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For the LTI system given below, there are three regions of convergence.

$$H(z)=\frac{5-3z^{-1}}{1-\frac53z^{-1}-\frac23z^{-2}}$$

a) Find all possible regions of convergence for this filter.

b) For each region of convergence, find the impulse response of the filter.

I tried this problem using the poles and came up with the following outcomes. Want to confirm

Are the ROCS $z<2$ and $z>-\frac{1}{3}$ and $z=3/5$? Any help in this regard is highly appreciated.

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It seems that you've correctly found the two poles: $z_{\infty,0}=2$ and $z_{\infty,1}=-\frac13$. The only other thing you need to know is that the possible ROCs are limited by the pole radii, and that there can be no poles inside an ROC.

Note that $z<2$ doesn't make much sense because $z$ is a complex number. If you mean $|z|<2$ then the pole at $z=-\frac13$ would be inside that ROC, so $|z|<2$ is not a valid ROC for the given transfer function.

The three possible ROCs for your example have the following forms:

  1. $|z|<a$
  2. $a<|z|<b$
  3. $|z|>b$

The numbers $a$ and $b$ are of course related to the poles of the transfer function. I trust that you can take it from here.

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