2.9 Inverse systems

Inverse systems

Many signal processing problems can be interpreted as trying to undo the action of some system. For example, echo cancellation, channel obvolution,etc. The problem is illustrated below.

If our goal is to design a system $H_I$ that reverses the action of $H$ , then we clearly need $H\left(z\right)H_I\left(z\right)=1$ . In the case where

then this can be achieved via

Thus, the zeros of $H\left(z\right)$ become poles of $H_I\left(z\right)$ , and the poles of $H\left(z\right)$ become zeros of $H_I\left(z\right)$ . Recall that $H\left(z\right)$ being stable and causal implies that all poles are inside the unit circle. If we want $H\left(z\right)$ to have a stable, causal inverse $H_I\left(z\right)$ , then we must have all zeros inside the unit circle, (since they become the poles of $H_I\left(z\right)$ .) Combining these, $H\left(z\right)$ is stable and causal with a stable and causal inverse if and only if all poles and zeros of $H\left(z\right)$ are inside the unit circle. This type of system is called a minimum phase system.