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This collection reviews fundamental concepts underlying the use of concise models for signal processing. Topics are presented from a geometric perspective and include low-dimensional linear, sparse, and manifold-based signal models, approximation, compression, dimensionality reduction, and Compressed Sensing.

For a wide variety of signal processing applications (including analysis, compression, noise removal, and so on) it is useful toconsider the representation of a signal in terms of some dictionary  [link] . In general, a dictionary Ψ issimply a collection of elements drawn from the signal space whose linear combinations can be used to represent or approximatesignals.

Considering, for example, signals in R N , we may collect and represent the elements of the dictionary Ψ as an N × Z matrix, which we also denote as Ψ . From this dictionary, a signal x R N can be constructed as a linear combination of the elements (columns) of Ψ . We write

x = Ψ α
for some α R Z . (For much of our notation in this section, we concentrate on signals in R N , though the basic concepts translate to other vector spaces.)

Dictionaries appear in a variety of settings. The most common may be the basis, in which case Ψ has exactly N linearly independent columns, and each signal x has a unique set of expansion coefficients α = Ψ - 1 x . The orthonormal basis (where the columns are normalized and orthogonal) is also ofparticular interest, as the unique set of expansion coefficients α = Ψ - 1 x = Ψ T x can be obtained as the inner products of x against the columns of Ψ . That is, α ( i ) = x , ψ i , i = 1 , 2 , , N , which gives us the expansion

x = i = 1 N x , ψ i ψ i .

We also have that x 2 2 = i = 1 N x , ψ i 2 .

Frames are another special type of dictionary  [link] . A dictionary Ψ is a frame if there exist numbers A and B , 0 < A B < such that, for any signal x

A x 2 2 z x , ψ z 2 B x 2 2 .
The elements of a frame may be linearly dependent in general (see [link] ), and so there may exist many ways to express a particular signal among the dictionary elements.However, frames do have a useful analysis/synthesis duality: for any frame Ψ there exists a dual frame Ψ ˜ such that
x = z x , ψ z ψ ˜ z = z x , ψ ˜ z ψ z .
In the case where the frame vectors are represented as columns of the N x Z matrix Ψ , the matrix Ψ ˜ containing the dual frame elements is simply the transpose of the pseudoinverse of Ψ . A frame is called tight if the frame bounds A and B are equal. Tight frames have the special properties of (i) being theirown dual frames (after a rescaling by 1 / A ) and (ii) preserving norms, i.e., i = 1 N x , ψ i 2 = A x 2 2 . The remainder of this section discusses several importantdictionaries.

A simple, redundant frame Ψ containing three vectors that span R 2 .

The canonical basis

The standard basis for representing a signal is the canonical (or “spike”) basis. In R N , this corresponds to a dictionary Ψ = I N (the N × N identity matrix). When expressed in the canonical basis, signals are often said tobe in the “time domain.”

Fourier dictionaries

The frequency domain provides one alternative representation to the time domain. The Fourier series and discrete Fourier transformare obtained by letting Ψ contain complex exponentials and allowing the expansion coefficients α to be complex as well. (Such a dictionary can be used to represent real or complexsignals.) A related “harmonic” transform to express signals in R N is the discrete cosine transform (DCT), in which Ψ contains real-valued, approximately sinusoidal functions and the coefficients α are real-valued as well.

Questions & Answers

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Damian Reply
research.net
kanaga
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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Damian Reply
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Akash Reply
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Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
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
Devang Reply
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
Abhijith Reply
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?
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.
Harper
Do you know which machine is used to that process?
s.
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SUYASH Reply
for screen printed electrodes ?
SUYASH
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s. Reply
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
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Sanket Reply
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
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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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