The GATE Grind

GATE 2023 EC – Question 30

Communications · Random Processes · 1 mark · Multiple select

For a real signal, which of the following is/are valid power spectral density/densities?

Diagram for GATE 2023 EC question 30
  1. $S_X(\omega)=\dfrac{2}{9+\omega^2}$
  2. $S_X(\omega)=e^{-\omega^2}\cos^2\omega$
  3. Plot (C): a rectangle of height +1 for positive $\omega$ and −1 for negative $\omega$
  4. Plot (D): a rectangle of height 1 on positive $\omega$ only, zero for negative $\omega$

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Show answer and explanation

Correct answer: (A) $S_X(\omega)=\dfrac{2}{9+\omega^2}$; (B) $S_X(\omega)=e^{-\omega^2}\cos^2\omega$

Explanation

A valid PSD of a real signal must be real, non-negative and even in $\omega$. (A) is even and positive: valid. (B) is even and non-negative: valid. (C) takes negative values and is odd: invalid. (D) is non-negative but not even (zero for $\omega<0$), which is impossible for a real signal: invalid.