Autocorrelation and PSD Let $X(t)$ and $Y(t)$ be two orthogonal processes with power spectral densities 
$$S_{xx}(f) = S_{yy}(f)=\begin{cases}
1-\lvert f\rvert, &amp; \lvert f\rvert&lt;1
\\[1ex]
0,&amp; \text{otherwise}
\end{cases}$$

Define a new process $Z(t) = Y(t)-X(t-1)$. Determine and sketch the power spectral density $S_{zz}(f)$. 

The solution is the following:
The autocorrelation function of the process is 
$$R_{zz}= E \bigg\{\big[Y(t-\tau)-X(t+\tau-1)\big]\big[Y(t)-X(t-1)\big] \bigg\} $$ 
And now comes the part which I do not understand at all. 
\begin{align}
R_{zz}&amp;=R_{yy}(\tau)-R_{yx}(\tau+1)-R_{xy}(\tau-1)+R_{xx}(\tau)\\
&amp;=R_{yy}(\tau) + R_{xx}(\tau)\ \quad\text{since}\quad R_{yx} = R_{xy} = 0 \quad\text{from orthogonality.}
\end{align}

Therefore, $S_{zz}(f)=2S_{yy}(f)=2S_{xx}(f)$ as shown below.


Can anyone explain it like I am five?
HINT:

Expand the terms inside the expectation $E\{\ldots\}$, and then use the fact that the expectation of a sum equals the sum of the individual expectations (due to linearity).