The asymptotic minimum variance estimate of stationary linear single output processes

The problem of minimum error variance estimation of single output linear stationary processes in the presence of weak measurement noise is considered. By applying domain analysis to the case of single input systems and white observation noise, explicit and simple expressions are obtained for the error covariance matrix of estimate and the optimal Kalman gains both for minimum- and nonminimum-phase systems. It is found that as the noise intensity approaches zero, the error covariance matrix of estimating the output and its derivatives becomes insensitive to uncertainty, in the system parameters. This matrix depends only on the shape of the high frequency tail of the power-density spectrum of the observation, and thus it can be easily determined from the system transfer function. The theory developed is extended to deal with white measurement noise in multiinput systems where an analog- to the single input nonminimum-phase case is established. The results are also applied to colored observation noise problems and a simple method to derive the minimum error covariance matrices and the optimal filter transfer functions is introduced.