Correlated Noise

In the case where the noise is not merely additive, but propagates through the system via some linear transformation, the Kalman filter is generalized to

$$\left\{ \begin{array}{l} \mathbf{x}_{k+1} = \mathbf{A}_{k} ... ... \mathbf{x}_{k} + \mathbf{V}_k \mathbf{v}_k \end{array}\right.$$ (2.110)

The process noise is correlated through a matrix $\mathbf{W}_k$ to the source, and the observation noise through a matrix $\mathbf{V}_k$.

In this case, it is possible to apply the same equations of the Kalman system by introducing the substitutions

$$\begin{array}{l} \mathbf{Q}'_k = \mathbf{W}_k \mathbf{Q}_k \... ..._k = \mathbf{V}_k \mathbf{R}_k \mathbf{V}^{\top}_k \end{array}$$ (2.111)

This result will be useful in the following section on the extended Kalman filter.

Clearly, if the matrices $\mathbf{W}_k$ and $\mathbf{V}_k$ are identities, meaning that the noise is simply additive, the expression simplifies and reverts to the form seen previously.