I am doing an exercise where I'm supposed to decide if a system is stable or not. The output is given by $y[n] = x[n- 1 ] + u [n + 2 ]$, with $u[n]$ the Heaviside step function. I know that a system is stable if and only if the impulse response is absolutely summable. So, since the impulse response is $h[n] = \delta[ n - 1] + u [n+2]$, I wrote the summation \begin{equation*} \sum _{n = -\infty} ^ {+\infty} |h[n]| =\sum_{n = -\infty} ^ {+\infty} |\delta[ n - 1] + u [n+2]| =2 +\sum_{n\neq 1, n\ge -2}^{+\infty}|u[n+2]|. \end{equation*} Since the second summation is infinity, I concluded that the system is not stable. However, my solution sheet says that the system is stable. Where am I getting wrong? Any help would be appreciated.
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since the impulse response is $h[n] = \delta[ n - 1] + u [n+2]$
It's not. Your system isn't LTI so it doesn't have an impulse response (in the LTI sense) and you can't use an LTI stability criteria.
You need to go back to the original definition of BIBO stability (Bounded Input -> Bounded Output).
$$ |y[n]| \leq B \; \forall \; n \in \mathbb{Z} $$
In this case it's trivial to show that if $|x[n]| < M$ then
$$|y[n]| < M+1$$
So the output is bounded if the input is bounded and the system is stable.
Where am I getting wrong?
You were assuming that the input gets convolved with the unit step. But that's not the case: the unit step is just added to the output.
