Minimum Variance Distortionless Response Estimators for Linear Discrete State-Space Models

Abstract

For linear discrete state-space models, under certain conditions, the linear least-mean-squares filter estimate has a convenient recursive predictor/corrector format, aka the Kalman filter. The purpose of this paper is to show that the linear minimum variance distortionless response (MVDR) filter shares exactly the same recursion, except for the initialization which is based on a weighted least-squares estimator. If the MVDR filter is suboptimal in mean-squared error sense, it is an infinite impulse response distortionless filter (a deconvolver) which does not depend on the prior knowledge (first- and second-order statistics) on the initial state. In other words, the MVDR filter can be pre-computed and its behaviour can be assessed in advance independently of the prior knowledge on the initial state.