The GATE Grind

GATE 2021 IN – Question 16

Signals and Systems · Laplace and z-transforms · 1 mark · Multiple choice

Let $u(t)$ be the unit step. The bilateral Laplace transform of $f(t)=e^tu(-t)$ is ______.

  1. $1/(s-1)$ with $\Re(s)<1$
  2. $1/(s-1)$ with $\Re(s)>1$
  3. $-1/(s-1)$ with $\Re(s)<1$
  4. $-1/(s-1)$ with $\Re(s)>1$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (C) $-1/(s-1)$ with $\Re(s)<1$

Explanation

Integrate over the negative time axis: F(s)=∫(−∞ to 0)e^((1−s)t)dt=1/(1−s)=−1/(s−1). Convergence at −∞ requires Re(1−s)>0, or Re(s)<1.

Therefore **C**, including the essential left-half-plane ROC.