# Summary of ROC Rules

This is a very short guide on how to find all possible outcomes of a system where Region of Convergence (ROC) and the original signal is not known.

### Summary of ROC Rules

- For a causal system the ROC extends outwards.
- For a non-causal system the ROC extends inwards.
- For a two-sided system, the ROC can extend inwards or outwards from every pole.
- The ROC cannot contain any poles
- The system is stable if the unity circle is included in the ROC

### One Pole System Example

$$X(z) = \frac{1}{1-0.5z^{-1}}$$

1. Causal and Stable 2. Non-Causal and unstable

$x[n] = 0.5^nu[n]$ $x[n] = -0.5^u[-n-1]$

Rule #1 and #5 Rule #2

$$X(z) = \frac{1}{1-2z^{-1}}$$

1. Causal and unstable 2. Non-Causal and stable

$x[n] = 2^nu[n]$ $x[n] = -2^u[-n-1]$

Rule #1 Rule #2 and #5

### Multiple Pole System Example

$$ X(z) = \frac{1}{1-0.5z^{-1}} + \frac{1}{1-2z^{-1}} $$

This system has four possible ROC's, that is: there are four systems in the time domain that shares this z-transform.

1. Causal: $$ x[n] = 0.5^nu[n] + 2^nu[n]$$

Rule #1 and #4

2. Non-causal $$x[n] = -0.5^nu[-n-1] - 2^nu[-n-1]$$

Rule #2 and #4

3. Two sided: $$x[n] = 0.5^nu[n] - 2^nu[-n-1]$$

Rule #3 and #5

4. Two sided: $$x[n] = -0.5^nu[-n-1] + 2^nu[n]$$

Rule #3

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