Is the sum of white noise and shifted white noise white noise again?
Calculate the autocorrelation of the process.

$$\begin{align}
R_{xx}[n] &amp;=\mathbb{E}[(W[k] + c W[k-1])(W[k-n] + c W[k-1-n])] \\
&amp;=\mathbb{E}[W[k]W[k-n]]+ \mathbb{E}[cW[k]W[k-1-n]]+\mathbb{E}[cW[k-1]W[k-n]]+\mathbb{E}[c^2W[k-1]W[k-1-n]] \\
&amp;=\sigma^2\delta[n]+c\sigma^2\delta[n+1]+c\sigma^2\delta[n-1]+c^2\sigma^2\delta[n]\\
&amp;=\sigma^2(1+c^2)\delta[n]+c\sigma^2\delta[n+1]+c\sigma^2\delta[n-1]
\end{align}$$

The definition of white noise implies that $R_{xx}[n]=\sigma^2\delta[n]$, which is not the case here.