# 0.4 Time-frequency dictionaries  (Page 2/3)

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It can be interpreted as a Fourier transform of $\phantom{\rule{0.166667em}{0ex}}f$ at the frequency ξ , localized by the window $g\left(t-u\right)$ in the neighborhood of u . This windowed Fourier transform is highly redundant and represents one-dimensional signalsby a time-frequency image in $\left(u,\xi \right)$ . 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 $\phantom{\rule{0.166667em}{0ex}}f$ creates high-amplitude windowed Fourier coefficients $Sf\left(u,\xi \right)$ at frequencies $\xi \left(u\right)$ 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 $\phantom{\rule{0.166667em}{0ex}}f$ 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

${\int }_{-\infty }^{+\infty }\psi \left(t\right)\phantom{\rule{0.166667em}{0ex}}dt=0,$

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

$\begin{array}{c}\hfill \mathcal{D}={\left\{{\psi }_{u,s},\left(t\right),=,\frac{1}{\sqrt{s}},\phantom{\rule{0.166667em}{0ex}},\psi ,\left(\frac{t-u}{s}\right)\right\}}_{u\in \mathbb{R},s>0}.\end{array}$

The continuous wavelet transform of $\phantom{\rule{0.166667em}{0ex}}f$ at any scale s and position u is the projection of $\phantom{\rule{0.166667em}{0ex}}f$ on the corresponding wavelet atom:

$\begin{array}{c}\hfill W\phantom{\rule{0.166667em}{0ex}}f\left(u,s\right)=⟨\phantom{\rule{0.166667em}{0ex}}f,{\psi }_{u,s}⟩={\int }_{-\infty }^{+\infty }f\left(t\right)\phantom{\rule{0.166667em}{0ex}}\frac{1}{\sqrt{s}}\phantom{\rule{0.166667em}{0ex}}{\psi }^{*}\left(\frac{t-u}{s}\right)\phantom{\rule{-0.166667em}{0ex}}dt.\end{array}$

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

## Varying time-frequency resolution

As opposed to windowed Fourier atoms, wavelets have a time-frequency resolution that changes.The wavelet ${\psi }_{u,s}$ has a time support centered at u and proportional to s . Let us choose a wavelet ψ whose Fourier transform $\stackrel{^}{\psi }\left(\omega \right)$ is nonzero in a positive frequency interval centered at η . The Fourier transform ${\stackrel{^}{\psi }}_{u,s}\left(\omega \right)$ is dilated by $1/s$ and thus is localized in a positive frequencyinterval centered at $\xi =\eta /s$ ; its size is scaled by $1/s$ . In the time-frequency plane, the Heisenberg boxof a wavelet atom ${\psi }_{u,s}$ is therefore a rectangle centered at $\left(u,\eta /s\right)$ , 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] .

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
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?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
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?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
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
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
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?
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 ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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