<< Chapter < Page
  Wavelets and wavelet transforms     Page 26 / 28
Chapter >> Page >

Discrete multiresolution analysis, the discrete-time wavelet transform, and the continuous wavelet transform

Up to this point, we have developed wavelet methods using the series wavelet expansion of continuous-time signals called the discrete wavelettransform (DWT), even though it probably should be called the continuous-time wavelet series. This wavelet expansion is analogous tothe

Local Basis
Local Basis

Fourier series in that both are series expansions that transform continuous-time signals into a discrete sequence of coefficients. However, unlike the Fourier series, the DWT can be made periodic ornonperiodic and, therefore, is more versatile and practically useful.

In this chapter we will develop a wavelet method for expanding discrete-time signals in a series expansion since, in most practicalsituations, the signals are already in the form of discrete samples. Indeed, we have already discussed when it is possible to use samples ofthe signal as scaling function expansion coefficients in order to use the filter bank implementation of Mallat's algorithm. We find there is anintimate connection between the DWT and DTWT, much as there is between the Fourier series and the DFT. One expands signals with the FS but oftenimplements that with the DFT.

To further generalize the DWT, we will also briefly present the continuous wavelet transform which, similar to the Fourier transform, transforms afunction of continuous time to a representation with continuous scale and translation. In order to develop the characteristics of these variouswavelet representations, we will often call on analogies with corresponding Fourier representations. However, it is important tounderstand the differences between Fourier and wavelet methods. Much of that difference is connected to the wavelet being concentrated in bothtime and scale or frequency, to the periodic nature of the Fourier basis, and to the choice of wavelet bases.

Discrete multiresolution analysis and the discrete-time wavelet transform

Parallel to the developments in early chapters on multiresolution analysis, we can define a discrete multiresolution analysis (DMRA) for l 2 , where the basis functions are discrete sequences [link] , [link] , [link] . The expansion of a discrete-time signal in terms of discrete-time basis function is expressed in a form parallel to [link] as

f ( n ) = j , k d j ( k ) ψ ( 2 j n - k )

where ψ ( m ) is the basic expansion function of an integer variable m . If these expansion functions are an orthogonal basis (or form a tight frame), the expansion coefficients (discrete-time wavelet transform) arefound from an inner product by

d j ( k ) = f ( n ) , ψ ( 2 j n - k ) = n f ( n ) ψ ( 2 j n - k )

If the expansion functions are not orthogonal or even independent but do span 2 , a biorthogonal system or a frame can be formed such that a transform and inverse can be defined.

Because there is no underlying continuous-time scaling function or wavelet, many of the questions, properties, and characteristics of the analysisusing the DWT in  Chapter: Introduction to Wavelets , Chapter: A multiresolution formulation of Wavelet Systems , Chapter: Regularity, Moments, and Wavelet System Design , etc. do not arise. In fact, because of the filter bank structure for calculating the DTWT,the design is often done using multirate frequency domain techniques, e.g., the work by Smith and Barnwell and associates [link] . The questions of zero wavelet moments posed by Daubechies,which are related to ideas of convergence for iterations of filter banks, and Coifman's zero scalingfunction moments that were shown to help approximate inner products by samples, seem to have no DTWT interpretation.

Questions & Answers

what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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
How can I make nanorobot?
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
how can I make nanorobot?
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.
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?