GATE 2016 EC – Question 45
Consider the signal
$$x[n] = 6\delta[n+2] + 3\delta[n+1] + 8\delta[n] + 7\delta[n-1] + 4\delta[n-2].$$
If $X(e^{j\omega})$ is the discrete-time Fourier transform of $x[n]$, then $\frac{1}{\pi}\int_{-\pi}^{\pi} X(e^{j\omega})\sin^2(2\omega) \, d\omega$ is equal to ________
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Correct answer: 7.9 to 8.1
Explanation
Write $\sin^2(2\omega) = \frac{1 - \cos(4\omega)}{2}$. The integral of $X(e^{j\omega})$ over a full period is $2\pi\,x[0]$. The $\cos(4\omega)$ term picks out $x[4]$ and $x[-4]$, which are both 0. So the result is $\frac{1}{\pi} \times \frac{1}{2} \times 2\pi\,x[0] = x[0] = 8$.