GATE 2024 EE – Question 33

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

Let $X(\omega)$ be the Fourier transform of the signal

$$x(t)=e^{-t^4}\cos t,\qquad-\infty<t<\infty.$$

The value of the derivative of $X(\omega)$ at $\omega=0$ is ______ (rounded off to 1 decimal place).

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Correct answer: -0.01 to 0.01

Explanation

$\dfrac{dX}{d\omega}\Big|_{\omega=0}=-j\int_{-\infty}^\infty t\,x(t)\,dt$. Here $x(t)$ is even, so $t\,x(t)$ is odd and the integral vanishes: the derivative is 0.0.