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Eigenfunctions of LTI Systems

Module by: Justin Romberg

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

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 ℋ ℋ operating on a continuous input ft f t to produce continuous time output yt y t
ℋft=yt ℋ f t y t (1)
transfn.png
Figure 1: ℋft=yt ℋ f t y t . f f and t t are continuous time (CT) signals and ℋ ℋ is an LTI operator.
is mathematically analogous to an NNxNN matrix A A operating on a vector x∈ ℂ N x ℂ N to produce another vector b∈ ℂ N b ℂ N (see Matrices and LTI Systems for an overview).
Ax=b A x b (2)
transfn2.png
Figure 2: Ax=b A x b where x x and b b are in ℂ N ℂ N and A A is an N N x N N matrix.
Just as an eigenvector of A A is a v∈ ℂ N v ℂ N such that Av=λv A v λ v , λ∈ℂ λ ,
transfn3.png
Figure 3: Av=λv A v λ v where v∈ ℂ N v ℂ N is an eigenvector of A A.
we can define an eigenfunction (or eigensignal) of an LTI system ℋ ℋ to be a signal ft f t such that
∀λ,λ∈ℂ:ℋft=λft λ λ ℋ f t λ f t (3)
transfn4.png
Figure 4: ℋft=λft ℋ f t λ f t where f f is an eigenfunction of ℋ ℋ.
Eigenfunctions are the simplest possible signals for ℋ ℋ 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 , s∈ℂ s are eigenfunctions for all LTI systems.
ℋⅇst= λ s ⅇst ℋ s t λ s s t (4)
transfn5.png
Figure 5: ℋⅇst= λ s ⅇst ℋ s t λ s s t where ℋ ℋ is an LTI system.
Note: While ∀s,s∈ℂ: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 ℋ ℋ:
ℋⅇst=∫-∞∞hτⅇst-τdτ=∫-∞∞hτⅇstⅇ-sτdτ=ⅇst∫-∞∞hτⅇ-sτdτ ℋ 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
ℋⅇst= λ s ⅇst ℋ 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 ℋ ℋ. To make these dependencies explicit, we will use the notation Hs≡ λ s H s λ s .
transfn6.png
Figure 6: ⅇst s t is the eigenfunction and Hs H s are the eigenvalues.
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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