Modeling, Estimation and Optimal Filtration in Signal Processing

Appendix I

Calculation of the Kalman Gain using the Mehra Approach

Calculation of the autocorrelation function of the innovation sequence

Let us first recall the notations, introduced in Chapter 5, which are used in this appendix and in Chapter 6. (k/k – 1) denotes the a priori estimation of the state vector at instant k:

and P(k/k – 1) denotes the associated autocorrelation matrix:

Furthermore, the Kalman gain at instant k is given by:

whereas the innovation is defined as follows:

Using equation [I.4], the autocorrelation function of this innovation , is given by:

For the particular case j = 0, we can show that:

To simplify the expression of the innovation's autocorrelation function [I.5] ...