GATE EC 2025 Set 1 Q32 — Random Processes

Consider a real-valued random process

$$f(t)=\sum_{n=1}^{N} a_{n} \: p(t-n T),$$

where $T>0$ and $N$ is a positive integer. Here, $p(t)=1$ for $t \in[0,0.5 T]$ 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.

1. The mean of the process $f(t)$ is independent of time $t$.
2. The autocorrelation function $E[f(t) f(t+\tau)]$ is independent of time $t$ for all $\tau$.

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