# Performance Analyses of a RAIM Algorithm for Kalman Filter with GPS and NavIC Constellations
## 4 \(\boldsymbol{\
### Simulation Results  
Having modeled \(\boldsymbol{\Psi}\) with underlying parameters residing within a range, results of extensive simulations are shown to justify that \(\boldsymbol{\Psi}_{\text{max}}\) can be used in place of \(\boldsymbol{\Psi}\) to determine test statistics and PLs. In other words, it is illustrated that \(a_{\boldsymbol{\Theta},\ 2}^{\text{max}}\) or \(a_{\boldsymbol{\Omega}}^{\text{max}}\) with \(\boldsymbol{\Theta}\) formed as \((c\mathbf{I}+\boldsymbol{\Psi}_{\text{max}})^{-1}\) is less than or equal to one (see after Equation (5)). In addition, \(a_{\boldsymbol{\Theta},\ 2}^{\text{min}}\) or \(\lambda_{\boldsymbol{\Omega}}^{\text{min}}\) is shown to be more than the minimum eigenvalue of \(\boldsymbol{\hat{S}}^{T}\boldsymbol{\Psi}_{\text{max}}\boldsymbol{\hat{S}}^{ S}\boldsymbol{\Omega}^{T}\boldsymbol{\Theta}\boldsymbol{\hat{S}}^{S}\) (=\(\boldsymbol{\Omega}_{1}\)) times a known scale factor. This is needed to ensure that the calculated \(\Gamma\) of Equation (11) is larger than its actual value (see Equation (10) and the paragraph after Equation (11) for justification of this). The actual value of \(\Gamma\) would need the actual \(\boldsymbol{\Psi}\), which is not known. For the simulation results of this section, lower limits of the scale factors used to account for uncertain parameters are: \(\varrho_{\text{tr, min}}=0.5\); \(\varrho_{\text{tra, min}}=0.5\); \(\mathcal{K}_{\text{tr, min}}=0.25\); \(\mathcal{K}_{\text{ura, min}}=0.25\); \(\mathcal{K}_{\text{m, min}}=0.25\); and \(\mathcal{K}_{\text{ml-n, min}}=0.25\). For the selected lower limits, \(\mathcal{K}_{\text{min}}\) can be obtained a priori by running matlab_program_figures_1_2.m. \(\mathcal{K}_{\text{wh}}\) is 5 for white noise overbound. Scale factors are uniformly distributed between their lower and upper limits. While they are changed between simulations, for a given satellite they are assumed to be constant over \(M\) epochs (for \(\alpha^{\prime}_{1}\)) or \((N-M)\) epochs (for \(\alpha^{\prime}_{2}\)). \(M=10\), \(N=20\), and the time interval between two epochs \(\Delta t=1\) s. Fourteen GPS and seven NavIC satellites are considered. A brief description of the algorithm needed for simulations is provided in Appendix A. Matlab implementation of the algorithm can be found in the Supplementary Materials (matlab_program_algorithm_1_ts1_figure_3.m and matlab_program_algorithm_1_ts2_figure_3.m), where all inputs can be changed as needed, and results with the changed values can be generated. The algorithm also has a provision to remove a satellite at the \(M\)th or \((N-M)\)th epoch to check the change in visibility and remove white noise injection from a random intermediate time epoch. Satellites are excluded when the elevation angle is below the 5\({}^{\circ}\) mask angle. A satellite can be chosen to come into visibility after a random number of epochs. LOS vectors of \(\mathbf{H}\) are randomly selected without considering actual GPS and NavIC satellite orbits. Hence, results of these simulations are likely to hold for other constellations if error model parameters, frequency bands, and \(a_{\min}\) and \(a_{\max}\) are suitably changed. Although there is an inherent relationship between elevation angles and \(\mathbf{H}\), it is not important here as long as the structure of the simulated \(\mathbf{H}\) resembles that of the actual measurement model matrix. That is, the LOS vector constraint of each row is satisfied, the clock bias columns are properly formed, and the rank of \(\mathbf{H}\) is 5. With this, valid measurement model matrices would just be a subset of the wide range of \(\mathbf{H}\) matrices considered in the simulations. Figure 3 depicts the difference between the maximum eigenvalues of \(\mathbf{\Omega}_{1}\) and \(\mathbf{\Omega}\) for both \(\alpha^{\prime}_{1}\) and \(\alpha^{\prime}_{2}\) when \(\theta\) decreases with time (see subplots 1 and 3--one below the other on the left).