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GATE 2026 IN – Question 43

Signals and Systems · Laplace and z-transforms · 2 marks · Multiple choice

Given two signals $x(t)=u(t-2)-u(t-3)$ and $y(t)=e^{-4t}u(t)$, if $z(t)=x(t)*y(t)$, then $Z(s)$ is ________.

  1. $\frac{e^{-3s}(e^s-1)}{s(s+4)}$
  2. $\frac{e^{-s}(e^{-3s}-1)}{(s+3)(s+4)}$
  3. $\frac{e^{-3s}(1-e^s)}{(s+3)(s+4)}$
  4. $\frac{e^{-3s}(1-e^s)}{s(s+4)}$

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Correct answer: (A) $\frac{e^{-3s}(e^s-1)}{s(s+4)}$

Explanation

The transforms are $X(s)=(e^{-2s}-e^{-3s})/s$ and $Y(s)=1/(s+4)$. Convolution in time becomes multiplication in the Laplace domain.

Thus $Z(s)=(e^{-2s}-e^{-3s})/[s(s+4)]=e^{-3s}(e^s-1)/[s(s+4)]$. This is **A**.