Question

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

Hint:

The autocorrelation function of a random process can depend on time if the process is not stationary or if the function involved in the process varies over time. In this case, the autocorrelation function depends on the time \( t \) because the non-zero values of \( p(t) \) are time-dependent.

Answer 1

The Correct Option is A

Step 1: Mean of the process \( f(t) \)
The mean of \( f(t) \) is computed as: \[ E[f(t)] = E\left[\sum_{n=1}^{N} a_n p(t - nT)\right]. \] Since \( a_n \) are zero-mean independent random variables, the expectation of each \( a_n \) is 0. Thus: \[ E[f(t)] = 0 \quad {for all } t. \] Therefore, the mean of the process \( f(t) \) is indeed independent of time \( t \), and (i) is TRUE. 
Step 2: Autocorrelation function
The autocorrelation function is: \[ R_f(\tau) = E[f(t)f(t+\tau)] = E\left[\sum_{n=1}^{N} a_n p(t - nT) \sum_{m=1}^{N} a_m p(t + \tau - mT)\right]. \] Since \( a_n \) are independent, the autocorrelation will depend on \( t \) because the function \( p(t) \) is not constant over time (it is non-zero only in the range \( [0, 0.5T] \)). Thus, the autocorrelation function is not independent of time. So, (ii) is FALSE. 
Conclusion:
Statement (i) is TRUE because the mean is constant and independent of time.
Statement (ii) is FALSE because the autocorrelation function depends on time \( t \). Thus, the correct answer is (A): (i) is TRUE and (ii) is FALSE.