# doubt of intersection of ROC Z Transform

dear friends of StackExchange. I have a doubt of the intersection of two ROC. I have H(Z), X(Z) and and i have to determine:

\begin{align} Y(Z)= H(Z)X(Z)\end{align} $$\displaystyle$$

$ROC$ H(Z) $\cap$ $ROC$ X(Z)
$$\displaystyle$$ poles of $H(Z): p1= \left | -\frac{1}{4} \right |$ ; $p2= \left | \frac{3}{2} \right |$ $$\displaystyle$$

The system is supposed stable. Ops, sorry i forgot to write it in the initial post, under this assumption the corresponding $ROC$ of $H(Z)$ is : \begin{align} \quad&\ \left | \frac{1}{4} \right | <z< \left | \frac{3}{2} \right | \end{align} $$\displaystyle$$ pole of $X(Z): p3= \left | 2 \right |$ $$\displaystyle$$ For stability condition: \begin{align} \quad&\ z< \left |2 \right | \end{align} The intersection area is: $$\displaystyle$$

$Y(Z): ROC$ H(Z) $\cap$ $ROC$ Y(Z) = \begin{align} \quad&\ \left | \frac{1}{4} \right | < \left | z \right | < \left | \frac{3}{2} \right | \end{align}

i see it drawing the circumference and tracing the circumferences related to the poles and their corresponding ROC .

In order to answer this question we would need to know the ROCs of $H(z)$ and $Y(z)$. The pole radii only determine the limits of the possible ROCs. E.g., for $H(z)$ there are three different possible ROCs:

\begin{align}1.\quad&|z|>\frac{3}{2}\\ 2.\quad&\frac32>|z|>\frac14\\ 3.\quad&|z|<\frac14\end{align}\tag{1}

For $Y(z)$, there are two possible ROCs:

\begin{align}1.\quad&|z|<2\\ 2.\quad&|z|>2\end{align}\tag{2}

Note that each of these ROCs corresponds to a different time domain sequence.

Certain combinations of the ROCs in $(1)$ and $(2)$ overlap, others don't. Below are shown the combinations that result in an overlap of ROCs, and the resulting combined ROC. The first number is the number of the ROC of $H(z)$ given in $(1)$, the second number is the ROC of $Y(z)$ given in $(2)$:

\begin{align}1+2:\quad&|z|>2\\ 1+1:\quad&2>|z|> \frac32\\ 2+1:\quad&\frac32>|z|>\frac14\\ 3+1:\quad&|z|<\frac14\end{align}\tag{3}

All other combinations of ROCs result in no overlap.

• Thank you so much, i missed condition for initial post! I edited now – P_B Dec 17 '15 at 19:43