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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.

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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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