# How is the ROC of a transform function determined?

Suppose I have $$x[n]$$ and $$y[n]$$, and I calculate their respective Z-transforms $$X(z)$$ and $$Y(z)$$ as well as their respective ROCs.

Calculating $$H(z)$$ is as simple as calculating the quotient of $$\frac{Y(z)}{X(z)}$$, but how do I get $$H(z)$$'s ROC?

The ROC of $$Y(z)=X(z)H(z)$$ is the intersection of the ROCs of $$X(z)$$ and $$H(z)$$, unless there are pole-zero cancellations, which change the ROC of the product. In the given example we have to take into account the pole-zero cancellation at $$z_0=2$$. We can define a new function $$X'(z)=X(z)(1-2z^{-1})$$ with the pole at $$z_0=2$$ removed. We do not need to define a new $$H'(z)$$ with the corresponding zero removed, because that zero doesn't influence the ROC anymore (since the corresponding pole has already been removed). The ROC of $$Y(z)$$ is now the intersection of the ROCs of $$X'(z)$$ ($$|z|>\frac13$$) and $$H(z)$$. Consequently, the only possible ROC for $$H(z)$$ is $$|z|>\frac23$$. This means that the corresponding system is causal and stable, because the ROC is outside the pole with the largest magnitude and the ROC contains the unit circle.