GATE EC 2025 Set 1 — Question 42
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Communications → Random Processes → Random Process & Stationarity
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Question
Consider a real-valued random process where and is a positive integer. Here, for and otherwise. The coefficients are pairwise independent, zero-mean unit-variance random variables. Read the following statements about the random process and choose the correct option.(i) The mean of the process is independent of time .
(ii) The autocorrelation function is independent of time for all . (Here, is the expectation operation.)
(ii) The autocorrelation function is independent of time for all . (Here, is the expectation operation.)
Correct answer
(A) (i) is TRUE and (ii) is FALSE
Solution
The random process is defined as . We are given that and for all , and the coefficients are pairwise independent.Statement (i): Mean of the processSince for all , the mean is independent of time . Thus, statement (i) is TRUE.Statement (ii): Autocorrelation functionSince are zero-mean and pairwise independent, .To check if this is independent of , consider :This sum represents a sequence of pulses of width occurring at intervals of . Specifically, when for some , and otherwise. Since the value of the autocorrelation function depends on the specific time , it is not independent of time. Thus, statement (ii) is FALSE.Therefore, (i) is TRUE and (ii) is FALSE.
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