Chandrasekahr filtering and smoothing methods given covariance model and properties of X and Y functions

Abstract

This paper is concerned with the Chandrasekhar filtering and smoothing methods given the covariance model of continous stationary stochastic signals in the presence of white Gaussian noises. Two types of Riccati differential equations systems are presented. One is for calculating the Chandrasekhar's X-function and the other is for the Chandrasekhar's Y-function. The explicit relationships between the two solutions to the Riccati differential equations systems and the Chandrasekhar's X- and Y-functions are shown. Also derived are the relations between the Chandrasekhar's X-function and Y-function. In addition, another type of Chandrasekhar's X- and Y-functions that is different from the original ones in terms of numerical stability and accuracy and the corresponding Riccati differential equations systems are presented. Similar formulas are also given for this type of Chandrasekhar's X- and Y-functions.