# Why is being jointly WSS important in signal estimation with LMMSE estimator?

Assume that we have $y_k = s_k + n_k$.

We have observed $y_k$ and want to estimate $s_k$. The goal is to use LMMSE (Linear MMSE) estimator to find $y_k$ and on the other hand, we know that our filter is causal, it means we have

$$\hat{s_k} = \sum\limits_{m=-\infty}^k h_{k-m}y_m\text,$$ but why should $s_k$ and $y_k$ be jointly WSS?

Recall that in the derivation of the linear MMSE estimator we force the error term $\hat{s}_k-s_k$ to be orthogonal to the signal $y_k$ i.e. $E[(\hat{s}_k-s_k)y_{k-j}] = 0$ for all $j$. The assumption of joint wide sense stationarity allows one to write $E[s_k y_{k-j}]$ as a function of the lag $j$, i.e. the covariance isn't a function of time $k$, just the lag. As a result, the optimal filter coefficients $h_k$ can be obtained by solving a system of linear equations.