Performance sensitivity of the Kalman filter to choice of state and observation vectors

The problem of applying Kalman filtering techniques to the processing of phased array data is addressed. The primary objective is to obtain a lucid explanation of why a previously developed filter design [1] encounters numerical difficulty upon initialization for a certain combination of state and observation vectors, while operating smoothly when one component of each is altered. The analysis of the factors contributing to computational difficulty suggests a method of alleviating the difficulty, but diminishes the optimality of the filter's output estimates. The difficulty results from the combination of an ill-conditioned observation covariance matrix, truncation error in representing the elements of the covariance matrix at long ranges, and the particular form of asymptotic behavior exhibited by the inverse covariance matrix.