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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

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+_
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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
im not good at math so would this help me
Rachael Reply
I'm not good at math so would you help me
what is the problem that i will help you to self with?
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
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what is the function of carbon nanotubes?
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
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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