Consider for simplicity a Kalman filter applied to the one-dimensional state space model
$x_{n}=f_{n}x_{n-1}+q_{n}$
$y_{n}=h_{n}x_{n}+r_{n}$
with white noise errors. Assume that $r_n=e_n-e_{n-1}$ with $e_{n}\in\mathrm{WN}(0,\sigma_{e}^{2})$. How should one adjust the standard Kalman filter (our some other filter) with respect to the time correlation in $r_{n}$, i.e., $Cov(r_{n},r_{n-1})=-\sigma_{e}^{2}$? The standard method seems to be to introduce $r_{n}$ as an AR-process in the state vector, but this is inconsequent since it would imply $Cov(r_{n},r_{n-2})\neq0$ which is not true. It is possible to adjust the filter in some more rigorous way?