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GATE 2016 EC (EC3) – Question 17

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

If $x(t)=[\sin t/(\pi t)]*[\sin t/(\pi t)]$, where * denotes convolution, x(t) is

  1. $\sin t/(\pi t)$
  2. $\sin(2t)/(2\pi t)$
  3. $2\sin t/(\pi t)$
  4. $[\sin t/(\pi t)]^2$

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Show answer and explanation

Correct answer: (A) $\sin t/(\pi t)$

Explanation

The continuous-time Fourier transform of $\sin t/(\pi t)$ is a unit-height rectangle for |ω|<1.

Time convolution corresponds to multiplying the transforms. Squaring this 0-or-1 rectangle leaves it unchanged except at irrelevant endpoint conventions. Its inverse transform is the original sinc signal, so **A**.

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