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Full wavelet packet decomposition

In order to generate a basis system that would allow a higher resolution decomposition at high frequencies, we will iterate (split and down-sample)the highpass wavelet branch of the Mallat algorithm tree as well as the lowpass scaling function branch. Recall that for the discrete wavelettransform we repeatedly split, filter, and decimate the lowpass bands. The resulting three-scale analysis tree (three-stage filter bank) is shown in Figure: Three-Stage Two-Band Analysis Tree . This type of tree results in a logarithmic splitting of the bandwidths and tiling of the time-scale plane, as shown in [link] .

If we split both the lowpass and highpass bands at all stages, the resulting filter bank structure is like a full binary tree as in [link] . It is this full tree that takes O ( N log N ) calculations and results in a completely evenly spaced frequency resolution. In fact, its structure is somewhat similar to the FFT algorithm.Notice the meaning of the subscripts on the signal spaces. The first integer subscript is the scale j of that space as illustrated in [link] . Each following subscript is a zero or one, depending the path taken through the filter bank illustrated in Figure: Three-Stage Two-Band Analysis Tree . A “zero" indicates going through a lowpass filter (scaling functiondecomposition) and a “one" indicates going through a highpass filter (wavelet decomposition). This is different from the convention forthe M > 2 case in [link] .

The full binary tree for the three-scale wavelet packet transform.
The full binary tree for the three-scale wavelet packet transform.

[link] pictorially shows the signal vector space decomposition for the scaling functions and wavelets. [link] shows the frequency response of the packet filter bank much as Figure: Frequency Bands for the Analysis Tree did for M = 2 and [link] for M = 3 wavelet systems.

[link] shows the Haar wavelet packets with which we finish the example started in Section: An Example of the haar Wavelet System . This is an informative illustration that shows just what “packetizing" does to the regular wavelet system. Itshould be compared to the example at the end of  Chapter: A multiresolution formulation of Wavelet Systems . This is similar to the Walsh-Haddamar decomposition,and [link] shows the full wavelet packet system generated from the Daubechies φ D 8 ' scaling function. The “prime" indicates this is the Daubechies system with thespectral factorization chosen such that zeros are inside the unit circle and some outside. This gives the maximum symmetry possiblewith a Daubechies system. Notice the three wavelets have increasing “frequency." They are somewhat like windowed sinusoids, hence thename, wavelet packet. Compare the wavelets with the M = 2 and M = 4 Daubechies wavelets.

Adaptive wavelet packet systems

Normally we consider the outputs of each channel or band as the wavelet transform and from this have a nonredundant basis system. If, however,we consider the signals at the output of each band and at each stage or scale simultaneously, we have more outputs than inputs and clearly have aredundant system. From all of these outputs, we can choose an independent subset as a basis. This can be done in an adaptive way, depending on thesignal characteristics according to some optimization criterion. One possibility is the regular wavelet decomposition shown in Figure: Frequency Bands for the Analysis Tree .

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
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
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
Daniel
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
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
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
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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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