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GATE 2014 EE (EE1) – Question 45

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

$F(\omega)=\int_{-\infty}^{\infty}f(t)e^{-j\omega t}dt$ and $g(t)=\int_{-\infty}^{\infty}F(u)e^{-jut}du$. What is the relation between g(t) and f(t)?

  1. g(t) is always proportional to f(t).
  2. g(t) is proportional to f(t) if f(t) is even.
  3. g(t) is proportional to f(t) only if f(t) is sinusoidal.
  4. g(t) is never proportional to f(t).

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Show answer and explanation

Correct answer: (B) g(t) is proportional to f(t) if f(t) is even.

Explanation

With the stated angular-frequency convention, Fourier duality gives g(t)=2πf(−t). If f is even, f(−t)=f(t), so g is proportional to f, establishing **B**. Evenness is sufficient, not necessary:an odd function is also proportional with factor−2π.