# 8.1 One-Dimensional Finite Difference Kalman Filter

## 8.1.1 One-Dimensional Finite Difference State Prediction

The scalar derivation for the EKF state prediction equation in terms of was given in (7.10)), repeated here for clarity

(8.1)

As noted previously, this is accurate to third order in c.

Replacing the second-order derivative with its finite difference approximation from (2.47), (8.1) becomes

(8.2)

Gathering terms, this can be rewritten as

(8.3)

Let

(8.4)

so that

(8.5)

and define the finite difference evaluation points as

(8.6)

where

(8.7)

See Section 2.2 for an explanation of the notation . Using (8.4) and (8.6), (8.3) becomes

(8.8)

Replacing the continuous c defined in (7.3) with ...