# Connexions

You are here: Home » Content » Eigenfunctions of LTI Systems

### Recently Viewed

This feature requires Javascript to be enabled.

### Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.

# Eigenfunctions of LTI Systems

Module by: Justin Romberg. E-mail the author

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

Note: You are viewing an old version of this document. The latest version is available here.

## 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)

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)

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

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)

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)

### 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 .

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.

## Content actions

### Give feedback:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks