8.3 An Alternate Derivation of the Multidimensional Finite Difference Covariance Prediction Equations

Applying the finite difference equation (2.73), the alternate EKF state error covariance equation (7.69) becomes

(8.48) equation

From (7.67) we have img. Using the same method as above, (8.48) transforms into

(8.49) equation

As in the one-dimensional case, this latter form for the predictive state covariance matrix is only partially of second order. It will be shown below that (8.37) and (8.49) are identical to the unscented Kalman filter.

Note that (8.43) can be written in the alternate form

(8.50) equation

In addition, (8.47) can be rewritten as

(8.51) equation

Because the finite-difference filter is almost identical to the unscented Kalman filter, discussed in the next chapter, we will only present the results for the DIFAR case study for the unscented Kalman filter.

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