GATE EC 2017 Set 1 — Question 51
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Communications → Random Processes → Autocorrelation & PSD
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Question
Let be a wide sense stationary random process with the power spectral density as shown in Figure (a), where is in Hertz (Hz). The random process is input to an ideal lowpass filter with the frequency responseas shown in Figure (b). The output of the lowpass filter is .
Let be the expectation operator and consider the following statements:I.
II.
III. Select the correct option:
II.
III. Select the correct option:
Correct answer
(A) only I is true
Solution
1.Mean of the processes: For a Wide Sense Stationary (WSS) process with mean , the power spectral density contains an impulse at . The given is continuous at and has no impulse, implying .
2.Mean of the output: For an LTI system with frequency response , the mean of the output is . Since from the definition of the filter, . Thus, is true. Statement I is TRUE.
3.Power of : The average power of is given by the area under its PSD:
4.Power of : The PSD of the output is . For the ideal lowpass filter with cutoff Hz:
5.Evaluation of Statements II and III:
- Statement II: and . They are not equal. Statement II is FALSE.
- Statement III: . Statement III is FALSE.
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