# 14.4 Eigenfunctions of lti systems

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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 $ℋ$ operating on a continuous input $f(t)$ to produce continuous time output $y(t)$

$ℋ(f(t))=y(t)$ ℋ f t y t . f and t are continuous time (CT) signals and ℋ is an LTI operator.

is mathematically analogous to an $N$ x $N$ matrix $A$ operating on a vector $x\in {ℂ}^{N}$ to produce another vector $b\in {ℂ}^{N}$ (seeMatrices and LTI Systemsfor an overview).

$Ax=b$ A x b where x and b are in ℂ N and A is an N x N matrix.

Just as an eigenvector of $A$ is a $v\in {ℂ}^{N}$ such that $Av=\lambda v$ , $\lambda \in \mathbb{C}$ , A v λ v where v ℂ N is an eigenvector of A . we can define an eigenfunction (or eigensignal ) of an LTI system $ℋ$ to be a signal $f(t)$ such that
$\forall \lambda , \lambda \in \mathbb{C}\colon ℋ(f(t))=\lambda f(t)$ ℋ f t λ f t where f is an eigenfunction of ℋ .

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

## Eigenfunctions of any lti system

The class of LTI systems has a set of eigenfunctions in common: the complex exponentials $e^{st}$ , $s\in \mathbb{C}$ are eigenfunctions for all LTI systems.

$ℋ(e^{st})={\lambda }_{s}e^{st}$ ℋ s t λ s s t where ℋ is an LTI system.

While $\{\forall s, s\in \mathbb{C}\colon e^{st}\}$ 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 $e^{st}$ and the impulse response $h(t)$ of $ℋ$ :

$ℋ(e^{st})=\int_{()} \,d \tau$ 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, ${\lambda }_{s}()$ . Therefore
$ℋ(e^{st})={\lambda }_{s}e^{st}$
The eigenvalue ${\lambda }_{s}()$ is a complex number that depends on the exponent $s$ and, of course, the system $ℋ$ . To make these dependencies explicit, we will use the notation $H(s)\equiv {\lambda }_{s}()$ . s t is the eigenfunction and H s are the eigenvalues.

Since the action of an LTI operator on its eigenfunctions $e^{st}$ 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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