Random Processes: Filtering, Estimation, and Detection

8 Optimum Linear Systems:The Kalman Approach

8.1 INTRODUCTION

In the previous chapter optimal linear systems were obtained from the classical Wiener framework to perform operations of filtering prediction and smoothing. The optimal filter was determined in terms of various cross-correlation and autocorrelation functions, or equivalently cross and auto spectral densities, and thus can be considered primarily a frequency domain approach. The Weiner results presented were primarily for wide sense stationary processes; however, for nonstationary processes integral equations were obtained for the solutions. For the case of vector process estimation, the spectral factorization problem for the estimation of wide sense stationary processes proved to be mathematically intractable, and for estimation of nonstationary processes, the resulting integral equations could not be solved by the standard Laplace transform techniques and their solutions were virtually impossible to obtain.

The Kalman filter addresses both the problem of nonstationarity and vector estimation, and it gives the solution in the time domain by virtue of formulating the problem in the state space. Thus rather than the autocorrelation functions or power spectral densities being given, the required information for the Kalman filter solution is given in the state variable format. This information is provided in terms of the state transition, excitation, and measurement matrices of the given state space formulation. This chapter ...