Initial-value solutions for on-line smoothing and filtering of linear stationary signals

This note presents an initial-value solution for the impulse response function specifying the linear least-squares estimates of a linear scalar stationary signal with general covariance functions in the presence of a white Gaussian noise. The method employed is an invariant imbedding. The resulting initial-value solution enables us to design a new on-line filter, and fixed-point, initial-point, and fixed-lag smoothers based on covariance specification.