In many applications, discrete-time random signals arise through periodic sampling of co...

In many applications, discrete-time random signals arise through periodic sampling of continuous-time random signals. We are concerned in this problem with a derivation of the sampling theorem for random signals. Consider a continuous-time, stationary, random process defined by the random variables {x_a(t)}, where t is a continuous variable. The autocorrelation function is defined as

and the power density spectrum is

A discrete-time random process obtained by periodic sampling is defined by the set of random variables {x[n]}, where x[n] = x_a(nT ) and T is the sampling period.

(a) What is the relationship between Φ_xx[n] and Φ_xcxc(τ)?

(b) Express the power density spectrum of the discrete-time process in terms of the power density spectrum of the continuous-time process.

(c) Under what condition is the discrete-time power density spectrum a faithful representation of the continuous-time power density spectrum?