0
$\begingroup$

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?

$\endgroup$
4
$\begingroup$

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.

| improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.