14.4 Eigenfunctions of lti 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 $\mathscr{H}$ operating on a continuous input $f(t)$ to produce continuous time output $y(t)$

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).

Just as an eigenvector of $A$ is a $v\in {ℂ}^N$ such that $Av=\lambda v$ , $\lambda \in \mathbb{C}$ ,

Eigenfunctions are the simplest possible signals for $\mathscr{H}$ 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.

We can prove [link] by expressing the output as a convolution of the input $e^{st}$ and the impulse response $h(t)$ of $\mathscr{H}$ :

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.