Question

If \( u(t) \) is the unit step function, then the region of convergence (ROC) of the Laplace transform of the signal \( x(t) = {e^t^2 [ u(t-1) - u(t-10) ] \) is:}

• A. \( -\inftyLt;\text{Re}(s)Lt;\infty \)
• B. \( \text{Re}(s) \geq 10 \)
• C. \( \text{Re}(s) \leq 1 \)
• D. \( 1 \leq \text{Re}(s) \leq 10 \)

Hint:

For bounded finite-duration signals, the ROC is always the entire \( s \)-plane.

Answer 1

The Correct Option is A

Step 1: Signal analysis.
The given signal is:
\[ x(t) = e^t [ u(t-1) - u(t-10) ]. \] This is a bounded signal for the time interval \( t \in [1, 10] \).
Step 2: Determining ROC.
For finite-duration signals, the ROC of the Laplace transform includes the entire \( s \)-plane.
Hence, the correct option is (1) \( -\inftyLt;\text{Re}(s)Lt;\infty \).