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or

c ˜ j ( k ) = m h ˜ ( m - 2 k ) c j + 1 ( m )

or

c ˜ j ( k ) = n c j ( k + P n )

where for [link]

c j ( k ) = m h ( m - 2 k ) c j + 1 ( m )

The corresponding relationships for the wavelet coefficients are

d ˜ j ( k ) = m h 1 ( m - 2 k ) c ˜ j + 1 ( m ) = m h ˜ 1 ( m - 2 k ) c j + 1 ( m )

or

d ˜ j ( k ) = n d j ( k + 2 j P n )

where

d j ( k ) = m h 1 ( m - 2 k ) c j + 1 ( m )

These are very important properties of the DWT of a periodic signal, especially one artificially constructed from a nonperiodic signal in orderto use a block algorithm. They explain not only the aliasing effects of having a periodic signal but how to calculate the DWT of a periodicsignal.

Structure of the periodic discrete wavelet transform

If f ( t ) is essentially infinite in length, then the DWT can be calculated as an ongoing or continuous process in time. In other words,as samples of f ( t ) come in at a high enough rate to be considered equal to c J 1 ( k ) , scaling function and wavelet coefficients at lower resolutions continuously come out of the filter bank. This is best seenfrom the simple two-stage analysis filter bank in Section: Three-Stage Two-Band Analysis Tree . If samples come in at what is called scale J 1 = 5 , wavelet coefficients at scale j = 4 come out the lower bank at half the input rate. Wavelet coefficients at j = 3 come out the next lower bank at one quarter the input rate and scaling function coefficients at j = 3 come out the upper bank also at one quarter the input rate. It is easy to imagine morestages giving lower resolution wavelet coefficients at a lower and lower rate depending on the number of stages. The last one will always be thescaling function coefficients at the lowest rate.

For a continuous process, the number of stages and, therefore, the level of resolution at the coarsest scale is arbitrary. It is chosen to be thenature of the slowest features of the signals being processed. It is important to remember that the lower resolution scales correspond to aslower sampling rate and a larger translation step in the expansion terms at that scale. This is why the wavelet analysis system gives good timelocalization (but poor frequency localization) at high resolution scales and good frequency localization (but poor time localization) at low orcoarse scales.

For finite length signals or block wavelet processing, the input samples can be considered as a finite dimensional input vector, the DWT as asquare matrix, and the wavelet expansion coefficients as an output vector. The conventional organization of the output of the DWT places the outputof the first wavelet filter bank in the lower half of the output vector. The output of the next wavelet filter bank is put just above that block.If the length of the signal is two to a power, the wavelet decomposition can be carried until there is just one wavelet coefficient and one scalingfunction coefficient. That scale corresponds to the translation step size being the length of the signal. Remember that the decomposition does nothave to carried to that level. It can be stopped at any scale and is still considered a DWT, and it can be inverted using the appropriatesynthesis filter bank (or a matrix inverse).

More general structures

The one-sided tree structure of Mallet's algorithm generates the basic DWT. From the filter bank in Section: Three-Stage Two-Band Analysis Tree , one can imagine putting a pair of filters and downsamplers at the output of the lower wavelet bankjust as is done on the output of the upper scaling function bank. This can be continued to any level to create a balanced tree filter bank. Theresulting outputs are “wavelet packets" and are an alternative to the regular wavelet decomposition. Indeed, this “growing" of the filter banktree is usually done adaptively using some criterion at each node to decide whether to add another branch or not.

Still another generalization of the basic wavelet system can be created by using a scale factor other than two. The multiplicity-M scaling equationis

φ ( t ) = k h ( k ) φ ( M t - k )

and the resulting filter bank tree structure has one scaling function branch and M - 1 wavelet branches at each stage with each followed by a downsampler by M . The resulting structure is called an M -band filter bank, and it too is an alternative to the regular wavelet decomposition. This is developed in Section: Multiplicity-M (M-band) Scaling Functions and Wavelets .

In many applications, it is the continuous wavelet transform (CWT) that is wanted. This can be calculated by using numerical integration to evaluatethe inner products in [link] and [link] but that is very slow. An alternative is to use the DWT to approximate samples of the CWT much asthe DFT can be used to approximate the Fourier series or integral [link] , [link] , [link] .

As you can see from this discussion, the ideas behind wavelet analysis and synthesis are basically the same as those behind filter bank theory.Indeed, filter banks can be used calculate discrete wavelet transforms using Mallat's algorithm, and certain modifications and generalizationscan be more easily seen or interpreted in terms of filter banks than in terms of the wavelet expansion. The topic of filter banks in developed in Chapter: Filter Banks and the Discrete Wavelet Transform and in more detail in Chapter: Filter Banks and Transmultiplexers .

Questions & Answers

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
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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.
Bharti
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Damian Reply
absolutely yes
Daniel
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Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
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
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
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.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
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s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
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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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