Preservation of displacement ranks and the numerical stability of time recursive fast algorithms

The authors investigate the role played by the preservation of the displacement rank in the update of covariance matrix time-recursive fast algorithms. Standard fast recursive-least-squares algorithms assume the constancy in time of the displacement rank and act on a supposed reduced set of generators. The authors show that some closure relationships have to be maintained in order to preserve the low displacement structure, and they advocate a preliminary canonical reduction procedure. A state-space interpretation of the Kalman gain is introduced that makes it possible to compute it from past prediction errors.