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Modeling, Estimation and Optimal Filtration in Signal Processing by Mohamed Najim

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Appendix J

Calculation of the Kalman Gain (the Carew and Belanger Method)

Introduction

In this appendix, we present the Carew and Belanger approach [1] as an alternative to the Mehra method to obtain the optimal Kalman gain. This approach is based on the prediction of the state vector using a sub-optimal gain. We first present some basic definitions and notations, used in the rest of the appendix; then, we consider the calculation of the innovation sequence's autocorrelation function for the sub-optimal case; and finally, using this calculation, we establish the relation between the optimal and sub-optimal cases.

Notations and definitions

The superscript * denotes the sub-optimal case and the superscript ^ denotes the optimal case. Thus, for example, images*(k/k–1) is the a priori estimation of the state vector images(k) when the gain K* is sub-optimal whereas images(k/k – 1) denotes the a priori estimation of images(k) for optimal Kalman gain images.

We now introduce the following notations:

P*(k/k-1) denotes the autocorrelation ...

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