## 2 Mathematical Modeling
### Dilution of Precision
\begin{array}{cccc}-\frac{y_{\rm E}-y_{\rm S,1}}{z}&\frac{ x_{\rm E}-x_{\rm S,1}}{z}&0\\ -\frac{y_{\rm E}-y_{\rm S,2}}{z}&\frac{x_{\rm E}-y_{\rm S,2}}{z}&0\\ \vdots&\vdots&\vdots\\ -\frac{y_{\rm E}-y_{\rm S,k}}{z}&\frac{x_{\rm E}-x_{\rm S,2}}{z}&0\\ \vdots&\vdots&\vdots\\ \frac{y_{\rm E}-y_{\rm S,2}}{z}&\frac{x_{\rm E}-x_{\rm S,2}}{z}&0\\ \frac{\left(x_{\rm E}-x_{\rm S,1}\right)\left(x_{\rm E}-x_{\rm S,1}\right)}{z \mu^{2}}&\frac{\left(y_{\rm E}-y_{\rm S,1}\right)\left(x_{\rm E}-z_{\rm S,1} \right)}{z\mu^{2}}&-\frac{z_{\rm S}}{\mu^{2}}\\ \frac{\left(x_{\rm E}-x_{\rm S,2}\right)\left(x_{\rm E}-z_{\rm S,2}\right)}{z \mu^{2}_{\rm S,2}}&\frac{\left(y_{\rm E}-y_{\rm S,2}\right)\left(z_{\rm E}-z_{ \rm S,2}\right)}{z\mu^{2}_{\rm S,2}}&-\frac{z_{\rm S}}{\mu^{2}_{\rm S,2}}\\ \vdots&\vdots&\vdots\\ \frac{\left(x_{\rm E}-y_{\rm S,k}\right)\left(x_{\rm E}-z_{\rm S,k}\right)}{z \mu^{2}_{\rm S,k}}&\frac{\left(y_{\rm E}-y_{\rm S,k}\right)\left(z_{\rm E}-z_{ \rm S,k}\right)}{z\mu^{2}_{\rm S,k}}&-\frac{z_{\rm S}}{\mu^{2}_{\rm S,k}}\\ \vdots&\vdots&\vdots\\ \frac{\left(x_{\rm E}-x_{\rm S,n}\right)\left(z_{\rm E}-z_{\rm S,n}\right)}{ z\mu^{2}_{\rm S,n}}&\frac{\left(y_{\rm E}-y_{\rm S,n}\right)\left(z_{\rm E}-z_{ \rm S,n}\right)}{z\mu^{2}_{\rm S,n}}&-\frac{z_{\rm S}}{\mu^{2}_{\rm S,n}}\\ \end{array} \right] \tag{24}\]  
Under the assumptions in (14), it is possible to simplify the expression in (12) as in (25). \[\mathbf{F}=\frac{1}{\sigma^{2}}\mathbf{H}^{\rm T}\mathbf{H} \tag{25}\]  
Combining (25) and (11), (26) is obtained. \[PDOP=\sqrt{\mathit{Tr}\left(\mathbf{F^{-1}}\right)}=\sigma\sqrt{\mathit{Tr} \left(\left(\mathbf{H}^{\rm T}\mathbf{H}\right)^{-1}\right)} \tag{26}\]  
It is remarked that for some authors, especially in the GNSS literature (e.g., [37]), only the geometric part of (26) (i.e., the value of the square root that only depends on the Geometric Design Matrix \(\mathbf{H}\)) is considered for the definition of the DOP, here indicated as \(PDOP^{\prime}\), as shown in (27). \[PDOP^{\prime}=\frac{PDOP}{\sigma}=\sqrt{\mathit{Tr}\left(\left(\mathbf{H}^{ \rm T}\mathbf{H}\right)^{-1}\right)} \tag{27}\]  
The expression in (27) leads to the position error \(\sigma_{\rm p}\), which can be written [38] as shown in (28). \[\sigma_{\rm p}=PDOP=PDOP^{\prime}\sigma \tag{28}\]Since the purpose of this study is the minimization of the position error \(\sigma_{\rm p}\), and assuming that \(\sigma=\text{const. }\), the particular convention used for the definition of the DOP does not affect the result because \(\min\left(\sigma_{\rm p}\right)=\min(PDOP)=\min(PDOP^{\prime})\). One of the consequences of all the assumptions that leads to \(\sigma=\text{const. }\) is that the problem of minimization of \(PDOP\) (or \(PDOP^{\prime}\)) is independent of both the array geometry and the frequency of the signal. The authors highlight that assuming \(\sigma=\text{const. }\) is a very strong hypothesis that comes from a high-level modeling of the system of satellites, and real-case applications must take into account the variations in the standard deviation \(\sigma\), which depends on the actual fine design of the receiver and on \(SNR\). Another observation is that the value to be optimized considered in model (11) combined with the assumptions made so far in Sections 2.1 and 2.2 is instantaneous and related to a single measurement per receiver.