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GATE 2021 EE – Question 42

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

Let $f(t)$ be an even function, i.e. $f(-t)=f(t)$ for all $t$. Let the Fourier transform of $f(t)$ be defined as $F(\omega)=\int_{-\infty}^{\infty}f(t)e^{-j\omega t}dt$. Suppose $\dfrac{dF(\omega)}{d\omega}=-\omega F(\omega)$ for all $\omega$, and $F(0)=1$. Then

  1. $f(0)<1$
  2. $f(0)>1$
  3. $f(0)=1$
  4. $f(0)=0$

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Show answer and explanation

Correct answer: (A) $f(0)<1$

Explanation

The equation gives $F(\omega)=e^{-\omega^2/2}$. Then $f(0)=\frac{1}{2\pi}\int F(\omega)d\omega=\frac{\sqrt{2\pi}}{2\pi}=0.399<1$.