## 3 Kalman Filter Algorithms for the Case of Correlated Noise Terms
### Contemporaneous-Form Kalman Filter  
Derivations of this version of the Kalman filter can be found in the textbook by [PERSON] [1], based on orthogonal projection, or the textbook by [PERSON] and [PERSON] [22], based on a least-mean-squares approach; the algorithm is also given by several other authors, such as [PERSON] [2], [PERSON] [23], and [PERSON] and [PERSON] [21]. Its equations are given as follows:  
\[\mathbf{x}_{t|t-1} =\mathsf{A}_{t}\mathbf{x}_{t-1|t-1} \tag{12}\] \[\mathsf{P}_{t|t-1} =\mathsf{A}_{t}\mathsf{P}_{t-1|t-1}\mathsf{A}_{t}^{\mathsf{T}}+ \mathsf{Q}_{t}\] (13) \[\mathsf{V}_{t} =\mathsf{C}_{t}\mathsf{P}_{t|t-1}\mathsf{C}_{t}^{\mathsf{T}}+ \mathsf{R}_{t}+\mathsf{C}_{t}\mathsf{S}_{t}+\left(\mathsf{C}_{t}\mathsf{S}_{t} \right)^{\mathsf{T}}\] (14) \[\mathsf{K}_{t} =\left(\mathsf{P}_{t|t-1}\mathsf{C}_{t}^{\mathsf{T}}+\mathsf{S}_ {t}\right)\mathsf{V}_{t}^{-1}\] (15) \[\mathbf{x}_{t|t} =\mathbf{x}_{t|t-1}+\mathsf{K}_{t}(\mathbf{y}_{t}-\mathsf{C}_{t} \mathbf{x}_{t|t-1})\] (16) \[\mathsf{P}_{t|t} =\mathsf{P}_{t|t-1}-\mathsf{K}_{t}\mathsf{V}_{t}\mathsf{K}_{t}^{ \mathsf{T}} \tag{17}\]  
Compared to the standard Kalman filter with uncorrelated noise terms, only the expressions for the innovation covariance, Equation (14), and for the optimal Kalman gain, Equation (15), contain additional terms. It should be emphasised that for the computation of the innovation likelihood, Equation (14) is not to be used for the innovation covariance; rather, the standard expression, \(\mathsf{C}_{t}\mathsf{P}_{t|t-1}\mathsf{C}_{t}^{\mathsf{T}}+\mathsf{R}_{t}\), has to be used.  
Also the contemporaneous form information filter is needed:  
\[\mathsf{B}_{t} =\mathsf{F}_{t-1|t-1}+\mathsf{A}_{t}^{\mathsf{T}}\mathbf{\Theta}_{t} \mathsf{A}_{t} \tag{18}\] \[\mathbf{\chi}_{t|t-1} =\mathbf{\Theta}_{t}\mathsf{A}_{t}\mathsf{B}_{t}^{-1}\mathbf{\chi}_{t-1|t -1}\] (19) \[\mathsf{F}_{t|t-1} =\mathbf{\Theta}_{t}-\mathbf{\Theta}_{t}\mathsf{A}_{t}\mathsf{B}_{t}^{-1} \mathsf{A}_{t}^{\mathsf{T}}\mathbf{\Theta}_{t}\] (20) \[\mathbf{\Gamma}_{t} =(\mathsf{R}_{t}-\mathsf{S}_{t}^{\mathsf{T}}\mathsf{F}_{t|t-1} \mathsf{S}_{t})^{-1}\] (21) \[\mathsf{G}_{t} =\mathsf{C}_{t}+\mathsf{S}_{t}^{\mathsf{T}}\mathsf{F}_{t|t-1}\] (22) \[\mathbf{\chi}_{t|t} =(\mathsf{I}^{(m)}+\mathsf{G}_{t}^{\mathsf{T}}\mathsf{T}_{t} \mathsf{S}_{t}^{\mathsf{T}})\mathbf{\chi}_{t|t-1}+\mathsf{G}_{t}^{\mathsf{T}} \mathbf{\Gamma}_{t}\mathbf{y}_{t}\] (23) \[\mathsf{F}_{t|t} =\mathsf{F}_{t|t-1}+\mathsf{G}_{t}^{\mathsf{T}}\mathbf{\Gamma}_{ t}\mathsf{G}_{t} \tag{24}\]  
where \(\mathsf{I}^{(m)}\) denotes the \((m\times m)\)-dimensional unity matrix, and the quantities denoted by the symbols \(\mathsf{B}_{t}\), \(\mathbf{\Gamma}_{t}\), and \(\mathsf{G}_{t}\) have been introduced in order to simplify the equations.  
The information filter algorithm in Equations (18)-(24) can be condensed into two equations (plus the two equations defining \(\mathbf{\Gamma}_{t}\) and \(\mathsf{G}_{t}\)), directly describing the update of the filtered information state estimate and its information matrix:  
\[\mathbf{\chi}_{t|t} =(\mathsf{I}^{(m)}+\mathsf{G}_{t}^{\mathsf{T}}\mathbf{\Gamma}_{ t}\mathsf{S}_{t}^{\mathsf{T}})\mathbf{\Theta}_{t}\mathsf{A}_{t}\big{(}\mathsf{F}_{t-1|t -1}+\mathsf{A}_{t}^{\mathsf{T}}\mathbf{\Theta}_{t}\mathsf{A}_{t}\big{)}^{-1}\mathbf{ \chi}_{t-1|t-1}+\mathsf{G}_{t}^{\mathsf{T}}\mathbf{\Gamma}_{t}\mathbf{y}_{t} \tag{25}\] \[\mathsf{F}_{t|t} =\mathbf{\Theta}_{t}-\mathbf{\Theta}_{t}\mathsf{A}_{t}\big{(}\mathsf{F}_{ t-1|t-1}+\mathsf{A}_{t}^{\mathsf{T}}\mathbf{\Theta}_{t}\mathsf{A}_{t}\big{)}^{-1} \mathbf{\Lambda}_{t}^{\mathsf{T}}\mathbf{\Theta}_{t}+\mathsf{G}_{t}^{\mathsf{T}} \mathbf{\Gamma}_{t}\mathsf{G}_{t} \tag{26}\]