## 2 Collective Detection Approach
### Working Principle of Collective Detection  
The CD approach has been proposed because of its use as both a Direct Positioning method and a High-Sensitivity acquisition method. The direct positioning algorithms are based on a set of individual correlograms formed by code delay/Doppler for the satellites potentially visible. The direct positioning search is carried out in the position/clock bias domain with each potential position and clock bias coordinate is mapped to the code delay and Doppler space for each satellite. The mapping of the signal code delay to the position/clock bias domain of the user (MS) is done differentially with the respect to the pseudorange measurements provided by the reference station (RS), as shown in Figure 2.  
The acquisition search grid is set in a space defined by 3D position coordinates \((\Delta N,\Delta E,\Delta D)\) which are the algebraic distances between the MS and the RS in North, East and Down directions, and the relative clock bias of the receiver to the reference station (\(\Delta B\)). The pseudorange seen by the MS for the satellite \(k\) can be calculated by [7]:  
\[\rho_{\text{MS},k}=\rho_{\text{RS},k}+\Delta\rho_{k} \tag{1}\]  
where \(\rho_{\text{RS},k}\) is the pseudorange seen by the RS, and \(\Delta\rho_{k}\) represents the difference in pseudorange calculated from the MS w.r.t the one measured by the RS for the satellite \(k\). In CD, the uncertainty space is centered on the initial position and clock bias, and can be defined w.r.t. the accuracy of the initial knowledge. The range-offset at a position separated by \((\Delta N,\Delta E,\Delta D,\Delta B)\) from the RS is expressed in terms of the position and the clock bias [2]:  
\[\Delta\rho_{k}(\Delta N,\Delta E,\Delta D,\Delta B)=-\cos(az_{k})\cos(el_{k}) \Delta N-\sin(az_{k})\cos(el_{k})\Delta E+\sin(el_{k})\Delta D\ +\ c\cdot\Delta B \tag{2}\]  
where \(az_{k}\) and \(el_{k}\) are, respectively, the azimuth and elevation angles of satellite \(k\) as seen by the RS. The coordinates \(\Delta N\), \(\Delta E\) and \(\Delta D\) represent the 3D position displacement of the MS with respect to the RS in a North-East-Down (NED) local coordinate frame. The last term \(c\cdot\Delta B\) represents the pseudorange variation due to the clock bias of the MS, \(c\) being the speed of light. In order to considerably decrease the total number of points to evaluate, the circular search area proposed in [7] is used in this paper. The local horizontal search is a polar Rho-theta referential instead of a North-East referential. In this way, \(\Delta N\) and \(\Delta E\) are expressed in terms of \(R\) and \(\theta\)[7]:  
\[\left\{ \begin{array}{l}\Delta N=R.\cos(\theta)\\ \Delta E=R.\sin(\theta)\end{array} \right. \tag{3}\]  
So, from (2) we obtain:  
\[\Delta\rho_{k}(R,\theta,\Delta D,\Delta B)=-R\cos(el_{k})\cos(az_{k}-\theta)+ \sin(el_{k})\Delta D+c\cdot\Delta B \tag{4}\]  
Thus, the estimated code delay for satellite \(k\) for a hypothetical location \(\left(R_{i},\theta_{j},\Delta D_{mt},\Delta B_{n}\right)\) is given by [7]:  
Figure 2: Mapping of the MS code delay search to position/clock bias and pseudorange domains. \[\hat{\tau}_{k}=\frac{\left[\rho_{\text{RS},k}+\Delta\rho_{k}\big{(}R_{i},\theta_{j}, \Delta D_{m},\Delta B_{n}\big{)}\right]_{c\cdot T_{code}}}{c\cdot T_{code}}\cdot N _{code} \tag{5}\]  
where \(T_{code}\) is the signal spreading code period (i.e., 1 ms for GPS L1 C/A code), \(N_{code}\) is the number of code chips per period, and \([. ]_{c\cdot T_{code}}\) represents the modulo \(c\cdot T_{code}\) operation such that \(\hat{\tau}_{k}\in[0,N_{code}-1]\) chip.