# A response of an unstable system

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 !

• lay tek Feb 18, 2023 at 12:03
• @OverLordGoldDragon -- kewl Feb 18, 2023 at 22:19

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".
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.