# 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)?

• It's better to give a specific example. Anyway, when you have $H(z)$, you can determine the ROC by looking at the poles locations (in the complex plane) such that in the ROC their must be no poles. Jun 15, 2017 at 11:26

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:

1. the ROC is an annulus ($R_1<|z|<R_2$): $h[n]$ is a two-sided sequence (non-causal)
2. the ROC is defined by $|z|>R$: $h[n]$ is a right-sided sequence
3. the ROC is defined by $|z|<R$: $h[n]$ 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.

• this is to say: one pair input/output isn't enough to determine impulse response (how the system really works). I am not very sure about it... isn't the system linear? Jun 15, 2017 at 13:57
• @Surferonthefall: If you just have the Z-transforms (without ROC), then there's usually no unique solution. But if you know that the system is causal then the ROC is implied and you get a unique solution. Jun 15, 2017 at 15:06
• Since you have the input and output in the time domain, the region of convergence for both $Y(z)$ and $X(z)$ are known.
– msm
Jun 16, 2017 at 17:03