I have a signal given by the following equation:
$$y_k = X_k S_0 + \sum_{l=0 \& l\neq k}^{N-1}S_{l-k}+n_k$$
where $X_k$ are independent and identically distributed random variables. $n_k$ is a complex gaussian noise sample. $X$ and $n$ are independent.
Since $X_k$ takes on real values (say +1,-1) therefore, the characteristic function (CF) of the real part of the above equation is
$$\phi(\omega) = e^{j\omega R\{S_0\}}\prod_{l=0 \& l \neq k}^{N-1} \cos(\omega R\{S_{l-k}\}) e^{-(\omega^2 \sigma^2)/2}$$
$R$ stands for real. I do not understand how the first and the second terms of the CF are obtained. I know that the CF of the sum of independent RVs is equal to the product of their individual CFs but from do we get the cos term and why do the $X$ terms vanished ?