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The multiresolution decomposition seems to separate components of a signal in a way that is superior to most other methods for analysis, processing,or compression. Because of the ability of the discrete wavelet transform to decompose a signal at different independent scales and to do it in avery flexible way, Burke calls wavelets “The Mathematical Microscope" [link] , [link] . Because of this powerful and flexible decomposition, linear and nonlinear processing of signals in the wavelettransform domain offers new methods for signal detection, filtering, and compression [link] , [link] , [link] , [link] , [link] , [link] . It also can be used as the basis for robust numerical algorithms.

You will also see an interesting connection and equivalence to filter bank theory from digital signal processing [link] , [link] . Indeed, some of the results obtained with filter banks are the same as with discrete-timewavelets, and this has been developed in the signal processing community by Vetterli, Vaidyanathan, Smith and Barnwell, and others. Filter banks,as well as most algorithms for calculating wavelet transforms, are part of a still more general area of multirate and time-varying systems.

The presentation here will be as a tutorial or primer for people who know little or nothing about wavelets but do have a technical background. Itassumes a knowledge of Fourier series and transforms and of linear algebra and matrix theory. It also assumes a background equivalent to a B.S.degree in engineering, science, or applied mathematics. Some knowledge ofsignal processing is helpful but not essential. We develop the ideas in terms of one-dimensional signals [link] modeled as real or perhaps complex functions of time, but the ideas and methods have also proveneffective in image representation and processing [link] , [link] dealing with two, three, or even four or more dimensions. Vector spaces have proved to be a natural setting for developing both the theory andapplications of wavelets. Some background in that area is helpful but can be picked up as needed [link] . The study and understanding of wavelets is greatly assisted by using some sort of wavelet software system to work outexamples and run experiments. Matlab T M programs are included at the end of this book and on our web site (noted at the end of thepreface). Several other systems are mentioned in Chapter: Wavelet-Based Signal Processing and Applications .

There are several different approaches that one could take in presenting wavelet theory. We have chosen to start with the representation of asignal or function of continuous time in a series expansion, much as a Fourier series is used in a Fourier analysis. From this seriesrepresentation, we can move to the expansion of a function of a discrete variable (e.g., samples of a signal) and the theory of filter banks toefficiently calculate and interpret the expansion coefficients. This would be analogous to the discrete Fourier transform (DFT) and itsefficient implementation, the fast Fourier transform (FFT). We can also go from the series expansion to an integral transform called thecontinuous wavelet transform, which is analogous to the Fourier transform or Fourier integral. We feel starting with the series expansion gives thegreatest insight and provides ease in seeing both the similarities and differences with Fourier analysis.

Questions & Answers

how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
I'm not sure why it wrote it the other way
I got X =-6
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
Idrissa Reply
im all ears I need to learn
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what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
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Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Wavelets and wavelet transforms. OpenStax CNX. Aug 06, 2015 Download for free at https://legacy.cnx.org/content/col11454/1.6
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