4.3 Eigenvectors of lsi systems

Suppose that $h$ is the impulse response of an LSI system. Consider an input $x\left[n\right]=z^n$ where $z$ is a complex number. What is the output of the system? Recall that $x*h=h*x$ . In this case, it is easier to use the formula:

where

In the event that $H\left(z\right)$ converges, we see that $y\left[n\right]$ is just a re-scaled version of $x\left[n\right]$ . Thus, $x\left[n\right]$ is an eigenvector of the system $H$ , right? Not exactly, but almost... technically, since $z^n\notin {\ell }_2\left(\mathbb{Z}\right)$ it isn't really an eigenvector. However, most DSP texts ignore this subtlety. Theintuition provided by thinking of $z^n$ as an eigenvector is worth the slight abuse of terminology.

Next time we will analyze the function $H\left(z\right)$ in greater detail. $H\left(z\right)$ is called the $z$ -transform of $h$ , and provides an extremely useful characterization of a discrete-time system.