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Change of signal

If both x and b in [link] are considered to be signals in the same coordinate or basis system, the matrix operator A is generally square. It may or may not be of full rank and it may or may not have a variety of otherproperties, but both x and b are viewed in the same coordinate system and therefore are the same size.

One of the most ubiquitous of these is convolution where the input to a linear, shift invariant system with impulse response h ( n ) is calculated by [link] if A is the convolution matrix and x is the input [link] .

y 0 y 1 y 2 = h 0 0 0 0 h 1 h 0 0 h 2 h 1 h 0 x 0 x 1 x 2 .

It can also be calculated if A is the arrangement of the input and x is the the impulse response.

y 0 y 1 y 2 = x 0 0 0 0 x 1 x 0 0 x 2 x 1 x 0 h 0 h 1 h 2 .

If the signal is periodic or if the DFT is being used, then what is called a circulate is used to represent cyclic convolution. An example for N = 4 is the Toeplitz system

y 0 y 1 y 2 y 3 = h 0 h 3 h 2 h 1 h 1 h 0 h 3 h 2 h 2 h 1 h 0 h 3 h 3 h 2 h 1 h 0 x 0 x 1 x 2 x 3 .

One method of understanding and generating matrices of this sort is to construct them as a product of first a decomposition operator, then amodification operator in the new basis system, followed by a recomposition operator. For example, one could first multiply a signal by the DFToperator which will change it into the frequency domain. One (or more) of the frequency coefficients could be removed (set to zero) and theremainder multiplied by the inverse DFT operator to give a signal back in the time domain but changed by having a frequency component removed. Thatis a form of signal filtering and one can talk about removing the energy of a signal at a certain frequency (or many) because of Parseval's theorem.

It would be instructive for the reader to make sense out of the cryptic statement “the DFT diagonalizes the cyclic convolution matrix" to add tothe ideas in this note.

Factoring the matrix A

For insight, algorithm development, and/or computational efficiency, it is sometime worthwhile to factor A into a product of two or more matrices. For example, the D F T matrix [link] illustrated in [link] can be factored into a product of fairly sparce matrices. If fact, the fast Fourier transform (FFT) can be derived byfactoring the DFT matrix into N log ( N ) factors (if N = 2 m ), each requiring order N multiplies. This is done in [link] .

Using eigenvalue theory [link] , a full rank square matrix can be factored into a product

A V = V Λ

where V is a matrix with columns of the eigenvectors of A and Λ is a diagonal matrix with the eigenvalues along the diagonal. The inverse is a method to “diagonalize" a matrix

Λ = V - 1 A V

If a matrix has “repeated eigenvalues", in other words, two or more of the N eigenvalues have the same value but less than N indepentant eigenvectors, it is not possible to diagonalize the matrix butan “almost" diagonal form called the Jordan normal form can be acheived. Those details can be found in most books on matrix theory [link] .

A more general decompostion is the singular value decomposition (SVD) which is similar to the eigenvalue problem but allows rectangular matrices.It is particularly valuable for expressing the pseudoinverse in a simple form and in making numerical calculations [link] .

State equations

If our matrix multiplication equation is a vector differential equation (DE) of the form

x ˙ = A x

or for difference equations and discrete-time signals or digital signals,

x ( n + 1 ) = A x ( n )

an inverse or even pseudoinverse will not solve for x . A different approach must be taken [link] and different properties and tools from linear algebra will be used. The solution of this first order vector DE is acoupled set of solutions of first order DEs. If a change of basis is made so that A is diagonal (or Jordan form), equation [link] becomes a set on uncoupled (or almost uncoupled in the Jordan form case) first order DEs and we know the solution of a first orderDE is an exponential. This requires consideration of the eigenvalue problem, diagonalization, and solution of scalar first order DEs [link] .

State equations are often used to model or describe a system such as a control system or a digital filter or a numerical algorithm [link] , [link] .

Questions & Answers

do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
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Daniel
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Maciej
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Abigail
Do somebody tell me a best nano engineering book for beginners?
s. Reply
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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.
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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
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Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
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
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
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Damian Reply
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Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
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Porter
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Yasmin
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I'm interested in nanotube
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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...
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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
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anybody can imagine what will be happen after 100 years from now in nano tech world
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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
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name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
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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
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Source:  OpenStax, Basic vector space methods in signal and systems theory. OpenStax CNX. Dec 19, 2012 Download for free at http://cnx.org/content/col10636/1.5
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