## 3 Smart Collaborative Positioning Using CD Approach
### Best Satellites Selection Algorithm in CD Process
\tag{18}\]  
where \(x\) represents the value of \(C/N_{0}\) (dB-Hz), \(\alpha\) is a parameter to adjust the curve of the weighting function \(w_{n}(C/N_{0})\) as defined in [15], \(V_{f}\) controls the value of the weighting function \(w_{n}\) for \(x=P\), i.e., \(w_{n}(P)=\frac{1}{V_{f}}\), \(P\) is the value of \(C/N_{0}\) for which the weighting function \(w_{n}\) is forced in order to obtain the weight defined by parameter \(V_{f}\), and \(\gamma\) is the threshold value is defined (step 1.i. of the algorithm description). According to [15], the optimal values obtained for the different parameters of the function \(w_{n,1}\) are: \(\left( \begin{array}{cc}\alpha,\;V_{f},\;P\end{array} \right)=\left(80,\;30, \;20\right)\). The second criterion is the satellite selection method based on maximal elevation angle during the observation period. For the elevation angle parameter, the weighting function is defined as [15]:  
\[w_{n,2}(x)=\left\{ \begin{array}{ll}\frac{\sin^{2}(x)}{\sin^{2}(10^{\circ})},& x<10^{\circ}\\ 1&,\;\;x\geq 10^{\circ}\end{array} \right. \tag{19}\]  
According to step 3 in the best satellite selection algorithm, if the satellites selected by each RS have the same parameters by calculating the cost function, then select the satellites that make up the best GDOP among the selected satellites as:  
\[\min\{\mathrm{GDOP}(x)\}\leq 6 \tag{20}\]  
Finally, the collective detection metric can be expressed in function of the best satellites as:  
\[D_{\mathrm{CD}}\big{(}R_{i},\theta_{j},\Delta B_{n}\big{)}=\sum_{k_{bs}}D_{ind }\Big{(}\tau^{n}_{k_{bs}},\hat{f}^{n}_{d_{k_{bs}}}\Big{)}=\sum_{k_{bs}}\left\{ \sum_{n=1}^{Rx}w_{n}\Big{|}S_{\delta}\Big{(}\tau^{n}_{k_{bs}},\hat{f}^{n}_{d_{ k_{bs}}}\Big{)}\Big{|}^{2}\right\} \tag{21}\]  
where \(k_{bs}\) represents the number of the best satellites, and \(\hat{\tau}^{n}_{k_{bs}}\) represent the estimated code delay corresponding to the best satellites \(k_{bs}\) for a hypothetical location \(\big{(}R_{i},\theta_{j},\Delta D_{m},\Delta B_{l}\big{)}\) w.r.t each RS as:  
\[ \begin{array}{ll}\hat{\tau}^{1}_{k_{bs}}=\frac{\left[\rho_{\mathrm{BS1},k_{bs _{bs}}}+\Delta\rho_{k_{bs_{bs}},1}\big{(}R_{i},\theta_{j},\Delta D_{m},\Delta B _{l}\big{)}\right]_{c\cdot T_{code}}}{c\cdot T_{code}}\cdot N_{code}\\ \hat{\tau}^{2}_{k_{bs}}=\frac{\left[\rho_{\mathrm{BS2},k_{bs_{bs}}}+\Delta \rho_{k_{bs_{bs}},2}\big{(}R_{i},\theta_{j},\Delta D_{m},\Delta B_{l}\big{)} \right]_{c\cdot T_{code}}}{c\cdot T_{code}}\cdot N_{code}\\ \cdots\\ \hat{\tau}^{n}_{k}=\frac{\left[\rho_{\mathrm{BSn},k}+\Delta\rho_{k_{bs}}(R_{i},\theta_{j},\Delta D_{m},\Delta B_{l})\right]_{c\cdot T_{code}}}{c\cdot T_{ code}}\cdot N_{code}\end{array}  \tag{22}\]  
where \(\Delta\rho_{k_{bs},1}\) and \(\Delta\rho_{k_{bs},2}\) are the difference in pseudororange from the MS w.r.t the one measured by the RS1 and RS2, respectively, for the best satellites \(k_{bs}\), as:  
\[ \begin{array}{ll}\Delta\rho_{k_{bs},1}=-R_{1}\cos\big{(}el_{k_{bs},1}\big{)} \cos\big{(}az_{k_{bs},1}-\theta_{1}\big{)}+c\cdot\Delta B_{1}\\ \Delta\rho_{k_{bs},2}=-R_{2}\cos\big{(}el_{k_{bs},2}\big{)}\cos\big{(}az_{k_{bs},2}-\theta_{2}\big{)}+c\cdot\Delta B_{2}\\ \cdots\\ \Delta\rho_{k_{bs},n}=-R_{n}\cos\big{(}el_{k_{bs},n}\big{)}\cos\big{(}az_{k_{bs},n}-\theta_{n}\big{)}+c\cdot\Delta B_{n}\end{array}  \tag{23}\]Thus, the acquisition problem is to search for the optimal vector \(X=(R,\theta,\Delta B)\) in the search space \(R\in[0\) m, \(10,000\) m\(]\), \(\theta\in[0^{\circ},\ 360^{\circ}]\), and \(\Delta B\in[-150,000\) m, \(150,000\) m\(]\) which maximizes the criterion function:  
\[ \begin{array}{l}J=\text{fun}\big{(}D_{\text{CD}}\big{(}R_{i},\theta_{j}, \Delta B_{l}\big{)}\big{)}\\ =\text{fun}\Bigg{(}\sum\limits_{b_{\text{bs}}}D_{ind}\Big{(}\xi_{b_{\text{bs}} }^{n},\hat{f}_{d_{\text{bs}}}^{n}\Big{)}\Bigg{)}\\ =\text{fun}\Bigg{(}\sum\limits_{b_{\text{bs}}}\sum\limits_{n=1}^{R_{\text{bs}}} w_{n}\Big{|}S_{\delta}\Bigg{(}\xi_{b_{\text{bs}}}^{n},\hat{f}_{d_{\text{bs}}}^{n} \Big{)}\Big{|}^{2}\Bigg{)}\end{array}