Solution 12 :

In this solution, we use the idea of the problem 11 of this module

Our argument is based on the regions of convergence of Laplace transforms and the corresponding implications for the rational systems. We recall that the region of convergence (ROC) of the the Laplace transform is always bounded by two vertical lines.

For a system to be causal, the region of convergence of the Laplace transform must include the part of the s-plane as

For a system to be stable, the imaginary axis has to be a part of the ROC.

So, a causal and stable system has its ROC as the right side of a pole of the transfer function H(s), with the poles all lying in the left half plane s<0 (else the imaginary axis will not be in the ROC).

Now, from the problem above, we see that the zeroes of the transfer function become the poles of its inverse. So, for the inverse to be stable and causal, all the poles of the inverse (i.e. the zeroes of the original transfer function) must also be in the left half plane s<0.

This completes the argument.