A unified solution to the singular and nonsingular linear minimum-variance estimation problem

A unified solution is derived for the transfer function matrix of the minimum-variance estimator of the state of a linear continuous-time system with noisy or noise-free measurements. For the noisy measurement case, the Kalman-Bucy filter standard equations are used to give the general format of the transfer function. The Kalman-Bucy gain, using the spectral factorization on the algebraic in the S plane, is derived and substituted in the transfer-function expression. The singular case is then shown to be a special case of the noisy measurement estimator