GATE 2023 EC Q.30: solution and answer

Official answer: A, BSource: Official answer key, GATE 2023 EC (IIT Kanpur)

1. A valid power spectral density of a real signal must satisfy three conditions: (i) $S_X(\omega )\ge 0$ for all $\omega$; (ii) $S_X(-\omega )=S_X(\omega )$, because $R_X(\tau )$ is real and even for a real signal; (iii) ${\int }_{-\infty }^{\infty }{S}_X(\omega )d\omega <\infty$, because the average power is finite. The spectrum here is the two-sided PSD in rad/s.

2. (A) $S_X(\omega )=\frac{2}{9+{\omega }^2}$ is positive and even in $\omega$. Also $${\int }_{-\infty }^{\infty }\frac{2d\omega }{9+{\omega }^2}=\frac{2}{3}{\left[{\tan }^{-1}\frac{\omega }{3}\right]}_{-\infty }^{\infty }=\frac{2\pi }{3}<\infty .$$ It is the transform of $\frac{1}{3}{e}^{-3|\tau |}$, since $e^{-a|\tau |}\leftrightarrow \frac{2a}{a^2+{\omega }^2}$ with $a=3$. Valid.

3. (B) $S_X(\omega )=e^{-{\omega }^2}{\cos }^2\omega$ is a product of non-negative terms, so it is $\ge 0$. It is even, because ${\cos }^2(-\omega )={\cos }^2\omega$. It is integrable, since $0\le S_X(\omega )\le e^{-{\omega }^2}$ and $\int e^{-{\omega }^2}d\omega =\sqrt{\pi }$. Valid.

4. (C) The plot is $+1$ over a band of positive $\omega$ and $-1$ over a band of negative $\omega$. It takes negative values, which violates $S_X(\omega )\ge 0$, and it is odd rather than even. Not valid.

5. (D) The plot is non-zero only for $\omega >0$, so $S_X(-\omega )\ne S_X(\omega )$. A one-sided spectrum corresponds to a complex autocorrelation, which cannot occur for a real signal. Not valid.

6. Hence the valid power spectral densities are (A) and (B).