# Influence of Noise-Limited Censored Path Loss on Model Fitting and Path Loss-Based Positioning +
## 5 Model Fitting Results
### Example 3  
The 3 GPP TR 38.901 channel model includes PL models for various scenarios from 0.5 to 100 GHz [14]. In this paper, we use this model for the urban microcell scenario at 2 and 28 GHz frequencies. The model describes the PL distributions for LOS and NLOS and the LOS probability model. The LOS probability depends on the link distance:  
\[P_{\rm LOS}=\left\{ \begin{array}{rl}&1\,,\,d<18\,\rm{m}\\ &\frac{18\,\rm{m}}{d}+\exp\left(-\frac{d}{36\,\rm{m}}\right)\left(1-\frac{18\, \rm{m}}{d}\right)\,,\,18\,\rm{m}<d.\end{array} \right. \tag{12}\]  
The LOS probability is 100% up to 18 m, 50% at 52 m, 10% at 189 m, and 1% at 1800 m. In LOS, the expected PL is  
\[\overline{PL}_{\rm LOS}=\left\{ \begin{array}{rl}&PL_{1}\,,\,18\,\rm{m}<d<d^ {\prime}_{\rm BP}\\ &PL_{2}\,,\,d^{\prime}_{\rm BP}<d<5000\,\rm{m}\end{array} \right. \tag{13}\]  
\[PL_{1}=32.4+21\log_{10}(d)+20\log_{10}(f_{\rm{c}}) \tag{14}\]  
\[PL_{2}=32.4+40\log_{10}(d)+20\log_{10}(f_{\rm{c}})-9.5\log_{10}((d^{\prime}_{ \rm BP})^{2}+(h_{\rm BS}-h_{\rm UT})^{2}), \tag{15}\]  
where \(f_{\rm{c}}\) is frequency in GHz, \(h_{\rm BS}\) is the base station height, \(h_{\rm UT}\) is the user terminal height, and \(d^{\prime}_{\rm BP}\) is the break point defined as [14]  
\[d^{\prime}_{\rm BP}=4h^{\prime}_{\rm BS}h^{\prime}_{\rm UT}f_{\rm{c}}/c_{0}, \tag{16}\]  
where \(h^{\prime}_{\rm BS}=h_{\rm BS}-1\) m and \(h^{\prime}_{\rm UT}=h_{\rm UT}-1\) m, \(f_{\rm{c}}\) is the frequency in Hz, and \(c_{0}\) is the speed of light. We assume \(h_{\rm BS}=10\) m and \(h_{\rm UT}=2\) m. The breakpoint distances are 240 and 3400 m at 2 and 28 GHz, respectively. Therefore, within reasonable link distances, at 2 GHz the LOS PL model is a dual-slope model with path loss exponent 2.1 up to 240 m and 4.0 for longer link distances. In NLOS, the expected PL is  
\[\overline{PL}_{\rm NLOS}=\max{(\overline{PL}_{\rm LOS},\overline{PL}^{\prime }_{\rm NLOS})}, \tag{17}\]  
where  
\[\overline{PL}^{\prime}_{\rm NLOS}=35.3\log_{10}(d)+22.4+21.3\log_{10}(f_{ \rm{c}})-0.3(h_{\rm UT}-1.5\,\rm{m}), \tag{18}\]and \(\overline{PL}_{\text{LOS}}\) is given by (13). The NLOS PL model has a path loss exponent of 3.53 and stronger frequency dependency than the LOS model. The shadow fading is modeled as a zero-mean log-normal distribution with \(\sigma=4\) dB in LOS and \(\sigma=7.82\) dB in NLOS. Training data were created with a uniform distribution of distances between 20 and 3000 m and divided into LOS and NLOS according to the LOS probability. The training data, fitted models, and noise threshold levels are illustrated in Figure 4 for LOS and NLOS at 2 and 28 GHz. Noise threshold levels of 120 and 140 dB were used at 2 and 28 GHz, respectively. The parameter estimates are summarized in Table 2. In LOS, censored PL, with the chosen \(PL^{*}\) and maximum link distance range, is rare, and therefore, the OLS and MLE fitted models are quite similar. In NLOS, there is a clear difference between OLS and MLE, just as in Examples 1 and 2 presented in Sections 5.1 and 5.2. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & & \(\hat{\mathbf{\alpha}}\) & \(\hat{\mathbf{\beta}}\) & \(\hat{\mathbf{\sigma}}\) \\ \hline \multirow{4}{*}{2 GHz} & \multirow{2}{*}{LOS} & OLS & 2.8 & 27 & 4.5 \\ \cline{3-6}  & & MLE & 3.0 & 23 & 4.7 \\ \cline{3-6}  & & OLS & 2.2 & 56 & 6.3 \\ \cline{3-6}  & & MLE & 3.5 & 29 & 7.8 \\ \hline \multirow{4}{*}{28 GHz} & \multirow{2}{*}{LOS} & OLS & 2.1 & 61 & 4.0 \\ \cline{3-6}  & & MLE & 2.1 & 61 & 4.0 \\ \cline{3-6}  & & OLS & 2.1 & 81 & 6.2 \\ \cline{3-6}  & & MLE & 3.5 & 53 & 7.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Fitted OLS and MLE parameter estimates for Example 3 at 2 and 28 GHz. Figure 4: Fitting the path loss model to the training data with OLS (green) and MLE (red); Example 3, 2 GHz LOS, 2 GHz NLOS, 28 GHz LOS, 28 GHz NLOS. Training data with \(PL\) under the noise threshold are shown with black dots, solid lines show \(\overline{PL}(d)\), and the dash lines are \(\overline{PL}(d)\pm 1.96\delta\).