# The power spectral density of random process,which is the multiplication and the sum of the other iid random process

Assume $X_n$ is an iid gaussian random process with zero mean and variance $\sigma^2$, and $U_n$ be an iid binary random process with $P_r\{ U_{n}=1\}=P_r\{U_n=-1\}=0.5$, and $\{U_n\}$ is independent of $\{X_n\}$, now let $Z_n=X_n U_n$ ; $Y_n=X_n+U_n$ ; $W_n=X_n+U_0$,please find their power spectral density .

I take this question as reference How to prove the autocorrelation of this random variable is just related to time difference?

For $Z_n$,$Z_n$ has the same distribution as $X_n$,but multiply $0.5$,so $Z_n$

\begin{align*} R_{ZZ}(\tau) &= \left\{\begin{array}{ll} 0.5\sigma^2 & \text{if } m = n \\ 0 & \text{otherwise} \end{array}\right. \\ &= \left\{\begin{array}{ll} 0.5\sigma^2 & \text{if } m-n = 0 \\ 0 & \text{if } m-n \neq 0 \end{array}\right. \end{align*}

For $Y_n$,$Y_n$ has the same distribution as $X_n$,but plus $0.5$, that is, $R_{YY}(\tau)= 0.5+\sigma^2$ only when $m=n$,$\tau=m-n$

For $W_n$,$Y_n$ and $W_n$ are similar,because the only when $n=0$,the $R_{YY}(\tau)=0.5+\sigma^2$,otherwise,$R_{YY}(\tau)=0.5$

So $W_n$,$Y_n$ and $Z_n$ are all WSS process,and their psd should be

$S_{WW}(f)=\sigma^2+0.5$ for $n=0$ ,otherwise,$0.5$,in all frequency;

$S_{YY}(f)=\sigma^2+0.5$ for all frequency;

$S_{ZZ}(f)=\sigma^2$ for all frequency

Is my thinking right ? if wrong,please tell me where am i wrong and the solution,thanks a lot