GATE 2025 EC Q.42: solution and answer

GATE 2025 · ECQ.42 · 2 marks · MCQ

Consider a real-valued random process f(t) = sum_{n=1}^{N} a_n p(t - nT), where T > 0 and N is a positive integer. Here, p(t) = 1 for t in [0, 0.5T] and 0 otherwise. The coefficients a_n 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 f(t) is independent of time t. (ii) The autocorrelation function E[f(t)f(t+tau)] is independent of time t for all tau. (Here, E[.] is the expectation operation.)

(A)
(i) is TRUE and (ii) is FALSE
(B)
Both (i) and (ii) are TRUE
(C)
Both (i) and (ii) are FALSE
(D)
(i) is FALSE and (ii) is TRUE

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Answer and solution

ANSWER

Option (A): statement (i) is true and statement (ii) is false.

Official answer: ASource: Official answer key, GATE 2025 EC (IIT Roorkee)

Mean: $E [f (t)] = \sum_{n = 1}^{N} E [a_{n}] p (t - n T) = 0$ because every $a_{n}$ has zero mean. This is a constant, so it does not depend on $t$ . Statement (i) is TRUE.
Autocorrelation: $E [f (t) f (t + τ)] = \sum_{n = 1}^{N} \sum_{m = 1}^{N} E [a_{n} a_{m}] p (t - n T) p (t + τ - m T)$ .
For $n \neq = m$ , independence and zero mean give $E [a_{n} a_{m}] = E [a_{n}] E [a_{m}] = 0$ . For $n = m$ , unit variance and zero mean give $E [a_{n}^{2}] = 1$ . So $E [a_{n} a_{m}] = δ_{nm}$ and $R (t, t + τ) = n = 1 \sum N p (t - n T) p (t + τ - n T) .$
Take $τ = 0$ : $R (t, t) = \sum_{n = 1}^{N} p^{2} (t - n T)$ . Each pulse is nonzero only on $[n T, n T + 0.5 T]$ , and these intervals do not overlap because $0.5 T < T$ . So $R (t, t) = 1$ for $t \in [n T, n T + 0.5 T]$ and $R (t, t) = 0$ for $t \in (n T + 0.5 T, (n + 1) T)$ .
For example, $R (T, T) = 1$ (the $n = 1$ pulse is on), but $R (1.75 T, 1.75 T) = 0$ , since $0.75 T \in / [0, 0.5 T]$ and $- 0.25 T \in / [0, 0.5 T]$ and no other pulse is on. So the value changes with $t$ at $τ = 0$ .
The autocorrelation therefore depends on $t$ (the process is cyclostationary with period $T$ , not wide-sense stationary), so it is not independent of $t$ for all $τ$ . Statement (ii) is FALSE.
(i) TRUE and (ii) FALSE is option (A).

KEY IDEA

The mean is zero, so it is constant. The autocorrelation at $τ = 0$ is $\sum p^{2} (t - n T)$ , which switches between 1 and 0 as $t$ moves through each period. A time-dependent autocorrelation means the process is not wide-sense stationary.

WATCH OUT FOR

Seeing a constant mean and a zero-mean, uncorrelated set of coefficients and assuming the process is stationary. Stationarity also needs the autocorrelation to be free of $t$ , and a pulse train with a fixed timing grid fails that test.

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GATE 2025 EC Q.42: solution and answer

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