# Performance Analyses of a RAIM Algorithm for Kalman Filter with GPS and NavIC Constellations
## 5 Determination of \(\Gamma\)  
Substituting Equation (5) in the first term of Equation (6) on the right and having a similar representation for the second term, one can write  
\[\alpha_{j}^{2}=\underline{\eta}_{r}^{\ \ T}\Theta_{2}\underline{\eta}_{r}+ \underline{\eta}_{rr}^{\ \ \ T}\Theta_{5}\underline{\eta}_{rr} \tag{35}\]  
where \(\underline{\eta}_{r}\) has unit-variance independent elements. \(\Theta_{5}\) would be the identity matrix only if \(\widetilde{\mathbf{W}}_{rr,\ \ell}\) of Equation (6) is equal to the actual pseudorange rate error covariance matrix. However, \(\mathbf{W}_{rr,\ \ell}\) generally overbounds the actual error covariance. As a result, the maximum eigenvalue of \(\Theta_{5}\) is less than or equal to one, and the minimum eigenvalue is at some value greater than zero [29]. They are denoted as \(\lambda_{\Theta,\ 5}^{\max}\) and \(\lambda_{\Theta,\ 5}^{\min}\), respectively, and determined using prior knowledge of overbounding different satellite range rate errors. \(\widetilde{\Theta}\) of Equation (11) thus changes to  
\[\widetilde{\Theta}= \begin{bmatrix}\Theta_{2}&[0]\\ [0]&\Theta_{5}\end{bmatrix}  \tag{36}\]  
It is apparent that \(\lambda_{\widetilde{\Theta}}^{\max}=\max\{\lambda_{\Theta,\ 2}^{\max},\ \lambda_{\Theta,\ 5}^{\max}\}\) and \(\lambda_{\widetilde{\Theta}}^{\min}=\min\{\lambda_{\Theta,\ 2}^{\min},\ \lambda_{\Theta,\ 5}^{\min}\}\). The upper and lower bounds of \(\lambda_{\Theta,\ 2}^{\max}\) and \(\lambda_{\widetilde{\Theta},\ 2}^{\min}\), respectively, as determined in the previous section, are used in place of their actual, unknown values. With the formulation of \(\widetilde{\Theta}\) after Equation (11), where \(\lambda_{\widetilde{\Theta}}^{\max}=1\), \(\Gamma\) can be directly obtained from the lookup table. As \(\Gamma\) was shown to increase with decreasing \(\lambda_{\widetilde{\Theta}}^{\min}\), the next entry of the lookup table immediately higher than the calculated \(1/\lambda_{\widetilde{\Theta}}^{\min}\) is considered. With \(\widetilde{\Theta}\) of Equation (36), and the minimum and maximum eigenvalues defined after it, \(\Gamma\) is now calculated in the following way. In this case, \(\lambda_{\widetilde{\Theta}}^{\max}\) is not necessarily equal to one. First, \(\Gamma\) is obtained from the lookup table the same way as before, using \(P_{MD,\ i}^{alloc}\), DOF, and the next ratio entry higher than the calculated \(\sqrt{\lambda_{\widetilde{\Theta}}^{\max}}/\lambda_{\widetilde{\Theta}}^{ \min}\). This \(\Gamma\) is not the right one when  
Figure 3: Differences between maximum (and minimum) eigenvalues of \(\Omega_{1}\) and \(\Omega\) over five hundred thousand simulations for each \(a_{1}^{\prime}\) and \(a_{2}^{\prime}\)\(\lambda_{\mathbf{\Theta}}^{\max}\
eq 1\), and therefore is denoted by \(\Gamma^{\prime}\). In addition, for the lookup table, the detection threshold, \(T_{th,\,1}\), is now determined from the non-central chi-squared distribution with the required DOF and some predetermined natural bias [37]. Using \(\Gamma^{\prime}\), the \(\Gamma^{*}\) of Equation (11) is calculated as  
\[\sqrt{\Gamma^{*}}=\sqrt{\Gamma^{\prime}}-T_{th,\,1}+T_{th,\,1}\left(\frac{ \sqrt{\lambda_{\mathbf{\Theta}}^{\max}}}{\lambda_{\mathbf{\Theta}}^{\min}}\right)_{ \text{lookup}} \tag{37}\]  
where \(\left(\frac{\sqrt{\lambda_{\mathbf{\Theta}}^{\max}}}{\lambda_{\mathbf{\Theta}}^{\min} }\right)_{\text{lookup}}\) is the lookup table entry. Using Equation (11), the correct \(\Gamma\) is given as  
\[\sqrt{\Gamma}=\left(\sqrt{\Gamma^{*}}-T_{th,\,1}\left(\frac{\sqrt{\lambda_{ \mathbf{\Theta}}^{\max}}}{\lambda_{\mathbf{\Theta}}^{\min}}\right)_{\text{lookup}} \right)\sqrt{\lambda_{\mathbf{\Theta}}^{\max}}+T_{th,\,1} \tag{38}\]  
Substituting Equation (37) in the preceding equation  
\[\sqrt{\Gamma}=\left(\sqrt{\Gamma^{\prime}}-T_{th,\,1}\right)\sqrt{\lambda_{ \mathbf{\Theta}}^{\max}}+T_{th,\,1} \tag{39}\]  
If \(\sqrt{\Gamma^{\prime}}<T_{th,\,1}\), \(\sqrt{\Gamma^{\prime}}\) is taken as \(T_{th,\,1}\), which may occur at a very large DOF.