Calculate Average and Correlation of WSS Random Processes Given two stochastic processes, $X[n]$ and $Y[n]$, both being WSS (wide state stationary) and independents. What would be the Average and Autocorrelation function of $Z[n] = Y[n]  X[n]$?
Is the resulting $Z[n]$ process also WSS? If so, can the PSD (power spectral density) be calculated?
Thank you.
 A: Let $(X_n)$ and $(Y_n)$ be independent wide-sense stationary (WSS) processes with averages $\mu_X=EX_n$ and $\mu_Y=EY_n$, covariance functions $C_X(m)=E(X_{n+m}-EX_{n+m})(X_{n}-EX_{n})$ and $C_Y(m)=E(Y_{n+m}-EY_{n+m})(Y_{n}-EY_{n})$, correlation functions $R_X(m)=C_X(m)/C_X(0)$ and $R_Y(m)=C_Y(m)/C_Y(0)$, and power spectral densities (PSDs) $f$ and $g$, so that 
$R_X(m)=\int_{-\infty}^\infty e^{2\pi i mx} f(x)dx$ and $R_Y(m)=\int_{-\infty}^\infty e^{2\pi i mx} g(x)dx$. Let $Z_n=X_n Y_n$. 
Then the process $(Z_n)$ is also WSS, with the average 
$$\mu_Z=EX_n Y_n=EX_n\,EY_n=\mu_X \mu_Y,$$
covariance function 
\begin{align*}
 C_Z(m)&=E(X_{n+m}Y_{n+m}-\mu_X\mu_Y)(X_{n}Y_{n}-\mu_X\mu_Y) \\ 
 &=EX_{n+m}X_{n}\,EY_{n+m}Y_n-\mu_X^2\mu_Y^2\\ 
 &=(C_X(m)+\mu_X^2)(C_Y(m)+\mu_Y^2)-\mu_X^2\mu_Y^2\\ 
 &=C_X(m)C_Y(m)+\mu_Y^2C_X(m)+\mu_X^2C_Y(m),  
\end{align*}
correlation function 
\begin{equation*}
 R_Z(m)=\frac{C_Z(m)}{C_Z(0)}  
 =a\,R_X(m)R_Y(m)
 +b\,R_X(m)
 +c\,R_Y(m) 
\end{equation*}
with $a:=\frac{C_X(0)C_Y(0)}{C_Z(0)}$, $b:=\frac{C_X(0)\mu_Y^2}{C_Z(0)}$, $c:=\frac{C_Y(0)\mu_X^2}{C_Z(0)}$, 
and PSD 
\begin{equation*}
 h=a f*g+bf+cg,
\end{equation*}
where $f*g$ is the convolution of $f$ and $g$.  
