GATE EC 2023 Set 1 — Question 30
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Communications → Random Processes → Autocorrelation & PSD
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Question
For a real signal, which of the following is/are valid power spectral density/densities?
Correct answer
(A) S_X(ω) = (2)/(9 + ω²); (B) S_X(ω) = e^(-ω²) cos² ω
Solution
For a real-valued signal , its power spectral density must satisfy the following properties:
1.Non-negativity: for all . This is because power cannot be negative.
2.Even Symmetry: for all . This is because the autocorrelation function of a real signal is even, and its Fourier transform (the PSD) must also be even.
Evaluating the options:- Option (A): . This function is non-negative for all and is an even function (). Thus, it is a valid PSD.
- Option (B): . Since and , the product is non-negative. Both components are even functions, so their product is also even. Thus, it is a valid PSD.
- Option (C): The graph shows taking a negative value of . This violates the non-negativity property. Thus, it is invalid.
- Option (D): The graph shows a rectangular pulse that is non-zero only for positive values of . This is not an even function (). Thus, it is invalid for a real signal.
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