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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)?

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  • $\begingroup$ 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. $\endgroup$
    – Learn_and_Share
    Jun 15, 2017 at 11:26

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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.

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  • $\begingroup$ 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? $\endgroup$
    – Surfer on the fall
    Jun 15, 2017 at 13:57
  • $\begingroup$ @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. $\endgroup$
    – Matt L.
    Jun 15, 2017 at 15:06
  • $\begingroup$ Since you have the input and output in the time domain, the region of convergence for both $Y(z)$ and $X(z)$ are known. $\endgroup$
    – msm
    Jun 16, 2017 at 17:03

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