A response of an unstable system

Question

I'm dealing with the following question :

The system realizes an accumulator, therefore its impulse response (h[n]) is just the unit step function multiplied by some factor . In the next subsections , I'm asked to provide an answer in terms of X(ejw ) meaning I have to assume that the system is stable but it is obvious that a regular step impulse response represents an unstable system, how can this contradiction be settled to solve this question or perhaps there is something inherently inconsistent with the details provided . Thanks a lot for the help !

Answer 1

First, systems with poles right on the stability boundary (such as $\frac 1 s$ or $\frac{z}{z - 1}$) are not BIBO stable but they're not necessarily unstable, because -- especially for the single integrator case -- a zero input will result in a bounded output. The term I know to describe this is "metastable".

Second, you don't have to assume BIBO stability of all the parts to analyse a system. Control systems engineers use integrators in their controllers all the time, and are called on to stabilize unstable systems quite frequently. It turns out that all the analysis tools work out, if you're careful about your infinities (and if you're being detailed enough, your regions of convergence).

In the case of your problem, I would start by stating the assumption that $$\int_{-\infty}^\infty x_c(t) dt$$ is finite, then proceeding from there.