# Adaptive DCS-SOMP for Localization Parameter Estimation in 5G Networks
## 3 System Model  
As outlined in the 3 rd Generation Partnership Project (3 GPP) guidelines released in the TR 38.901 V17.0.0 [14], we analyze a MIMO system with \(N_{t}\) transmitter antennas at the base station (BS) and \(N_{r}\) receiver antennas at the MS. This system operates at carrier frequency \(f_{c}\) and with bandwidth \(B\). Similarly to [5], we consider the BS to send \(G\) signals to the receiver, where the \(g\)-th signal is composed of \(M_{t}\) symbols. Furthermore, we consider an indoor environment where there are \(L\) paths for the signals being transmitted from a single BS to the MS, such paths can be classified as Line of Sight (LoS) and single-bounce Non-Line of Sight (NLoS) paths. Multiple-bounce paths are not taken into account due to their limited reception strength at mmWave frequencies [15]. Additionally, we assume perfect clock synchronization, as in [16], enabling accurate ToA estimation. Figure 1 illustrates a typical Industry 4.0 scenario in a smart warehouse where a forklift moves products from the conveyor belt to the shelves. In such a scenario, the localization and tracking of the forklift can be achieved using geometry-based methods, which rely on accurate estimation of localization parameters [5; 17]. As in [14], we assume the channel's dependency on the response vector. Thus, for the \(n\)-th subcarrier (where \(n=0,...,N-1\)), the channel matrix \(\mathbf{H}\) can be represented as follows:  
\[\mathbf{H}[n]=\sum_{l=1}^{L}\rho_{l}h_{l}e\frac{-j2\pi(n-1)\tau_{l}}{NT_{s}} \sqrt{N_{l}}\mathbf{a}_{l}^{H}(\phi_{l}^{az},\phi_{l}^{el})\sqrt{N_{r}} \mathbf{a}_{r}(\theta_{l}^{az},\theta_{l}^{el}), \tag{1}\]  
where, for the \(l\)-th path, \(\rho_{l}\) represents the pathloss, \(h_{l}\) stands for the complex channel gain, \(\tau_{l}\) is the ToA, \(\phi_{l}^{az}\) is the azimuth AoD, \(\phi_{l}^{el}\) is the elevation AoD, \(\phi_{l}^{az}\) is the azimuth AoA, \(\theta_{l}^{el}\) is the elevation AoA, and \(T_{s}=1/B\) denotes the sampling period. In order to take into \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Article** & **Method** & **Array** & **ToA** & **2D-AoD** & **2D-AoA** \\ \hline
[5] & DCS-SOMP & ULA & ✓ & \(\times\) & \(\times\) \\
[8] & MUSIC & ULA & ✓ & \(\times\) & ✓ \\
[10] & SBT & ULA & \(\times\) & \(\times\) & \(\times\) \\
[11] & MUSIC & URA & \(\times\) & ✓ & ✓ \\
[12] & SSFEAL & UCA & \(\times\) & \(\times\) & ✓ \\
**Our** & Adaptive & UCA & ✓ & ✓ & ✓ \\
**Proposal** & DCS-SOMP & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Related work overview. Figure 1: Indoor Localization Scenario with LoS and NLoS Paths: in a smart warehouse, a forklift is connected to a mmWave 5G network for product transport. account 2D-AoD, i.e., \((\phi^{az}_{l},\phi^{el}_{l})\) and 2D-AoA, i.e., \((\theta^{az}_{l},\theta^{el}_{l})\), we employ a Uniform Circular Array (UCA) as in [13], thus we define the response vectors \(\mathbf{a}_{t}\) and \(\mathbf{a}_{r}\), as follows:  
\[\mathbf{a}_{t}(\phi^{az}_{l},\phi^{el}_{l})=\Big{[}1,e^{j\frac{2\pi}{\lambda} rsin(\phi^{el}_{l})cos(\phi^{el}_{l}-sin(1))},\dots,e^{j\frac{2\pi}{\lambda} rsin(\phi^{el}_{l})cos(\phi^{m_{z}-sin(N_{t}-1))}}\Big{]}, \tag{2}\]  
\[\mathbf{a}_{r}(\theta^{az}_{l},\theta^{el}_{l})=\Big{[}1,e^{j\frac{2\pi}{ \lambda}rsin(\theta^{el}_{l})cos(\theta^{m_{z}-sin(1)})},\dots,e^{j\frac{2\pi }{\lambda}rsin(\theta^{el}_{l})cos(\theta^{m_{z}-sin(N_{t}-1))}}\Big{]}, \tag{3}\]  
where \(\lambda\) is the wavelength, and \(r\) is the radius of the UCA. When a massive antenna array is used, as described in [13], the radius can be defined as \(r=(N_{t}-1)\frac{\lambda/2}{2\pi}\) for \(\mathbf{a}_{t}(\theta^{az}_{l},\phi^{el}_{l})\), and as \(r=(N_{r}-1)\frac{\lambda/2}{2\pi}\) for \(\mathbf{a}_{r}(\theta^{az}_{l},\theta^{el}_{l})\).