Digital Communication Systems - Autocorrelation

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Homework Statement

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.

Homework Equations

Let τ denote a time shift; that is, t = t_2 and τ = t_1 - t_2

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.

 

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sorry I didn't mean to post