Q1: The autocorrelation function of an ergodic random process is given by __________ .

The expectations or ensemble average of a random process x(t) are averages "across the process". The DC value of x(t) is defined by the time average The other time average of particular interest is the autocorrelation function R_x(t, T) defined in terms of the sample function x(t) observed over the interval - T ≤ t ≤ T. Following equation, we may formally define the time-averaged autocorrelation function of a sample function x(t) as follows : This second time-average should also be viewed as a random variable with a mean and variance of its own. In a manner similar to ergodicity of the mean, we say that the process x(t) is ergodic in the autocorrelation function if the following two limiting conditions are satisfied : .