# Performance Analyses of a RAIM Algorithm for Kalman Filter with GPS and NavIC Constellations
## 4 \(\boldsymbol{\
### Uncertain \(a_{\ell q}\)
The actual \(\sigma^{(\ell)}(i))^{2}\), \(\sigma_{\text{act}}^{(\ell)}(i))^{2}\) is given by  
\[\sigma_{\text{act}}^{(\ell)}(i))^{2}=\Big{(}\text{mp}^{(\ell)}\big{(}\theta^ {(\ell)}(i)\big{)}\Big{)}^{2}\mathcal{K}_{\text{tr}}\sigma_{\text{tr}}^{2}+ \mathcal{K}_{\text{ura}}\sigma_{\text{ura}}^{2}+\mathcal{K}_{\text{ml-n}}\big{(} (\sigma_{\text{ml, }i}^{(\ell)})^{2}+(\sigma_{\text{n, }i}^{\ell})^{2}\Big{)} \tag{24}\]  
As Equation (20) has the modeled values of \(\sigma\) in the denominator, it reduces to  
\[a_{\ell q}=\Big{(}\exp^{(-\beta_{\ell q}^{\text{mod}}At)}\Big{)}^{q}\frac{( \sigma^{(\ell)}(i))^{2}}{\sigma^{(\ell)}(i)\sigma^{(\ell+q)}(i)} \tag{25}\]  
With the rearrangement of terms as before,  
\[a_{\ell q}=\Bigg{(}\exp^{(-\beta_{\ell q}^{\text{mod}}At)}\frac{\sigma^{(\ell )}(i)}{\sigma^{(\ell+1)}(i)}\Bigg{)}\dots\Bigg{(}\exp^{(-\beta_{\ell q}^{\text {mod}}At)}\frac{\sigma^{(\ell+q-1)}(i)}{\sigma^{(\ell+q)}(i)}\Bigg{)} \tag{26}\]  
The \(\ell\)th diagonal term of \(\boldsymbol{\Psi}_{\text{sub}}^{i}\) is  
\[a_{\ell 0}=\Bigg{(}\frac{\sigma_{\text{act}}^{(\ell)}(i)}{\sigma^{(\ell)}(i)} \Bigg{)}^{2} \tag{27}\]  
It should be noted that the term \((\sigma_{\text{act}}^{(\ell)}(i)/\sigma^{(\ell)}(i))\) is less than or equal to one as \(\sigma^{(\ell)}(i)\geq\sigma_{\text{act}}^{(\ell)}(i)\). If \(\mathcal{K}_{\text{min}}^{\prime}\) is the minimum of \(\mathcal{K}_{\text{tr, min}}\), \(\mathcal{K}_{\text{ura, min}}\), and \(\mathcal{K}_{\text{ml-n, min}}\), then using Equation (24), \(\mathcal{K}_{\text{min}}^{\prime}\leq a_{\ell 0}\leq 1\). Multiplying and dividing \(\Big{(}\exp^{(-\beta_{\ell q}\Delta t)}\Big{)}^{q}\) in Equation (26) and rearranging terms,  
\[a_{\ell q}=\Bigg{(}\exp^{(-\beta_{\ell q}\Delta t)}\frac{\sigma^{(\ell)}(i)}{ \sigma^{(\ell+1)}(i)}\Bigg{)}\dots\Bigg{(}\exp^{(-\beta_{\ell q}\Delta t)} \frac{\sigma^{(\ell+q-1)}(i)}{\sigma^{(\ell+q)}(i)}\Bigg{)}\Bigg{(}\frac{\exp ^{(-\beta_{\ell q}^{\text{mod}}At)}}{\exp^{(-\beta_{\ell q}\Delta t)}}\Bigg{)}^ {q} \tag{28}\]  
Using \(a_{\text{min}}\) and \(a_{\text{max}}\), and \(\exp^{(-\beta_{\ell q}^{\text{mod}}q\Delta t)}\leq\exp^{(-\beta_{\ell q}q \Delta t)}\) noted after Equation (23),  
\[a_{\text{min}}^{q}\Bigg{(}\min_{q,\,\theta\,\mathcal{K}_{\text{tr}}}\Bigg{(} \frac{\exp^{(-\beta_{\ell q}^{\text{mod}}\Delta t)}}{\exp^{(-\beta_{\ell q} \Delta t)}}\Bigg{)}^{q}\Bigg{)}\leq a_{\ell q}\leq a_{\text{max}}^{q} \tag{29}\]where \(\min_{q,\ \theta}\mathcal{K}_{,\ \varrho}\left(\exp^{(-\beta_{\theta_{q}}^{\text{mod} }\Delta t)}/\exp^{(-\beta_{\theta_{q}}\Delta t)}\right)^{\theta}\) is the minimum value obtained over all values of \(\mathcal{K}\), \(\varrho\), \(\theta\), and \(q\) (maximum \(q\) is \((M-1)\) for \(\alpha_{1}^{\prime}\) and \((N-M-1)\) for \(\alpha_{2}^{\prime}\)). For a given \(q\) and \(\theta\), \((\exp^{(-\beta_{\theta_{q}}^{\text{mod}}\Delta t)}/\exp^{(-\beta_{\theta_{q}} \Delta t)})^{q}\) is minimum when all \(\mathcal{K}\)s and \(\varrho\)s are set to their respective minimum values (see Equations (17), (19), (22), and (23)). This is greater than or equal to the value \(\mathcal{K}_{\text{min}}^{\prime\prime}\) obtained by setting all \(\mathcal{K}\)s to \(\mathcal{K}_{\text{min}}^{\prime}\). \(\mathcal{K}_{\text{min}}^{\prime\prime}\) is close but less than \(\mathcal{K}_{\text{min}}^{\prime}\). The minimum of all \(\mathcal{K}_{\text{min}}^{\prime\prime}\)s over all values of \(\theta\) and \(q\) is set as the lower bound, and denoted as \(\mathcal{K}_{\text{min}}\) (\(0<\mathcal{K}_{\text{min}}<1\)). It can be determined by following the instructions in matlab_program_figures_1_2.m and then running it. Next, a provision is made to include artificial white noise in \(a_{\ell q}\).