chapter: The Scaling Function and Scaling Coefficients, Wavelet and Wavelet Coefficients guarantee not that the wavelets will be a basis, but a tight frame. In practice, however, theyare usually a basis.

Sinc expansion as a tight frame example

An example of an infinite-dimensional tight frame is the generalized Shannon's sampling expansion for the over-sampled case [link] . If a function is over-sampled but the sinc functions remains consistentwith the upper spectral limit W , the sampling theorem becomes

g ( t ) = T W π n g ( T n ) sin ( ( t - T n ) W ) ( t - T n ) W

or using R as the amount of over-sampling

R W = π T , for R 1

we have

g ( t ) = 1 R n g ( T n ) sin ( π R T ( t - T n ) ) π R T ( t - T n )

where the sinc functions are no longer orthogonal now. In fact, they are no longer a basis as they are not independent. They are, however, a tightframe and, therefore, act as though they were an orthogonal basis but now there is a “redundancy" factor R as a multiplier in the formula.

Notice that as R is increased from unity, [link] starts as [link] where each sample occurs where the sinc function is one or zero but becomes an expansion with the shifts still being t = T n , however, the sinc functions become wider so that the samples are no longer at thezeros. If the signal is over-sampled, either the expression [link] or [link] could be used. They both are over-sampled but [link] allows the spectrum of the signal to increase up to the limit without distortion while [link] does not. The generalized sampling theorem [link] has a built-in filtering action which may be an advantage or it may not.

The application of frames and tight frames to what is called a redundant discrete wavelet transform (RDWT) is discussed later in Section: Overcomplete Representations, Frames, Redundant Transforms, and Adaptive Bases and their use in Section: Nonlinear Filtering or Denoising with the DWT . They are also needed for certain adaptive descriptionsdiscussed at the end of Section: Overcomplete Representations, Frames, Redundant Transforms, and Adaptive Bases where an independent subset of theexpansion vectors in the frame are chosen according to some criterion to give an optimal basis.

Conditional and unconditional bases

A powerful point of view used by Donoho [link] gives an explanation of which basis systems are best for a particular class of signals and whythe wavelet system is good for a wide variety of signal classes.

Donoho defines an unconditional basis as follows. If we have a functionclass F with a norm defined and denoted | | · | | F and a basis set f k such that every function g F has a unique representation g = k a k f k with equality defined as a limit using the norm, we consider the infinite expansion

g ( t ) = k m k a k f k ( t ) .

If for all g F , the infinite sum converges for all | m k | 1 , the basis is called an unconditional basis . This is very similar to unconditional or absolute convergence of a numerical series [link] , [link] , [link] . If the convergence depends on m k = 1 for some g ( t ) , the basis is called a conditional basis .

An unconditional basis means all subsequences converge and all sequences of subsequences converge. It means convergence does not depend on the orderof the terms in the summation or on the sign of the coefficients. This implies a very robust basis where the coefficients drop off rapidly forall members of the function class. That is indeed the case for wavelets which are unconditional bases for a very wide set of function classes [link] , [link] , [link] .

Unconditional bases have a special property that makes them near-optimal for signal processing in several situations. This property has to do withthe geometry of the space of expansion coefficients of a class of functions in an unconditional basis. This is described in [link] .

The fundamental idea of bases or frames is representing a continuous function by a sequence of expansion coefficients. We have seen that theParseval's theorem relates the L 2 norm of the function to the 2 norm of coefficients for orthogonal bases and tight frames [link] . Different function spaces are characterized by different norms on the continuous function. If we have an unconditional basis for thefunction space, the norm of the function in the space not only can be related to some norm of the coefficients in the basis expansion, but theabsolute values of the coefficients have the sufficient information to establish the relation. So there is no condition on the sign or phaseinformation of the expansion coefficients if we only care about the norm of the function, thus unconditional .

For this tutorial discussion, it is sufficient to know that there are theoretical reasons why wavelets are an excellent expansion system for awide set of signal processing problems. Being an unconditional basis also sets the stage for efficient and effective nonlinear processing of thewavelet transform of a signal for compression, denoising, and detection which are discussed in  Chapter: The Scaling Function and Scaling Coefficients, Wavelet and Wavelet Coefficients .

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
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
Google
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
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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 ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
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
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
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?

Ask