A communication signal $x(u,t)$ is transmitted to a receiver, which, because of two distinct paths of transmission, observes the received signal $y(u,t)$: $y(u,t) = x(u,t) + c x(u,t-t_m), t \in (-\infty, \infty)$, where $c$ and $t_m$ have known constant values in the range $|c| < 1$ and $t_m > 0$. The random process $x(u,t)$ is a wide-sense stationary (WSS) Gaussian random process with known power spectral density (PSD) $$ S_X(\omega) = \begin{cases} P & \text{for } 0 \leq \omega \leq 4\pi, \\ 0 & \text{otherwise}. \end{cases} $$ Is $y(u,t)$ a zero-mean random process?
The following is my try: $$ \begin{align} E[y(u, t)] & = E[x(u, t) + c x(u, t - t_m)] \\ & = E[x(u, t)] + c E[x(u, t - t_m)]. \end{align} $$ Then I don't know how to continue.