# 5

# Intermediate Topics on Kalman Filtering

## 5.1 ALTERNATIVE FORM OF THE DISCRETE KALMAN FILTER–THE INFORMATION FILTER

The Kalman filter equations given in Chapter 4 can be algebraically manipulated into a variety of forms. An alternative form that is especially useful will now be presented (1). We begin with the expression for updating the error covariance, Eq. (4.2.22), and we temporarily omit the subscripts to save writing:

Recall that the Kalman gain is given by Eq. (4.2.17):

Substituting Eq. (5.1.1) into Eq. (5.1.2),

Going back to the error covariance update:

Substituting Eq. (5.1.3) for the gain **K** in Eq. (5.1.4), we get

Factoring out **P** and rearranging the equation in terms of (**P**)^{−1}, we get

For the error covariance projection equation, we start with the usual prediction stage:

Using ...