Question

If \( R_1 \) is the region of convergence of \( x(n) \) and \( R_2 \) is the region of convergence of \( y(n) \), then the region of convergence of \( x(n) * y(n) \) is _______.

• A. \( R_1 + R_2 \)
• B. \( R_1 - R_2 \)
• C. \( R_1 \cup R_2 \)
• D. \( R_1 \cap R_2 \)

Hint:

For convolution, ROC = intersection of ROCs of the individual signals!

Answer 1

The Correct Option is D

Step 1: Concept of ROC (Region of Convergence)
For the convolution of two signals \( x(n) \) and \( y(n) \), the Z-transform of the result is the product of their individual Z-transforms: \[ Z\{x(n) * y(n)\} = X(z) \cdot Y(z) \] Step 2: Convolution Property
The ROC of the resulting convolution is the intersection of the ROCs of the individual signals. So: \[ ROC_{x*y} = R_1 \cap R_2 \] Step 3: Avoiding Misconceptions
- Union: not always analytic in the union
- Addition/Subtraction: not valid ROC operations
Conclusion:
Option (4) is correct — the ROC of convolution is intersection of individual ROCs.