Question

Let X(t) and Y(t) be both zero-mean and wide sense stationary random processes. Consider the random process Z(t) defined by

Z(t) = X(t) +Y(t)

Determine the autocorrelation function and the power spectral density of Z(t)

• If X(t) and Y(t) are jointly wide sense stationary.

Answer 1

The autocorrelation function of Z(t) is the sum of the autocorrelation functions of X(t) and Y(t). The power spectral density (PSD) of Z(t) is the sum of the individual PSDs for X(t) and Y(t).

Step-by-step explanation:

Given that both X(t) and Y(t) are zero-mean and wide sense stationary random processes, and Z(t) is defined as Z(t) = X(t) + Y(t), we can determine the autocorrelation function and power spectral density of Z(t) if X(t) and Y(t) are jointly wide sense stationary.

The autocorrelation function of Z(t) can be calculated as the sum of the autocorrelation functions of X(t) and Y(t) because X(t) and Y(t) are independent and wide sense stationary. Thus, the autocorrelation function of Z(t) is the sum of the autocorrelation functions of X(t) and Y(t).

The power spectral density (PSD) of Z(t) can be obtained by taking the Fourier transform of the autocorrelation function of Z(t). The PSD for Z(t) is the sum of the individual PSDs for X(t) and Y(t).