# Connected Objects Geo-Localization Based on SS-RSRP of 5G Networks
## System Model
### SS-RSRP Model  
Synchronization Signal-Reference Signal Received Power (SS-RSRP) is useful for 5G system analysis. It can also shed light on emerging 5G Public Networks (PN) operation and thus help develop local licensing regulations and interference control mechanisms. Generally SS-RSRP is measured through the use of pilots symbols, specific to the deployed cell commonly known as Cell specific Reference Signal (CRS) in 4G networks and altered in 5G by the Synchronization Signal Block (SSB). The accurate measurement of SS-RSRP mainly in micro cells and dense urban area and where interferences become a potential source of signal degradation, makes the accuracy of received SS-RSRP a challenging task. Henceforth, this subsection, will derive analytically the received SS-RSRP based on the full knowledge of RSS signal. Another approximation of \(PL\) is used in literature as \(PL=P_{E}-P_{ca}+G_{R_{x}}\), where \(P_{E}\) is the transmission power of \(T_{x}\) antenna, \(P_{ca}\) is the cable losses and \(G_{R_{x}}\) is the \(R_{x}\) antenna gain. The proposed model will be elaborated based on Figure 1, along with the following assumptions:  
1. We will assume in this paper, that there is no cable losses to simplify the calculation process. Hence \(PL=P_{E}+G_{R_{x}}\). 2. The antenna gain depends on the direction of arrival to the Macro-Cell (MC) \(G_{R_{x}}=10^{(\frac{-1800}{\pi\tau})^{2}}\)[25]. Currently OFDM is the potential waveform for 5G networks, hence we consider the \(k^{th}\)-OFDM symbol in baseband model. The transmitter and receiver are both equipped with one antenna \((n_{r}=n_{t})\). The channel type deployed in this paper is a selective Rayleigh channel. The analytical function of the received \(k^{th}\)-OFDM symbol is expressed as \(\mathbf{y}_{k}=\sum_{l=0}^{L-1}h_{k}(l)\mathbf{u}_{k}(k-l)+\mathbf{w}_{k}\), where \(\mathbf{u}_{k}\) is the transmitted symbol, \(\mathbf{w}_{k}\) is zero mean white Gaussian complex noise of variance \(\frac{N_{0}}{2}\) and \(L\) is the number of multipath. Let us define the equivalent channel matrix \(\mathcal{C}^{(n_{t}+n_{g})\times(n_{c}+n_{g})}\), where \(n_{c}\) is the number of subcarrier and \(n_{g}\) is the number of subcarrier reserved for the Cycle Prefix (CP) interval. Thus one can rewrite the \(k^{th}\)-OFDM received signal in matrix notation as \(\mathbf{y}_{k}=G_{k}\mathbf{u}_{k}+\mathbf{w}_{k}\), where \(\mathbf{y}_{k}\in\mathcal{C}^{(n_{c}+n_{g})\times 1}\) and \(\mathbf{w}_{k}\in\mathcal{C}^{(n_{t}+n_{g})\times 1}\) represents additive white Gaussian noise channel at time \(k\). When removing CP, the frequency domain signal is generated again by a DFT process and the whole mathematical expression is defined as \(\mathbf{z}_{k}=\frac{1}{\sqrt{n_{c}}}[(F\otimes I_{n_{c}})\xi_{2}]\mathbf{y}_{k}\), \(\xi_{2}\in\mathcal{C}^{n_{c}\times(n_{c}n_{g}+1)}\) is the cycle prefix removing matrix for the first \(n_{g}\) elements of \(\mathbf{y}_{k}\). This will lead to another expression of the DFT-output as \(\mathbf{z}_{k}=\big{[}(\mathbf{F}\otimes I_{n_{t}})\xi_{3}(\mathbf{F}^{-1}\otimes I_{n_{ t}})\big{]}\mathbf{x}_{k}+\mathbf{w}_{k}=H_{k}\mathbf{x}_{k}+\mathbf{w}_{k}\), \(\xi_{2}=[\mathbf{0}_{n_{t}n_{g}}\mathbf{I}_{n_{g}}]\otimes I_{n_{g}}\). The matrix \(\xi_{3}\in\mathcal{C}^{n_{c}\times n_{c}}\) is the circulant matrix, and \(\xi_{3}=\xi_{2}G_{k}\xi_{1}\) where \(\mathbf{w}_{k}\) is the noise in the frequency domain with zero mean and variance \(\sigma_{w}^{2}\) and \(H_{k}\) is \(\mathcal{C}^{n_{c}\times n_{c}}\) frequency domain matrix defined as \(\mathbf{H}_{k}=\big{[}(F\otimes I_{n_{c}})\xi_{3}(F^{-1}\otimes I_{n_{c}})\big{]}\). Figure 1: Angle orientation with Micro-cell. Now we can express analytically the channel matrix in frequency domain \(\mathbf{H}_{k}\) as [26; 27]:  
\[\mathbf{H}_{k}=\left( \begin{array}{ccc}\mathbf{H}_{k}(0)&&0\\ &\ddots&\\ 0&&\mathbf{H}_{k}(n_{c}n_{t})\end{array}