Question

The Region of Convergence (ROC) of the signal \( x(n) = \delta(n - k), k>0 \) is:

• A. \( z = \infty \)
• B. \( z = 0 \)
• C. Entire \( z \)-plane, except at \( z = 0 \)
• D. Entire \( z \)-plane, except at \( z = \infty \)

Hint:

For \( x(n) = \delta(n - k) \), the Z-transform is \( X(z) = z^{-k} \), with ROC excluding \( z = 0 \).

Answer 1

The Correct Option is C

Step 1: Find the Z-transform of \( x(n) \). Since \( x(n) = \delta(n - k) \), its Z-transform is: \[ X(z) = z^{-k}. \] 
Step 2: Find the ROC. - The function \( z^{-k} \) is well-defined for all \( z \neq 0 \). - So, the ROC is entire \( z \)-plane except \( z = 0 \). 
Step 3: Selecting the correct option. Since the correct ROC is entire \( z \)-plane except at \( z = 0 \), the answer is (C).