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GATE 2024 IN – Question 40

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

A signal has bilateral Laplace transform $X(s)=1/(s^2+13s+42)$ with region of convergence $-7<\operatorname{Re}(s)<-6$. If $u(t)$ is the unit step, which signal is $x(t)$?

  1. $-e^{-6t}u(t)-e^{-7t}u(-t)$
  2. $-e^{-6t}u(-t)-e^{-7t}u(t)$
  3. $e^{-6t}u(t)-e^{-7t}u(-t)$
  4. $-e^{-6t}u(-t)-e^{-7t}u(-t)$

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Show answer and explanation

Correct answer: (B) $-e^{-6t}u(-t)-e^{-7t}u(t)$

Explanation

Partial fractions give $X(s)=1/(s+6)-1/(s+7)$. The first term must be left-sided for $\operatorname{Re}(s)<-6$, giving $-e^{-6t}u(-t)$. The second must be right-sided for $\operatorname{Re}(s)>-7$, giving $-e^{-7t}u(t)$. Their ROC intersection is the stated strip.

Answer: **B**.