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Eigenfunctions of LTI Systems

Module by: Justin Romberg. E-mail the author

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

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Introduction

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 ft f t to produce continuous time output yt y t

ft=yt f t y t
(1)

Figure 1: ft=yt f t y t . f f and t t are continuous time (CT) signals and is an LTI operator.
Figure 1 (transfn.png)

is mathematically analogous to an NNxNN matrix A A operating on a vector x N x N to produce another vector b N b N (see Matrices and LTI Systems for an overview).

Ax=b A x b
(2)

Figure 2: Ax=b A x b where x x and b b are in N N and A A is an N N x N N matrix.
Figure 2 (transfn2.png)

Just as an eigenvector of A A is a v N v N such that Av=λv A v λ v , λC λ ,

Figure 3: Av=λv A v λ v where v N v N is an eigenvector of A A.
Figure 3 (transfn3.png)
we can define an eigenfunction (or eigensignal) of an LTI system to be a signal ft f t such that
λ ,λC:ft=λft λ λ f t λ f t
(3)

Figure 4: ft=λft f t λ f t where f f is an eigenfunction of .
Figure 4 (transfn4.png)

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

Eigenfunctions of any LTI System

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

est= λ s est s t λ s s t
(4)

Figure 5: est= λ s est s t λ s s t where is an LTI system.
Figure 5 (transfn5.png)

Note:

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

We can prove Equation 4 by expressing the output as a convolution of the input est s t and the impulse response ht h t of :

est=hτes(tτ)d τ =hτeste(sτ)d τ =esthτe(sτ)d τ s t τ h τ s t τ τ h τ s t s τ s t τ h τ s τ
(5)
Since the expression on the right hand side does not depend on t t, it is a constant, λ s λ s . Therefore
est= λ s est s t λ s s t
(6)
The eigenvalue λ s λ s is a complex number that depends on the exponent s s and, of course, the system . To make these dependencies explicit, we will use the notation Hs λ s H s λ s .

Figure 6: est s t is the eigenfunction and Hs H s are the eigenvalues.
Figure 6 (transfn6.png)

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

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