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To preface, this is not a homework related question but purely for self-study purposes.

If I am given the following Pole-Zero Plot: (Source: Berkeley Exam1)

  • How would I go about trying to determine whether the system is causal or stable?
  • Similarly, why should the region of convergence be $0.5 < |z| < 2 $ for stability?

I know that for stability, the ROC must include the unit circle. And for causality, $h[n]=0$ for $ n<0$. Other than that, I am having trouble applying those rules to a pole-zero plot.

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For systems with more than one pole (with different radii), there are $3$ types of ROCs:

  1. inside the circle with a radius corresponding to the smallest pole radius; the corresponding time-domain sequence is left-sided

  2. outside the circle with a radius corresponding to the largest pole radius; the corresponding time-domain sequence is right-sided

  3. in a ring defined by the radii $r_1$ and $r_2$ of any two poles with $r_1<r_2$, with no other pole at a radius in the interval $(r_1,r_2)$; the corresponding time-domain sequence is two-sided

In your example you have case $3$ above, because that's the only ROC that includes the unit circle, and, consequently, corresponds to a stable system. Both other possible ROCs (inside the circle with radius $0.5$ and outside the circle with radius $2$) do not include the unit circle, and, consequently, do not correspond to impulse responses of stable systems. Since a ROC equal to a ring between two poles corresponds to a two-sided sequence, the system can't be causal. A causal system has a ROC outside a circle (case $2$ in the list above).

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