1. The problem statement, all variables and given/known data I'm posting this question here at this point. I am having difficulty understanding autocorrelation in terms of solving the problem below. I don't seem to understand the math behind this. A white noise process W(t) with unity (N_0/2 = 1) power spectral density is input to a linear system. The output of the linear system is X(t), where X(t) = W(t) - W(t - 1) Determine the autocorrelation of X(t) and sketch it. 2. Relevant equations We can define the autocorrelation function of a stochastic process X(t) as the expectation of the product of two random variables X(t_1) and X(t_2), obtained by sampling the process X(t) at times t_1 and t_2 respectively. So we can write f_(X(t_1),X(t_2))(x_1,x_2) is the join probability density function of the process X_(t) sampled at times t_1 and t_2 M_(XX)(t_1,t_2) is used to emphasize the fact that this is a second order moment. For M_(XX)(t_1,t_2) to dependent on the time difference t_2 - t_1, we have R_(XX)(t_2 - t_1) Two different symbols for the autocorrelation function M_(XX)(t_1,t_2) and R_(XX)(t_2 - t_1) to denote that R_(XX)(t_2 - t_1) is the autocorrelation function specifically for a weak stationary process. Let τ denote a time shift; that is, t = t_2 and τ = t_1 - t_2 3. The attempt at a solution I understand that the first term on the last line is indeed equal to . I'm however unsure what to do with the other three terms. The solution sets the other three terms to different autocorrelation functions and I'm not sure how these other three terms are autocorrelation functions as well based off of the definition. Here's what the solution is. I don't understand how it went from the second line to the third line. Any help would be greatly appreciated. I also don't understand how how the solution goes from the third line to the fourth line. It seems to just simply replacing the autocorrelation functions with dirac delta functions. I'm not sure how these are equal in any way.