# Connected Objects Geo-Localization Based on SS-RSRP of 5G Networks
## 3 Distance Estimation
On the other hand, we deduce:

\[P_{\mathbf{R}}(\mathbf{r})=\frac{\mathbf{r}}{\sigma^{2}}\exp-(\frac{\mathbf{r}^{2}}{2\sigma^{ 2}})dr \tag{25}\]

The probability density followed by the distance \(\mathbf{R}\) is that of a Rayleigh variable as illustrated in Figure 5.

As it is shown in Figure 5, the precision of the proposed method is robust when we are approaching the barrier of one meter. The precision becomes weak while approaching the origin. At distance greater than 0.55 m the error probability tends to the zero. Figure 5: Error probability followed by the distance. ### Complexity and Performance Analysis

Our solution uses 2 algorithms. The theoretical complexity depends only on the second one since the first one is executed off-line and the complexity of our solution will be \(O(n^{2})\). The Algorithm described in [22] is based on SS-RSRP measurements and a step using Random forest classifier with the machine learning Library Sklearn. We are interested here in the Algorithm without considering the learning part which is also executed off-line. The theoretical complexity of this Algorithm is \(O(n^{2})\), [PERSON] et al. in [23] proposed a multiple-model PHD filter and map fusion for cooperative positioning and mapping in vehicular networks with 5G mmWave communication links. The proposed framework is based on 3 algorithms. The Map correction Algorithm uses repetitively and successively the two other algorithms. The theoretical complexity of these algorithms is \(O(n^{3})\). Presented works in Table 2 are all compliant with 5G standards but with different complexities. The theoretical complexity of [PERSON] et al. is higher than the other two algorithms and therefore the theoretical execution time is higher. Our work and that of [PERSON] are both based on SS-RSRP and give comparable theoretical complexities therefore comparable theoretical execution times. However, to obtain a reliable learning [PERSON]'s method requires a huge amount of data which makes it difficult to be used in real time. The use of the SS-RSRP and the absence of a learning phase are the main advantages of our work that allowed us to achieve the desired performance. ### CRLB Analysis

The Cramer-Rao Bound (CRB) is a second performance criterion that gives a lower bound to the mean square error of estimation in the set of unbiased estimates. It is used here as a performance measure for the proposed estimator. The observed signal is described by (16). It can be written as

\[D_{Psi}=\underbrace{d_{i}^{2}+d_{j}^{2}-2d_{i}d_{j}\cos(\theta_{i}-\theta_{j} )}_{D_{ij}}+w(n) \tag{26}\]

where \(w(n)\) is a white Gaussian noise following \(N(0,\sigma^{2})\). \(\sigma^{2}\) denotes the variance of the white Gaussian noise. In this equation we have four unknown variables \(\mathbf{\
abla}=[d_{i},d_{j},\theta_{i},\theta_{j}]\). To derive the CRLB, first, we write the likelihood function as

\[p(D_{ij};\mathbf{D}_{Psi})=\prod_{n=0}^{N-1}\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp \left[-\frac{(D_{ij}-D_{Psi})^{2}}{2\sigma^{2}}\right] \tag{27}\]

Since we have

\[=\frac{\mathbf{r}}{2\pi\sigma^{2}}\exp-(\frac{\mathbf{r}^{2}}{2\sigma^{ 2}}) \tag{21}\]0

The Fisher Information matrix \(\mathbf{I}(\Psi)\) doses exist and the CRLB = \(\frac{1}{I(\mathbf{\
abla})}\). In fact

\[=\frac{\mathbf{r}}{2\pi\sigma^{2}}\exp-(\frac{\mathbf{r}^{2}}{2\sigma^{ 2}}) \tag{21}\]1 \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Recent Works** & **5G Compliance** & **Complexity** & **Accuracy (m)** & **Additional Data** \\ \hline Our work & compliant & \(O(n^{2})\) & [0.55, 1] & No \\ [PERSON] et al. [22] & compliant & \(O(n^{2})\) & n/a & Learning Data \\ [PERSON] et al. [23] & compliant & \(O(n^{3})\) & n/a & No \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Performance and complexity comparison. After some mathematical manipulations we find

\[\left[I(\Psi)\right]^{-1}\leq\left( \begin{array}{c}\dfrac{\sigma^{2}}{2N}\\ \dfrac{\sigma^{2}}{2N}\\ 0\\ 0\end{array}