The GATE Grind

GATE 2022 EC – Question 15

Networks, Signals and Systems · Continuous-time Signals · 1 mark · Multiple choice

The Fourier transform $X(j\omega)$ of the signal

$$x(t)=\frac{t}{(1+t^2)^2}$$

is ________.

  1. $\frac{\pi}{2j}\,\omega e^{-|\omega|}$
  2. $\frac{\pi}{2}\,\omega e^{-|\omega|}$
  3. $\frac{\pi}{2j}\,e^{-|\omega|}$
  4. $\frac{\pi}{2}\,e^{-|\omega|}$

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Correct answer: (A) $\frac{\pi}{2j}\,\omega e^{-|\omega|}$

Explanation

The transform of $\frac{1}{1+t^2}$ is $\pi e^{-|\omega|}$. Since $\frac{d}{dt}\frac{1}{1+t^2}=\frac{-2t}{(1+t^2)^2}$, we have $x(t)=-\frac12\frac{d}{dt}\frac{1}{1+t^2}$, so $X(j\omega)=-\frac12(j\omega)\pi e^{-|\omega|}=\frac{\pi}{2j}\omega e^{-|\omega|}$.