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What is the causality & stability status for three cases shown (aso in attached photo) ?

$$H(z) = \frac{z(z-1)}{(z+1)(z+\frac{1}{3})} $$

for three possible regions of convergence as:

a-) |z| > 1

b-) |z| < 1/3

c-) 1/3 < |z| < 1

Am i ok as below?

a) Causal & marginally stable

b) anti-causal & unstable

c) non-causal & marginally stable

enter image description here

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Yes, all answers given by you are fine.

Causal: If ROC is outside the outermost pole

Left sided: If ROC is inside the innermost pole

Stable: If ROC contains the unit circle (marginally stable if it touches unit circle)

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I will only give you hints 1. Casual if |Z|>|a| 2. Stable if Roc contains unit circle So non causal if |Z|<|a|, unstable if Roc don't contain unit circle & marginally stable if poles are on unit circle.

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