GATE 2020 EC – Question 62
$X(\omega)$ is the Fourier transform of $x(t)$ shown below. The value of $\int_{-\infty}^{\infty}|X(\omega)|^2\,d\omega$ (rounded off to two decimal places) is ________.
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Correct answer: 58.50 to 58.80
Explanation
By Parseval's theorem, $\int|X(\omega)|^2d\omega=2\pi\int x^2(t)\,dt$. The signal is piecewise linear through $(-1,0),(0,1),(1,3),(2,1),(3,0)$. The energy over each segment is $\frac13$, $\frac{13}{3}$, $\frac{13}{3}$ and $\frac13$, for a total of $\frac{28}{3}$. So the integral is $2\pi\times\frac{28}{3}=58.64$.