On the general conditions of existence for linear MMSE filters: Wiener and Kalman

State estimation is a fundamental task in plethora of applications. Two important classes linear minimum mean square error (MMSE) estimators are Wiener filters (WFs) and Kalman (KFs), the latter being recursive form former for discrete state-space (LDSS) dynamic models. The set uncorrelation conditions regarding states measurements which allows to formulate general KF has been recently introduced literature, but both WFs KFs standard key assumption that measurement noise covariance invertible. In this contribution we provide i) an alternative derivation WF case non-invertible matrix, ii) proof under lately released, can always be formulated irrespective matrix rank. Notice these results also apply predictors smoothers, linearly constrained variance distortionless response (MVDR) filters, therefore remarkable result broad interest.