Correlated Noise

When the noise is not simply additive but propagates through the system via a linear transformation, the Kalman system generalizes to

$$\left\{ \begin{array}{l} \mathbf{x}_{k+1} = \mathbf{A}_{k} ... ... \mathbf{x}_{k} + \mathbf{V}_k \mathbf{v}_k \end{array}\right.$$ (3.28)

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

In this case, the same equations as for the Kalman system can be applied by introducing the substitutions

$$\begin{array}{l} \mathbf{Q}'_k = \mathbf{W}_k \mathbf{Q}_k \... ..._k = \mathbf{V}_k \mathbf{R}_k \mathbf{V}^{\top}_k \end{array}$$ (3.29)

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 identity matrices, that is, if the noise is simply additive, the expression simplifies and reduces to the form presented earlier.