# Performance Analyses of a RAIM Algorithm for Kalman Filter with GPS and NavIC Constellations
## 2 Overview of Existing KF RAIM
### Fault Detection Test Statistics  
Assuming only white noise in the GNSS measurements, three fault detection test statistics (\(\alpha_{j,\,k};\,j=1,...,3\)) at \(t_{k}\) are formulated in [26] as  
\[\alpha_{j,\,k}^{2}=\sum_{\ell=p_{j,\,k}}^{m_{j,\,k}}\left(\Delta\underline{ \rho}_{\ell}^{T}\mathbf{W}_{\ell}^{-1}(\mathbf{I}-\mathbf{G}_{\ell})\Delta \underline{\rho}_{\ell}+\Delta\underline{\underline{\rho}}_{\ell}^{T} \mathbf{W}_{rr,\,\,\ell}^{-1}(\mathbf{I}-\mathbf{G}_{\ell}^{rr})\Delta \underline{\underline{\rho}}_{\ell}\right) \tag{1}\]  
where \(\mathbf{I}\) represents the identity matrix of dimension \(n\times n\). \(n\) is the number of visible satellites at a given epoch and can change with time. \(\Delta\underline{\underline{\rho}}_{\ell}\) and \(\Delta\underline{\underline{\rho}}_{\ell}\) are pseudorange and pseudorange rate innovation vectors of the EKF, respectively, at time epoch \(\ell\). The limits of \(\ell\) on the right hand side, \(m_{j,\ k}\) and \(p_{j,\ k}\), change with \(j\) and \(k\). \(\mathbf{G}_{\ell}\) and \(\mathbf{G}_{\ell}^{rr}\) are given by WLS estimation as  
\[\mathbf{G}_{\ell} =\mathbf{H}_{\ell}(\mathbf{H}_{\ell}^{T}\mathbf{W}_{\ell}^{-1} \mathbf{H}_{\ell})^{-1}\mathbf{H}_{\ell}^{T}\mathbf{W}_{\ell}^{-1} \tag{2}\] \[\mathbf{G}_{\ell}^{rr} =\mathbf{H}_{\ell}(\mathbf{H}_{\ell}^{T}\mathbf{W}_{rr,\ \ell}{}^{-1} \mathbf{H}_{\ell})^{-1}\mathbf{H}_{\ell}^{T}\mathbf{W}_{rr,\ \ell}{}^{-1} \tag{3}\]  
where \(\mathbf{H}_{\ell}\) is the linearized measurement model matrix for pseudoranges (or pseudorange rates) and of dimension \(n\times 4\) for a single constellation and \(n\times 5\) for dual constellations. \(\mathbf{W}_{\ell}\) is the error covariance matrix of the pseudorange measurements at \(t_{\ell}\). \(\mathbf{W}_{rr,\ \ell}\) is its counterpart for pseudorange rates. WLS estimation-based fault detection in KF is used because it helps in the following ways. It results in the noise statistics of \(\alpha\)s being easily computed. Furthermore, each term under the summation of Equation (1) contributes the fault of its own epoch only, not of any other epochs. Thus, the computation of the failure mode slope (FMS) for mean position error bound is simplified. In addition, each \(\alpha\) can have a separate FMS. Consequently, three tests can be formulated. With three suitably designed test statistics, the PL is shown not to increase with time in [26], as opposed to the PL with a single statistic. The limitation of the above formulation is that measurement errors across time epochs are assumed to be uncorrelated, which is a gross simplification. To overcome this limitation, [16] assumes the following error models. Broadcast satellite orbit and clock error, residual tropospheric error, receiver noise, and multipath are together represented as a first order Gauss Markov (FOGM) random process with a 100 s time constant (carrier smoothing interval of code). From [27], individual error standard deviations are obtained for GPS carrier smoothed code for airborne users. The second constellation considered is NavIC. As for NavIC, the same equations are used with frequency bands properly changed. This is because equivalent error models are yet to be developed for this constellation. No ionospheric error is modeled, which can be eliminated with dual frequency measurements. Only white noise pertaining to 45 dB-Hz C/N\({}_{0}\) is assumed to be present in the pseudorange rate measurements.