# Connected Objects Geo-Localization Based on SS-RSRP of 5G Networks
## 3 Distance Estimation
Perfect knowledge of channel state has been supposed in all our performance evaluation trials. ### MSE Method

We used Matlab software Release 2019 Ubuntu version to compile Algorithms 1 and 2. Initially a realistic radio signal strength environment is generated using Algorithm 1. The two COs are supposed to be mobile within one meter square with a full coverage ensured by an MC. Afterward Algorithm 2 is enabled to generate the estimated distance between the two COs. We illustrated the whole process as in Figure 2. \begin{table}
\begin{tabular}{l l} \hline \hline
**System Parameter** & **Parameter Value** \\ \hline Channel mode & TDL-D \\ Bandwidth & 20 MHz \\ Number of subcarriers & 3300 \\ Extended Cyclic Prefix & 2048 \\ Subcarrier spacing & 60 kHz \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The 5G NR Rel.15 Simulation Parameters. Figure 2: The basic model of assessment. In order to scale the accuracy of the proposed Algorithm, we used the Minimum Mean Square Error (MSE). As it is illustrated in Figure 3 the MSE tends to zeros at the very first rounds of Algorithm 2. We notice that the convergence time for Algorithm 2 is not immediate, this is due to the needed calculation time required for RSSI generation in Algorithm 1 and also the processing time needed for Algorithm 2. Henceforth the MSE values become zeros after a short transitional phase. ### Accuracy Quantification

We can quantify the accuracy of our method as the deviation in horizontal and vertical coordinates \((\mathbf{x},\mathbf{y})\) compared to the estimated coordinates \((\mathbf{\hat{x}},\mathbf{\hat{y}})\). \(\mathbf{Q}=(\mathbf{x}-\mathbf{\hat{x}},\mathbf{y}-\mathbf{\hat{y}})=(\mathbf{X},\mathbf{Y})\). Normal distribution appears to be the most convenient choice, especially in the light of Figure 4. One can consider the deviations at \(\mathbf{x}\) and \(\mathbf{y}\) are independent random variables and follow Gaussian laws of zero mean and variance \(\sigma^{2}=0.1758\) m. The intervals next to the parameter are the 95% confidence intervals for the distribution parameters. These values are collected after Matlab processing of Figure 4. The density of joint probability of deviations \((\mathbf{X},\mathbf{Y})\) is given by:

\[P_{\mathbf{X}\mathbf{Y}}(\mathbf{x},\mathbf{y})=P_{\mathbf{X}}(\mathbf{x})P_{\mathbf{Y}}(\mathbf{y})=\frac{ \exp-(\frac{\mathbf{x}^{2}+\mathbf{y}^{2}}{2\sigma^{2}})}{2\pi\sigma^{2}} \tag{17}\]

Figure 4: Distribution of the obtained errors. Figure 3: Minimum Mean Square Error. The characterization of the precision in \(\mathbf{x}\) and in \(\mathbf{y}\) does not give information about the probability of the distance between the desired point and the point obtained. To obtain this probability, we change Cartesian coordinates to polar coordinates according to the transformation:

\[(\mathbf{X},\mathbf{Y})=(\mathbf{R}\cos(\Theta),\mathbf{R}\sin(\Theta)) \tag{18}\]

Thus the Jacobian of transformation is \(\mathbf{r}\):

\[\mathbf{r}=\left|\frac{\partial(x,y)}{\partial(\mathbf{r},\theta)}\right|=\left| \begin{array}{cc}\cos(\theta)&\sin(\theta)\\ -\mathbf{r}\sin(\theta)&\mathbf{r}\cos(\theta)\end{array} \right| \tag{19}\]

This will lead to :

\[P_{\mathbf{R}\mathbf{\Theta}}(r,\theta) =\frac{\mathbf{r}}{2\pi\sigma^{2}}\exp-\left[\frac{(\mathbf{r}\cos(\theta ))^{2}+(\mathbf{r}\sin(\theta))^{2}}{2\sigma^{2}}\right] \tag{20}\] \[=\frac{\mathbf{r}}{2\pi\sigma^{2}}\exp-(\frac{\mathbf{r}^{2}}{2\sigma^{ 2}}) \tag{21}\]

Hence to find the marginal, it is necessary to integrate on the other random variable, that is:

\[P_{\mathbf{\Theta}}(\theta) =\int_{r=0}^{+\infty}P_{\mathbf{R},\mathbf{\Theta}}(r,\theta) \tag{22}\] \[=\int_{r=0}^{+\infty}\frac{\mathbf{r}}{2\pi\sigma^{2}}\exp-(\frac{ \mathbf{r}^{2}}{2\sigma^{2}})dr\] (23) \[=\frac{1}{2\pi} \tag{24}\]

The angle is therefore uniformly distributed over \([0,2\pi]\).