# 0.10 Wavelet-based signal processing and applications  (Page 7/13)

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The new nonlinear method is entirely different. The spectra can overlap as much as they want. The idea is to have the amplitude, rather than thelocation of the spectra be as different as possible. This allows clipping, thresholding, and shrinking of the amplitude of the transform toseparate signals or remove noise. It is the localizing or concentrating properties of the wavelet transform that makes it particularly effectivewhen used with these nonlinear methods. Usually the same properties that make a system good for denoising or separation by nonlinear methods, makesit good for compression, which is also a nonlinear process.

## Denoising by thresholding

We develop the basic ideas of thresholding the wavelet transform using Donoho's formulations [link] , [link] , [link] . Assume a finite length signal with additive noise of the form

${y}_{i}={x}_{i}+ϵ{n}_{i},\phantom{\rule{1.em}{0ex}}i=1,...,N$

as a finite length signal of observations of the signal ${x}_{i}$ that is corrupted by i.i.d.  zero mean, white Gaussian noise ${n}_{i}$ with standard deviation $ϵ$ , i.e., ${n}_{i}\stackrel{iid}{\sim }\mathcal{N}\left(0,1\right)$ . The goal is to recover the signal $x$ from the noisy observations $y$ . Here and in the following, $v$ denotes a vector with the ordered elements ${v}_{i}$ if the index $i$ is omitted. Let $W$ be a left invertible wavelet transformation matrix of the discrete wavelet transform(DWT). Then Eq. [link] can be written in the transformation domain

$Y=X+N,\phantom{\rule{1.em}{0ex}}\text{or},\phantom{\rule{1.em}{0ex}}{Y}_{i}={X}_{i}+{N}_{i},$

where capital letters denote variables in the transform domain, i.e., $Y=Wy$ . Then the inverse transform matrix ${W}^{-1}$ exists, and we have

${W}^{-1}W=I.$

The following presentation follows Donoho's approach [link] , [link] , [link] , [link] , [link] that assumes an orthogonal wavelet transform with a square $W$ ; i.e., ${W}^{-1}={W}^{T}$ . We will use the same assumption throughout this section.

Let $\stackrel{^}{X}$ denote an estimate of $X$ , based on the observations $Y$ . We consider diagonal linear projections

$\text{Δ}=\text{diag}\left({\delta }_{1},...,{\delta }_{N}\right),\phantom{\rule{1.em}{0ex}}{\delta }_{i}\in \left\{0,1\right\},\phantom{\rule{1.em}{0ex}}i=1,...,N,$

which give rise to the estimate

$\stackrel{^}{x}={W}^{-1}\stackrel{^}{X}={W}^{-1}\text{Δ}Y={W}^{-1}\text{Δ}Wy.$

The estimate $\stackrel{^}{X}$ is obtained by simply keeping or zeroing the individual wavelet coefficients. Since we are interested inthe ${l}_{2}$ error we define the risk measure

$\mathcal{R}\left(\stackrel{^}{X},X\right)=E\left[\parallel ,\stackrel{^}{x},{-x\parallel }_{2}^{2}\right]=E\left[\parallel ,{W}^{-1},\left(\stackrel{^}{X}-X\right),{\parallel }_{2}^{2}\right]=E\left[\parallel ,\stackrel{^}{X},{-X\parallel }_{2}^{2}\right].$

Notice that the last equality in Eq. [link] is a consequence of the orthogonality of $W$ . The optimal coefficients in the diagonal projection scheme are ${\delta }_{i}={1}_{{X}_{i}>ϵ}$ ; It is interesting to note that allowing arbitrary ${\delta }_{i}\in \mathrm{I}\phantom{\rule{-1.99997pt}{0ex}}\mathrm{R}$ improves the ideal risk by at most a factor of 2 [link] i.e., only those values of $Y$ where the corresponding elements of $X$ are larger than $ϵ$ are kept, all others are set to zero. This leads to the ideal risk

${\mathcal{R}}_{id}\left(\stackrel{^}{X},X\right)=\sum _{n=1}^{N}min\left({X}^{2},{ϵ}^{2}\right).$

The ideal risk cannot be attained in practice, since it requires knowledge of $X$ , the wavelet transform of the unknown vector $x$ . However, it does give us a lower limit for the ${l}_{2}$ error.

Donoho proposes the following scheme for denoising:

1. compute the DWT $Y=Wy$
2. perform thresholding in the wavelet domain, according to so-called hard thresholding
$\stackrel{^}{X}={T}_{h}\left(Y,t\right)=\left\{\begin{array}{cc}Y,& |Y|\ge t\hfill \\ 0,& |Y|
or according to so-called soft thresholding
$\stackrel{^}{X}={T}_{S}\left(Y,t\right)=\left\{\begin{array}{cc}\text{sgn}\left(Y\right)\left(|Y|-t\right),& |Y|\ge t\hfill \\ 0,& |Y|
3. compute the inverse DWT $\stackrel{^}{x}={W}^{-1}\stackrel{^}{X}$

This simple scheme has several interesting properties. It's risk is within a logarithmic factor ( $logN$ ) of the ideal risk for both thresholding schemes and properly chosen thresholds $t\left(N,ϵ\right)$ . If one employs soft thresholding, then the estimate is with high probability at least assmooth as the original function. The proof of this proposition relies on the fact that wavelets are unconditional bases for a variety of smoothnessclasses and that soft thresholding guarantees (with high probability) that the shrinkage condition $|{\stackrel{^}{X}}_{i}|<|{X}_{i}|$ holds. The shrinkage condition guarantees that $\stackrel{^}{x}$ is in the same smoothness class as is $x$ . Moreover, the soft threshold estimate is the optimal estimate that satisfies the shrinkage condition. The smoothness property guarantees anestimate free from spurious oscillations which may result from hard thresholding or Fourier methods. Also, it can be shown that it is notpossible to come closer to the ideal risk than within a factor $logN$ . Not only does Donoho's method have nice theoretical properties, but italso works very well in practice.

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Renato
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?
Kyle
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Kyle
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Joe
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research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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Daniel
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Abigail
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NANO
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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
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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.
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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?
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Harper
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s.
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for screen printed electrodes ?
SUYASH
What is lattice structure?
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Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
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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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