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It can be interpreted as a Fourier transform of f at the frequency ξ , localized by the window g ( t - u ) in the neighborhood of u . This windowed Fourier transform is highly redundant and represents one-dimensional signalsby a time-frequency image in ( u , ξ ) . It is thus necessary to understand how to select many fewertime-frequency coefficients that represent the signal efficiently.

When listening to music, we perceive sounds that have a frequency that varies in time. Chapter 4 showsthat a spectral line of f creates high-amplitude windowed Fourier coefficients S f ( u , ξ ) at frequencies ξ ( u ) that depend on time u . These spectral components are detected and characterized byridge points, which are local maxima in this time-frequency plane. Ridge points define a time-frequency approximation support λ of f with a geometry that depends on the time-frequency evolution of the signal spectral components. Modifying thesound duration or audio transpositions are implemented by modifying the geometry of the ridge support in time frequency.

A windowed Fourier transform decomposes signals over waveforms that have the same time and frequency resolution. It is thus effectiveas long as the signal does not include structures having different time-frequency resolutions, some being very localizedin time and others very localized in frequency.  Wavelets address this issue by changing the time and frequency resolution.

Continuous wavelet transform

In reflection seismology, Morlet knew that the waveforms sent underground have a duration that is too longat high frequencies to separate the returns of fine, closely spaced geophysical layers. Such waveforms are called wavelets in geophysics. Instead of emitting pulses of equal duration,he thought of sending shorter waveforms at high frequencies. These waveforms were obtained by scaling the motherwavelet, hence the name of this transform. Although Grossmann was working in theoretical physics, he recognized in Morlet's approach some ideasthat were close to his own work on coherent quantum states.

Nearly forty years after Gabor, Morlet and Grossmann reactivated a fundamentalcollaboration between theoretical physics and signal processing, whichled to the formalization of the continuous wavelet transform(GrossmannM:84). These ideas were not totally new to mathematicians working in harmonic analysis, or to computer visionresearchers studying multiscale image processing. It was thus only the beginning of a rapid catalysis that brought togetherscientists with very different backgrounds.

A wavelet dictionary is constructed from a mother wavelet ψ of zero average

- + ψ ( t ) d t = 0 ,

which is dilated with a scale parameter s , and translated by u :

D = ψ u , s ( t ) = 1 s ψ t - u s u R , s > 0 .

The continuous wavelet transform of f at any scale s and position u is the projection of f on the corresponding wavelet atom:

W f ( u , s ) = f , ψ u , s = - + f ( t ) 1 s ψ * t - u s d t .

It represents one-dimensional signals by highly redundant time-scale images in ( u , s ) .

Varying time-frequency resolution

As opposed to windowed Fourier atoms, wavelets have a time-frequency resolution that changes.The wavelet ψ u , s has a time support centered at u and proportional to s . Let us choose a wavelet ψ whose Fourier transform ψ ^ ( ω ) is nonzero in a positive frequency interval centered at η . The Fourier transform ψ ^ u , s ( ω ) is dilated by 1 / s and thus is localized in a positive frequencyinterval centered at ξ = η / s ; its size is scaled by 1 / s . In the time-frequency plane, the Heisenberg boxof a wavelet atom ψ u , s is therefore a rectangle centered at ( u , η / s ) , with time and frequency widths, respectively,proportional to s and 1 / s . When s varies, the time and frequency width of this time-frequency resolution cell changes, butits area remains constant, as illustrated by [link] .

Questions & Answers

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
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?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
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Abhi
I rally confuse this number And equations too I need exactly help
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But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
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Tamia
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Uday
hi
salma
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?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
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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
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No. 7x -4y is simplified from 4x + (3y + 3x) -7y
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how do you translate this in Algebraic Expressions
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Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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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
what is nano technology
Sravani Reply
what is system testing?
AMJAD
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
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
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
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
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
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
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, A wavelet tour of signal processing, the sparse way. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10711/1.3
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