The GATE Grind

GATE 2023 EE – Question 18

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

The Fourier transform $X(\omega)$ of the signal $x(t)$ is given by

$X(\omega)=1$, for $|\omega|<W_o$; $\ \ =0$, for $|\omega|>W_o$

Which one of the following statements is true?

  1. $x(t)$ tends to be an impulse as $W_o\to\infty$.
  2. $x(0)$ decreases as $W_o$ increases.
  3. At $t=\dfrac{\pi}{2W_0}$, $x(t)=-\dfrac1\pi$
  4. At $t=\dfrac{\pi}{2W_0}$, $x(t)=\dfrac1\pi$

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Show answer and explanation

Correct answer: (A) $x(t)$ tends to be an impulse as $W_o\to\infty$.

Explanation

$x(t)=\dfrac{\sin(W_ot)}{\pi t}$. As $W_o\to\infty$ its spectrum becomes 1 for all $\omega$, so $x(t)\to\delta(t)$ (A true). $x(0)=W_o/\pi$ increases with $W_o$ (B false). At $t=\dfrac\pi{2W_o}$, $x=\dfrac{1}{\pi\cdot\pi/(2W_o)}=\dfrac{2W_o}{\pi^2}$, which is neither $\pm\frac1\pi$ (C, D false).