# Performance Analyses of a RAIM Algorithm for Kalman Filter with GPS and NavIC Constellations
## 3 Modified Pseudorange Error Models and \(\mathbf{\
If \(\underline{\mathbf{\eta}}(i)\) is the carrier-smoothed pseudorange measurement error of satellite \(i\), then it constitutes the following at an epoch \(\ell\):  
\[\underline{\mathbf{\eta}}^{(\ell)}(i)=\text{mp}^{(\ell)}\big{(}\theta^{(\ell)}(i )\big{)}\underline{\mathbf{\eta}}_{\text{tr}}(i)+\underline{\mathbf{\eta}}_{\text{ cc}}(i)+\underline{\mathbf{\eta}}_{\text{ml-n}}^{(\ell)}(i) \tag{16}\]where the superscript with parentheses denotes time epochs, and \(\text{mp}(\theta)\) is the mapping function of the MOPS model for elevation angle \(\theta\). The autocorrelation between \(\underline{\boldsymbol{\eta}}^{(\ell)}(i)\) and \(\underline{\boldsymbol{\eta}}^{(\ell+q)}(i)\) is  
\[ \begin{array}{ll}E\Big{(}\underline{\boldsymbol{\eta}}^{(\ell)}(i)\underline{ \boldsymbol{\eta}}^{(\ell+q)}(i)\Big{)}&=(\sigma^{(\ell)}(i))^{2}\exp^{(- \beta_{\ell}\Delta t)}=\text{mp}^{(\ell)}\big{(}\theta^{(\ell)}(i)\big{)}\text {mp}^{(\ell+q)}\big{(}\theta^{(\ell+q)}(i)\big{)}\sigma_{\text{tr}}^{2}\exp^{( -q\Delta t/\tau_{\text{tr}})}+\\ &\sigma_{\text{ura}}^{2}\exp^{(-q\Delta t/\tau_{\text{ma}})}+((\sigma_{\text{m },\,i}^{(\ell)})^{2}+(\sigma_{\text{n},\,i}^{(\ell)})^{2})\exp^{(-q\Delta t/ \tau_{\text{ma}})}\end{array}  \tag{17}\]  
where \(\Delta t=1\) s. \(E\) represents the expectation operation. \(\sigma^{(\ell)}(i)\) is the standard deviation of \(\underline{\boldsymbol{\eta}}^{(\ell)}(i)\). \(\beta\) denotes the reciprocal of the time constant of an equivalent FOGM process for \(\underline{\boldsymbol{\eta}}(i)\). \((\sigma^{(\ell)}(i))^{2}\) is  
\[(\sigma^{(\ell)}(i))^{2}=\Big{(}\text{mp}^{(\ell)}\big{(}\theta^{(\ell)}(i) \big{)}\Big{)}^{2}\sigma_{\text{tr}}^{2}+\sigma_{\text{ura}}^{2}+\big{(}( \sigma_{\text{m},\,i}^{(\ell)})^{2}+\Big{(}\sigma_{\text{n},\,i}^{(\ell)})^{ 2}\Big{)} \tag{18}\]  
Thus, based on the correlation between epochs \(\ell\) and \((\ell+q)\), \(\beta\) (denoted as \(\beta_{\ell q}\)) is calculated as  
\[\beta_{\ell q}=-\log\Bigg{(}\frac{E\Big{(}\underline{\boldsymbol{\eta}}^{( \ell)}(i)\underline{\boldsymbol{\eta}}^{(\ell+q)}(i)\Big{)}}{\Big{(}\sigma^{( \ell)}(i)\Big{)}^{2}}\Bigg{)}/(q\Delta t) \tag{19}\]  
It is evident from Equations (17) and (19) that \(\beta_{\ell q}\) is a function of \(q\) and \(\theta\). It should be noted that \(\big{(}\sigma_{\text{m},\,i}^{(\ell)}\big{)}^{2}+\big{(}\sigma_{\text{n},\,i }^{(\ell)}\big{)}^{2}\) is also a function of \(\theta\). Following the definition of \(\boldsymbol{\boldsymbol{\
abla}}\) for normalized innovations \(\Delta\underline{\boldsymbol{\hat{\rho}}}^{*}\), \(\boldsymbol{\
abla}_{\text{sub}}^{i}\) for satellite \(i\) mentioned after Equation (6) can be re-written as  
an epoch \(\ell\) with \(q\ =\ 1\,2,...\) is defined as  
\[a_{\ell q}=\frac{E\Big{(}\underline{\boldsymbol{\eta}}^{(\ell)}(i)\underline{ \boldsymbol{\eta}}^{(\ell+q)}(i)\Big{)}}{\sigma^{(\ell)}(i)\sigma^{(\ell+q)}(i )}=\Big{(}\exp^{(-\beta_{\ell q}\Delta t)}\Big{)}^{q}\frac{(\sigma^{(\ell)}(i) )^{2}}{\sigma^{(\ell)}(i)\sigma^{(\ell+q)}(i)} \tag{20}\]  
\(a_{\ell q}\) can be rearranged as  
\[a_{\ell q}=\Bigg{(}\exp^{(-\beta_{\ell q}\Delta t)}\frac{\sigma^{(\ell)}(i)}{ \sigma^{(\ell+1)}(i)}\Bigg{)}\Bigg{(}\exp^{(-\beta_{\ell q}\Delta t)}\frac{ \sigma^{(\ell+1)}(i)}{\sigma^{(\ell+2)}(i)}\Bigg{)}\ldots\Bigg{(}\exp^{(- \beta_{\ell q}\Delta t)}\frac{\sigma^{(\ell+q-1)}(i)}{\sigma^{(\ell+q)}(i)} \Bigg{)} \tag{21}\]  
Representing the terms within the parentheses as \(a_{l1,\,\ell q}\), \(a_{\ell q}=\prod_{\ell=\ell}^{\ell=\ell+q-1}a_{l1,\,\ell q}\). \(\ell q\) in the subscript of \(a_{l1,\,\ell q}\) denotes that \(\beta\) is obtained by considering the correlation between two epochs \(\ell\) and \(\ell\ +\ q\). Maximum \(a_{l1,\,\ell q}\) over \(\iota\ =\ \ell,\ldots,\ell+q-1\) is denoted as \(a_{l1,\,\ell q}^{\text{max}}\), and the minimum \(a_{l1,\,\ell q}\) is \(a_{l1,\,\ell q}^{\text{min}}\). \(a_{l1,\,\ell q}\)s are calculated with \(\beta_{\ell q}\) of Equation (19).