## 3 Kalman Filter Algorithms for the Case of Correlated Noise Terms
### Future-Form Kalman Filter  
This version of the Kalman filter is given in numerous textbooks, such as those by [PERSON] and [PERSON] [24], [PERSON] et al. [3], [PERSON] [20], and [PERSON] [25]. It can be derived by decorrelating the noise terms, such that, formally, the standard Kalman filter for uncorrelated noise terms can be applied. Its equations are given as follows:\[\tilde{A}_{t} =A_{t}-S_{t-1}\Phi_{t-1}C_{t-1} \tag{27}\] \[\tilde{Q}_{t} =Q_{t}-S_{t}\Phi_{t}S_{t}^{T}\] (28) \[\mathbf{x}_{t|t-1} =\tilde{A}_{t}\mathbf{x}_{t-1|t-1}+S_{t-1}\Phi_{t-1}\mathbf{y}_{t-1}\] (29) \[\mathrm{P}_{t|t-1} =\tilde{A}_{t}\mathrm{P}_{t-1|t-1}\tilde{A}_{t}^{T}+\tilde{Q}_{t-1}\] (30) \[\mathrm{V}_{t} =C_{t}\mathrm{P}_{t|t-1}C_{t}^{T}+R_{t}\] (31) \[\mathrm{K}_{t} =\mathrm{P}_{t|t-1}C_{t}^{T}\mathrm{V}_{t}^{-1}\] (32) \[\mathbf{x}_{t|t} =\mathbf{x}_{t|t-1}+\mathrm{K}_{t}(\mathbf{y}_{t}-\mathsf{C}_{t} \mathbf{x}_{t|t-1})\] (33) \[\mathrm{P}_{t|t} =\mathrm{P}_{t|t-1}-\mathrm{K}_{t}\mathrm{V}_{t}\mathrm{K}_{t}^{T} \tag{34}\]  
In this filter, \(\tilde{A}_{t}\), as defined by Equation (27), represents a _modified state transition matrix_, and \(\tilde{Q}_{t}\), as defined by Equation (28), represents a _modified dynamical noise covariance matrix_. Note that  
\[\tilde{Q}_{t}^{-1}=\mathbf{\Omega}_{t} \tag{35}\]  
where \(\mathbf{\Omega}_{t}\) is defined in Equation (8). As a further modification to the standard Kalman filter for uncorrelated noise terms, a new term, \(S_{t-1}\mathbf{\Phi}_{t-1}\mathbf{y}_{t-1}\), arises in the equation for the predicted state estimate, Equation (29); however, since this term is precisely known, it does not cause any further changes to the standard Kalman filter. As another subtle difference, in the case of the future-form Kalman filter, in Equation (30), the time index of the modified dynamical noise covariance matrix has to be chosen as \(t-1\) in contrast to Equation (13).  
Also, the future-form information filter is needed:  
\[\tilde{B}_{t} =\mathsf{F}_{t-1|t-1}+\tilde{A}_{t}^{T}\mathbf{\Omega}_{t-1} \tilde{A}_{t} \tag{36}\] \[\mathsf{F}_{t|t-1} =\mathbf{\Omega}_{t-1}-\mathbf{\Omega}_{t-1}\tilde{A}_{t}\tilde{B }_{t}^{-1}\tilde{A}_{t}^{T}\mathbf{\Omega}_{t-1}\] (37) \[\boldsymbol{\chi}_{t|t-1} =\mathbf{\Omega}_{t-1}\tilde{A}_{t}\tilde{B}_{t}^{-1}\boldsymbol {\chi}_{t-1|t-1}+\mathrm{F}_{t|t-1}S_{t-1}\mathbf{\Phi}_{t-1}\mathbf{y}_{t-1}\] (38) \[\boldsymbol{\chi}_{t|t} =\boldsymbol{\chi}_{t|t-1}+C_{t}^{T}\mathbf{\Phi}_{t}\mathbf{y}_{t}\] (39) \[\mathrm{F}_{t|t} =\mathsf{F}_{t|t-1}+C_{t}^{T}\mathbf{\Phi}_{t}C_{t} \tag{40}\]  
where the quantity denoted by the symbol \(\tilde{B}_{t}\) has been introduced in order to simplify the equations.  
Again, the information filter algorithm in Equations (36)-(40) can be condensed into two equations:  
\[\boldsymbol{\chi}_{t|t} =\mathbf{\Omega}_{t-1}\tilde{A}_{t}\big{(}\mathsf{F}_{t-1|t-1}+ \tilde{A}_{t}^{T}\mathbf{\Omega}_{t-1}\tilde{A}_{t}\big{)}^{-1}\boldsymbol{ \chi}_{t-1|t-1}\] \[+\Big{(}\mathbf{\Omega}_{t-1}-\mathbf{\Omega}_{t-1}\tilde{A}_{t} \big{(}\mathsf{F}_{t-1|t-1}+\tilde{A}_{t}^{T}\mathbf{\Omega}_{t-1}\tilde{A}_{t }\big{)}^{-1}\tilde{A}_{t}^{T}\mathbf{\Omega}_{t-1}\Big{)}S_{t-1}\mathbf{\Phi} _{t-1}\mathbf{y}_{t-1}+C_{t}^{T}\mathbf{\Phi}_{t}\mathbf{y}_{t} \tag{41}\] \[\mathsf{F}_{t|t} =\mathbf{\Omega}_{t-1}-\mathbf{\Omega}_{t-1}\tilde{A}_{t}\big{(} \mathsf{F}_{t-1|t-1}+\tilde{A}_{t}^{T}\mathbf{\Omega}_{t-1}\tilde{A}_{t}\big{)} ^{-1}\tilde{A}_{t}^{T}\mathbf{\Omega}_{t-1}+C_{t}^{T}\mathbf{\Phi}_{t}C_{t} \tag{42}\]  
It has to be emphasised that the filter algorithm in Equations (12)-(17) is not equivalent to the filter algorithm in Equations (27)-(34), and this applies likewise to the information filter algorithms. This is obvious from a direct comparison of the equations and can readily be confirmed by numerical examples.