Determine impulse response given input and output: which ROC? Let's suppose I have to find the impulse response of a discrete time LTI system given a specified input and its output through the system. I think I'm going to get the $\mathcal Z$-transform of input-output and then as everyone knows 
$$H(z)=\frac{Y(z)}{X(z)}$$ 
The problem is: I want it in time domain, so I need to inverse transform $H(z)$. How will I choose the ROC (hence causality / noncausality)?
The function $H(z)$ without a specified ROC is generally not sufficient for uniquely determining the time domain sequence $h[n]$. You need more information about the desired $h[n]$. There are three types of ROCs:


the ROC is an annulus ($R_1&lt;|z|&lt;R_2$): $h[n]$ is a two-sided sequence (non-causal)
the ROC is defined by $|z|&gt;R$: $h[n]$ is a right-sided sequence
the ROC is defined by $|z|&lt;R$: $h[n]$ is a left-sided sequence


If you're looking for a stable system, then the ROC needs to include the unit circle $|z|=1$. If that does not uniquely define the ROC, then you need to know if you're looking for a left-sided, a right-sided, or a two-sided sequence.