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An introduction to eigenvalues and eigenfunctions for Linear Time Invariant systems.


Hopefully you are familiar with the notion of the eigenvectors of a "matrix system," if not they do a quick review of eigen-stuff . We can develop the same ideas for LTI systems acting on signals. A linear time invariant (LTI) system operating on a continuous input f t to produce continuous time output y t

f t y t

f t y t . f and t are continuous time (CT) signals and is an LTI operator.

is mathematically analogous to an N x N matrix A operating on a vector x N to produce another vector b N (seeMatrices and LTI Systemsfor an overview).

A x b

A x b where x and b are in N and A is an N x N matrix.

Just as an eigenvector of A is a v N such that A v λ v , λ ,

A v λ v where v N is an eigenvector of A .
we can define an eigenfunction (or eigensignal ) of an LTI system to be a signal f t such that
λ λ f t λ f t

f t λ f t where f is an eigenfunction of .

Eigenfunctions are the simplest possible signals for to operate on: to calculate the output, we simply multiply the input by a complex number λ .

Eigenfunctions of any lti system

The class of LTI systems has a set of eigenfunctions in common: the complex exponentials s t , s are eigenfunctions for all LTI systems.

s t λ s s t

s t λ s s t where is an LTI system.

While s s s t are always eigenfunctions of an LTI system, they are not necessarily the only eigenfunctions.

We can prove [link] by expressing the output as a convolution of the input s t and the impulse response h t of :

s t τ h τ s t τ τ h τ s t s τ s t τ h τ s τ
Since the expression on the right hand side does not depend on t , it is a constant, λ s . Therefore
s t λ s s t
The eigenvalue λ s is a complex number that depends on the exponent s and, of course, the system . To make these dependencies explicit, we will use the notation H s λ s .

s t is the eigenfunction and H s are the eigenvalues.

Since the action of an LTI operator on its eigenfunctions s t is easy to calculate and interpret, it is convenient to represent an arbitrary signal f t as a linear combination of complex exponentials. The Fourier series gives us this representation for periodic continuous timesignals, while the (slightly more complicated) Fourier transform lets us expand arbitrary continuous time signals.

Questions & Answers

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.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
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
what is fullerene does it is used to make bukky balls
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
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