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Indeed, these results suggest that the largest Fourier or wavelet coefficients of smooth signals tend to concentrate at the coarsest scales (lowest-frequencies). In Linear Approximation from Approximation , we see that linear approximations formed from just the lowest frequencyelements of the Fourier or wavelet dictionaries (i.e., the truncation of the Fourier or wavelet representation to only the lowest frequency terms) provide very accurate approximations to smooth signals. Put differently, smooth signals live near the subspace spanned by just the lowest frequency Fourier or wavelet basis functions.

Sparse (nonlinear) models

Sparse signal models can be viewed as a generalization of linear models. The notion of sparsity comes from the fact that, by theproper choice of dictionary Ψ , many real-world signals x = Ψ α have coefficient vectors α containing few large entries, but across different signals the locations (indicesin α ) of the large entries may change. We say a signal is strictly sparse (or “ K -sparse”) if all but K entries of α are zero.

Some examples of real-world signals for which sparse models have been proposed include neural spike trains (in time), music andother audio recordings (in time and frequency), natural images (in the wavelet or curveletdictionaries  [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] ), video sequences (in a 3-D waveletdictionary  [link] , [link] ), and sonar or radar pulses (in a chirplet dictionary  [link] ). In each of these cases, the relevant information in a sparse representation of asignal is encoded in both the locations (indices) of the significant coefficients and the values to which they are assigned. This type of uncertainty is an appropriate model formany natural signals with punctuated phenomena.

Sparsity is a nonlinear model. In particular, let Σ K denote the set of all K -sparse signals for a given dictionary. It is easy to see that the set Σ K is not closed under addition. (In fact, Σ K + Σ K = Σ 2 K .) From a geometric perspective, the set of all K -sparse signals from the dictionary Ψ forms not a hyperplane but rather a union of K -dimensional hyperplanes, each spanned by K vectors of Ψ (see [link] (b)). For a dictionary Ψ with Z entries, there are Z K such hyperplanes. (The geometry of sparse signal collections has alsobeen described in terms of orthosymmetric sets; see  [link] .)

Signals that are not strictly sparse but rather have a few “large” and many “small” coefficients are known as compressible signals. The notion of compressibility can be made more precise by considering the rate at which the sorted magnitudes of the coefficients α decay, and this decay rate can in turn be related to the p norm of the coefficient vector α . Letting α ˜ denote a rearrangement of the vector α with the coefficients ordered in terms of decreasing magnitude, then the reordered coefficientssatisfy  [link]

α ˜ k α p k - 1 / p .
As we discuss in Nonlinear Approximation from Approximation , these decay rates play an important role in nonlinear approximation , where adaptive, K -sparse representations from the dictionary are used to approximate a signal.

We recall from [link] that for a smooth signal f , the largest Fourier and wavelet coefficients tend to cluster at coarse scales (low frequencies). Suppose, however, thatthe function f is piecewise smooth; i.e., it is C H at every point t R except for one point t 0 , at which it is discontinuous. Naturally, this phenomenon will be reflected in the transform coefficients.In the Fourier domain, this discontinuity will have a global effect, as the overall smoothness of the function f has been reduced dramatically from H to 0. Wavelet coefficients, however, depend only on local signalproperties, and so the wavelet basis functions whose supports do not include t 0 will be unaffected by the discontinuity. Coefficients surrounding the singularity will decay only as 2 - j / 2 , but there are relatively few such coefficients. Indeed, at each scale there are only O ( 1 ) wavelets that include t 0 in their supports, but these locations are highly signal-dependent. (For modeling purposes, these significantcoefficients will persist through scale down the parent-child tree structure.) After reordering by magnitude, the waveletcoefficients of piecewise smooth signals will have the same general decay rate as those of smooth signals. In Nonlinear Approximation from Approximation , we see that the quality of nonlinear approximations offered by wavelets for smooth 1-D signals is nothampered by the addition of a finite number of discontinuities.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Concise signal models. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10635/1.4
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