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


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 ( x ( t ) ) of the system for a given input x ( t ) is given by the continuous time convolution of the impulse response with the input

H ( x ( t ) ) = - h ( τ ) x ( t - τ ) d τ .

Now consider the input x ( t ) = e s t where s C . Computing the output for this input,

H ( e s t ) = - h ( τ ) e s ( t - τ ) d τ = - h ( τ ) e s t e - s τ d τ = e s t - h ( τ ) e - s τ d τ .


H ( e s t ) = λ s e s t


λ s = - h ( τ ) e - s τ d τ

is the eigenvalue corresponding to the eigenvector e s t .

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 s t for some s C . Furthermore, the above discussion has been somewhat formally loose as e s t 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.

Questions & Answers

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Introduction about quantum dots in nanotechnology
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s. Reply
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are you nano engineer ?
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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?
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s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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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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