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GATE 2015 EE (EE2) – Question 39

Signals and Systems · Applications of Fourier Transform for continuous and discrete time signals, Laplace Transform and Z transform · 2 marks · Multiple choice

The signal is $x(t)=e^{j10t}$ for |t|≤1 and zero otherwise. Its Fourier transform is

  1. $2\sin(\omega-10)/(\omega-10)$
  2. $2e^{j10}\sin(\omega-10)/(\omega-10)$
  3. $2\sin\omega/(\omega-10)$
  4. $e^{j10\omega}2\sin\omega/\omega$

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Correct answer: (A) $2\sin(\omega-10)/(\omega-10)$

Explanation

Apply the Fourier-transform definition over the nonzero support: $X(\omega)=\int_{-1}^{1}e^{-j(\omega-10)t}dt$.

Symmetric integration yields $2\sin(\omega-10)/(\omega-10)$, with limiting value 2 at ω=10. This is **A**; modulation shifts the rectangular pulse spectrum without adding an unrelated constant phase factor.

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