3.4 Eigenfunctions of continuous time lti systems

 Page 1 / 1
An introduction to eigenvalues and eigenfunctions for continuous time linear time invariant systems.

Introduction

Prior to reading this module, the reader should already have some experience with linear algebra and should specifically be familiar with the eigenvectors and eigenvalues of linear operators. A linear time invariant system is a linear operator defined on a function space that commutes with every time shift operator on that function space. Thus, we can also consider the eigenvector functions, or eigenfunctions, of a system. It is particularly easy to calculate the output of a system when an eigenfunction is the input as the output is simply the eigenfunction scaled by the associated eigenvalue. As will be shown, continuous time complex exponentials serve as eigenfunctions of linear time invariant systems operating on continuous time signals.

Eigenfunctions of lti systems

Consider a linear time invariant system $H$ with impulse response $h$ operating on some space of infinite length continuous time signals. Recall that the output $H\left(x\left(t\right)\right)$ of the system for a given input $x\left(t\right)$ is given by the continuous time convolution of the impulse response with the input

$H\left(x\left(t\right)\right)={\int }_{-\infty }^{\infty }h\left(\tau \right)x\left(t-\tau \right)d\tau .$

Now consider the input $x\left(t\right)={e}^{st}$ where $s\in \mathbb{C}$ . Computing the output for this input,

$\begin{array}{cc}\hfill H\left({e}^{st}\right)& ={\int }_{-\infty }^{\infty }h\left(\tau \right){e}^{s\left(t-\tau \right)}d\tau \hfill \\ & ={\int }_{-\infty }^{\infty }h\left(\tau \right){e}^{st}{e}^{-s\tau }d\tau \hfill \\ & ={e}^{st}{\int }_{-\infty }^{\infty }h\left(\tau \right){e}^{-s\tau }d\tau .\hfill \end{array}$

Thus,

$H\left({e}^{st}\right)={\lambda }_{s}{e}^{st}$

where

${\lambda }_{s}={\int }_{-\infty }^{\infty }h\left(\tau \right){e}^{-s\tau }d\tau$

is the eigenvalue corresponding to the eigenvector ${e}^{st}$ .

There are some additional points that should be mentioned. Note that, there still may be additional eigenvalues of a linear time invariant system not described by ${e}^{st}$ for some $s\in \mathbb{C}$ . Furthermore, the above discussion has been somewhat formally loose as ${e}^{st}$ may or may not belong to the space on which the system operates. However, for our purposes, complex exponentials will be accepted as eigenvectors of linear time invariant systems. A similar argument using continuous time circular convolution would also hold for spaces finite length signals.

Eigenfunction of lti systems summary

As has been shown, continuous time complex exponential are eigenfunctions of linear time invariant systems operating on continuous time signals. Thus, it is particularly simple to calculate the output of a linear time invariant system for a complex exponential input as the result is a complex exponential output scaled by the associated eigenvalue. Consequently, representations of continuous time signals in terms of continuous time complex exponentials provide an advantage when studying signals. As will be explained later, this is what is accomplished by the continuous time Fourier transform and continuous time Fourier series, which apply to aperiodic and periodic signals respectively.

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
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
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
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?
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.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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
what is biological synthesis of nanoparticles
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!   By  By Rhodes By Prateek Ashtikar  By  By