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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

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Damian Reply
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Do somebody tell me a best nano engineering book for beginners?
s. Reply
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Devang Reply
are you nano engineer ?
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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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.
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s. Reply
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s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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Sanket Reply
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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what is the application of nanotechnology?
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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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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silver nanoparticles could handle the job?
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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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