GATE 2024 EE – Question 16

Signals and Systems · Applications of Fourier Transform for continuous and discrete time signals, Laplace Transform and Z transform · 1 mark · Multiple choice

If $u(t)$ is the unit step function, then the region of convergence (ROC) of the Laplace transform of the signal

$$x(t)=e^{t^2}\left[u(t-1)-u(t-10)\right]$$

is

$-\infty<\text{Re}(s)<\infty$
$\text{Re}(s)\ge10$
$\text{Re}(s)\le1$
$1\le\text{Re}(s)\le10$

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Correct answer: (A) $-\infty<\text{Re}(s)<\infty$

Explanation

$x(t)$ is non-zero only on the finite interval $[1,10]$ and is bounded there, so $\int_1^{10}e^{t^2}e^{-st}dt$ converges for every $s$. The ROC is the entire $s$-plane.