# Performance Analyses of a RAIM Algorithm for Kalman Filter with GPS and NavIC Constellations
## 4 \(\boldsymbol{\
### Simulation Results
\(\lambda^{\min}_{\mathbf{\Omega},1}\) and \(\lambda^{\max}_{\mathbf{\Omega},1}\) are the minimum and maximum eigenvalues of \(\mathbf{\Omega}_{1}\), respectively. In each simulation, the initial \(\theta\) of a satellite is chosen with a uniform random variable between 5.1\({}^{\circ}\) and 90\({}^{\circ}\). Satellite speed is constant at its maximum value. Five hundred thousand simulations are divided into ten batches for ease of plotting in Matlab, with each batch having 50,000 runs. The minimum and maximum differences over 50,000 simulations in a batch are shown as the end points of a vertical line. Within a batch, 10,000 runs have white noise injected up to some random intermediate epochs. Since the minimum difference between \(\lambda^{\max}_{\mathbf{\Omega},1}\) and \(\lambda^{\max}_{\mathbf{\Omega}}\) is greater than zero in all batches, \(\lambda^{\max}_{\mathbf{\Omega}}\) is less than \(\lambda^{\max}_{\mathbf{\Omega},1}\), which in turn is less than or equal to one by proper calculation of \(c\). The differences between \(1/(\lambda^{\min}_{\mathbf{\Omega},1}K_{\min})\) and \(1/\lambda^{\min}_{\mathbf{\Omega}}\) for \(\alpha^{\prime}_{1}\) (and between \(1/(\lambda^{\min}_{\mathbf{\Omega},1}\frac{K_{\min}}{(K_{\mathrm{vh}}\ +\ 1)})\) and \(1/\lambda^{\min}_{\mathbf{\Omega}}\) for \(\alpha^{\prime}_{2}\)) are illustrated in the second (and fourth) subplot of the same figure. In this case, the longer vertical line of each batch belongs to the 10,000 simulations with artificial white noise. It is evident that \(1/\lambda^{\min}_{\mathbf{\Omega}}<1/(\lambda^{\min}_{\mathbf{\Omega},1}K_{ \min})\) for \(\alpha^{\prime}_{1}\), and \(1/\lambda^{\min}_{\mathbf{\Omega}}<1/(\lambda^{\min}_{\mathbf{\Omega},1}\frac {K_{\min}}{(K_{\mathrm{vh}}\ +\ 1)})\) for \(\alpha^{\prime}_{2}\). \(\lambda^{\min}_{\mathbf{\Omega}}\) is verified to be always positive. It can also be concluded that \(\sqrt{\lambda^{\max}_{\mathbf{\Omega}}}/\lambda^{\min}_{\mathbf{\Omega}}< \sqrt{\lambda^{\max}_{\mathbf{\Omega},1}}/(\lambda^{\min}_{\mathbf{\Omega},1} K_{\min})\) for \(\alpha^{\prime}_{1}\). Similarly, for \(\alpha^{\prime}_{2}\), \(\sqrt{\lambda^{\max}_{\mathbf{\Omega}}}/\lambda^{\min}_{\mathbf{\Omega}}\)\(<\sqrt{\lambda^{\max}_{\mathbf{\Omega},1}}/(\lambda^{\min}_{\mathbf{\Omega},1}\frac {K_{\min}}{(K_{\mathrm{vh}}\ +\ 1)})\). Hence, using \(\mathbf{\Psi}_{\max}\) in place of \(\mathbf{\Psi}\), \(\alpha^{\prime}_{1}\), \(\alpha^{\prime}_{2}\) and PLs can be calculated. Although not shown, the same conclusion can be derived with increasing elevation angle and changing satellite speeds, and verified with the provided Matlab programs. It should be noted that when \(c\) is adaptively calculated for \(\alpha^{\prime}_{1}\), \(\lambda^{\max}_{\mathbf{\Omega}}\) is less than \(\lambda^{\max}_{\mathbf{\Omega},1}\) and one only if \(c\) is more than or equal to a minimum value, \(c_{\min}\). It is found that \(c_{\min}\) is a function of \(M\) and elevation angle. Appendix B explains how \(c_{\min}\) is obtained. During adaptive calculation, the initial guess of \(c\) is taken as \(c_{\min}\) or higher. Next, a tighter value of \(\Gamma\) is determined. The effect of \(\mathbf{W}_{rr,\ \ell}\) overbounding the actual pseudorange rate (or precise range rate) error covariance is also discussed.