<< Chapter < Page Chapter >> Page >
Exit function, wavelet, and wavelet
(a) Exit function ν 0 ( t ) , ψ 0 ( t ) and ψ 0 ( t + 1 ) (b) Wavelet ψ 1 ( t ) and ψ 1 ( t + 1 ) (c) Wavelet ψ 2 ( t ) and ψ 2 ( t + 1 )

Filter banks and wavelets—summary

Filter banks are structures that allow a signal to be decomposed into subsignals—typically at a lower sampling rate.If the original signal can be reconstituted from the subsignals, the filter bank is said to be aperfect reconstruction (PR) filter bank. For PR, the analysis and synthesis filters have to satisfy a set of bilinear constraints.These constraints can be viewed from three perspectives, viz., the direct, matrix, and polyphase formulations. In PR filter bankdesign one chooses filters that maximize a “goodness" criterion and satisfy the PR constraints.

Unitary filter banks are an important class of PR filter banks—they give orthogonal decompositions of signals. For unitary filter banks,the PR constraints are quadratic in the analysis filters since the synthesis filters are index-reverses of the analysis filters.All FIR unitary filter banks can be explicitly parameterized. This leads to easy design (unconstrained optimization) and efficientimplementation. Sometimes one can impose structural constraints compatible with the goodness criterion. For example, modulated filter banks require that theanalysis and synthesis filters are modulates of single analysis and synthesis prototype filter respectively. Unitary modulated filterbanks exist and can be explicitly parameterized. This allows one to design and implement filter banks with hundreds of channels easilyand efficiently. Other structural constraints on the filters (e.g., linear phase filters) can be imposed and lead toparameterizations of associated unitary filter banks. Cascades of filter banks (used in a tree structure) can be used torecursively decompose signals.

Every unitary FIR filter bank with an additional linear constraint on the lowpass filter is associatedwith a wavelet tight frame. The lowpass filter is associated with the scaling function, and the remaining filters are each associated withwavelets. The coefficients of the wavelet expansion of a signal can be computed using a tree-structure where the filter bank isapplied recursively along the lowpass filter channel. The parameterization of unitary filter banks, with a minor modification,gives a parameterization of all compactly supported wavelet tight frames. In general, wavelets associated with a unitary filter bank areirregular (i.e., not smooth). By imposing further linear constraints (regularity constraints) on the lowpass filter,one obtains smooth wavelet bases. Structured filter banks give rise to associated structured waveletbases; modulated filter banks are associated with modulated wavelet bases and linear phase filter banks are associated with linear-phasewavelet bases. Filter banks cascade—where all the channels are recursively decomposed, they are associated with wavelet packet bases.

From a time-frequency analysis point of view filter banks trees can be used to give arbitraryresolutions of the frequency. In order to obtain arbitrary temporal resolutions one has to use local bases or switch between filter banktrees at points in time. Techniques for time-varying filter banks can be used to generate segmented wavelet bases (i.e., a different waveletbases for disjoint segments of the time axis). Finally, just as unitary filter banks are associated with wavelet tight frames,general PR filter banks, with a few additional constraints, are associated with wavelet frames (or biorthogonal bases).

Questions & Answers

Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
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.
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket 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
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!
QuizOver.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?