<< Chapter < Page Chapter >> Page >
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 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.

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Signals and systems' conversation and receive update notifications?

Ask