## 2 Mathematical Modeling  
### Spatial Representation  
To solve the mathematical problem, the spatial representation of the objects of interest is needed. The signal emitter is denoted by \(E\) and the satellites are denoted by \(S_{1},S_{2},\ldots,S_{k},\ldots,S_{n}\), where \(n\) is the total number of satellites. Both the signal emitter and the receiving satellites are represented as geometrical points without spatial extension. In general, when the signal emitter is located on Earth, it is typically represented by geodetic latitude \(\Lambda\), longitude \(\lambda\), and height \(h\) (LLH) as a set of coordinates: \(\mathbf{x}_{E}|_{\text{LLH}}=(\Lambda,\lambda,h)^{\text{T}}\). It can be recalled that geodetic latitude is defined as the angle between the equatorial plane and the normal unit vector, which is normal at the considered point on the representative ellipsoid of Earth's surface. Instead, the \(k\)-th satellite is generally expressed in perifocal coordinate system (PQW) using orbital parameters, as shown in (1), where \(a_{k}\) is the semi-major axis, \(e_{k}\) is the orbit eccentricity, and \(v_{k}\) is the true anomaly [28,29]. The perifocal coordinate system is a coordinate system used to represent the orbit of a satellite. It has its center in the Earth, which is in one of the foci of the elliptic orbit of the satellite, and is denoted by the PQW acronym because the unit vectors are commonly denoted by the letters \(\mathbf{p}\), \(\mathbf{q}\), and \(\mathbf{w}\); in particular, \(\mathbf{p}\) is directed towards the periapsis of the orbit, \(\mathbf{q}\) belongs to the plane of the orbit and is spaced by \(90^{\circ}\) from \(\mathbf{p}\), while the third one is obtained from the cross product \(\mathbf{w}=\mathbf{p}\times\mathbf{q}\) and is orthogonal to the plane of the orbit. \[\left.\mathbf{x}_{\text{S}_{k}}\right|_{\text{PQW}}=\frac{a_{k}\big{(}1-e_{k} ^{2}\big{)}}{1+e_{k}\cos(v_{k})} \begin{pmatrix}\cos\left(v_{k}\right)\\ \sin\left(v_{k}\right)\\ 0\end{pmatrix} \quad k=1,2,\ldots,n \tag{1}\]  
For a satellite \(S_{k}\), the data conversion from PQW to ECI occurs through the matrices of rotation \(\mathbf{R}_{(\Omega_{k}3)}\), \(\mathbf{R}_{(i_{k}1)}\), and \(\mathbf{R}_{(\omega_{k}3)}\), represented in (2). In particular, for the \(k\)-th satellite,\(i_{\text{k}}\) is the orbit inclination, \(\Omega_{\text{k}}\) is the right ascension of the ascending node, and \(\omega_{\text{k}}\) is the argument of perigee; the second argument of the matrices is related to the axis of rotation. \[ \begin{split}\mathbf{R}_{(\omega_{\text{k}},3)}&= \begin{bmatrix}\cos\left(\omega_{\text{k}}\right)&-\sin\left(\omega_{\text{k}} \right)&0\\ \sin\left(\omega_{\text{k}}\right)&\cos\left(\omega_{\text{k}}\right)&0\\ 0&0&1\end{bmatrix}&k=1,2,\ldots,n\\ \mathbf{R}_{(i_{\text{k}},1)}&=\begin{bmatrix}1&0&0\\ 0&\cos\left(i_{\text{k}}\right)&-\sin\left(i_{\text{k}}\right)\\ 0&\sin\left(i_{\text{k}}\right)&\cos\left(i_{\text{k}}\right)\end{bmatrix}&k=1,2, \ldots,n\\ \mathbf{R}_{(\Omega_{\text{k}},3)}&=\begin{bmatrix}\cos\left(\Omega_{\text{k}} \right)&-\sin\left(\Omega_{\text{k}}\right)&0\\ \sin\left(\Omega_{\text{k}}\right)&\cos\left(\Omega_{\text{k}}\right)&0\\ 0&0&1\end{bmatrix}&k=1,2,\ldots,n\\ \end{split}  \tag{2}\]  
PQW-to-ECI conversion is then reported as in (3). \[\mathbf{x}_{\text{S}}\big{|}_{\text{ECI}}=\mathbf{R}_{(\Omega_{\text{k}},3)} \mathbf{R}_{(i_{\text{k}},1)}\mathbf{R}_{(\omega_{\text{k}},3)}\mathbf{x}_{ \text{S}}\big{|}_{\text{PQW}}\quad k=1,2,\ldots,n \tag{3}\]  
For the emitter, data conversion follows two steps: from LLH to ECEF (Earth-Centered Earth-Fixed) and from ECEF to ECI. Denoting with \(e_{\oplus}\) the eccentricity of Earth's ellipsoid and with \(a_{\oplus}\) the ellipsoidal equatorial radius, the LLH-to-ECEF conversion [30] is reported as in (4). The value of the function \(N(\Lambda)\) is reported in (5). \[\mathbf{x}_{\text{E}}\big{|}_{\text{ECEF}}=