The GATE Grind

GATE 2025 EC – Question 42

Communications · Random Processes · 2 marks · Multiple choice

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, $E[\cdot]$ is the expectation operation.)

  1. (i) is TRUE and (ii) is FALSE
  2. Both (i) and (ii) are TRUE
  3. Both (i) and (ii) are FALSE
  4. (i) is FALSE and (ii) is TRUE

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Show answer and explanation

Correct answer: (A) (i) is TRUE and (ii) is FALSE

Explanation

(i) $E[f(t)]=\sum_nE[a_n]p(t-nT)=0$ for all $t$, which is independent of $t$: TRUE. (ii) Because the $a_n$ are uncorrelated, $E[f(t)f(t+\tau)]=\sum_np(t-nT)\,p(t+\tau-nT)$. For $\tau=0$ this is $\sum_np(t-nT)$, which is 1 when $t$ lies inside a pulse and 0 outside, so it depends on $t$: FALSE. The process is cyclostationary, not wide-sense stationary.