Quick questions regarding RoCs with Z transforms

[pjcircle]

Hi I was wondering when there is only one pole for a function's z-transform then the RoC can only be either left or right sided not two sided and then two sided requires having two or more poles so you actually have an "inner" region to be between then? Or does the entire z-plane just not including the ring of that one pole count as 2sided?

 

[jvicens]

Hello,

Thank you for your question. The number of poles in a function's z-transform does indeed affect the region of convergence (ROC). If there is only one pole, the ROC can only be either left-sided or right-sided, as you mentioned. This is because the presence of a pole at a specific point in the z-plane creates a singularity, meaning that the function is not defined at that point.

In order for a function's z-transform to have a two-sided ROC, there must be at least two poles in the z-plane. This is because the presence of multiple poles creates a "ring" in the z-plane, where the function is not defined. This ring acts as a barrier, separating the two-sided ROC from the rest of the z-plane.

So, in summary, the entire z-plane excluding the ring of the one pole would not count as a two-sided ROC. The two-sided ROC would only be the area between the poles, while the one-sided ROC would be the rest of the z-plane.

I hope this helps clarify the concept of ROC and how it is affected by the number of poles in a function's z-transform. Let me know if you have any further questions.