# Adaptive DCS-SOMP for Localization Parameter Estimation in 5G Networks
## 4 Proposed Method
### Adaptive Search in DCS-SOMP Approach for 3D Parameter Estimation  
We introduce an adaptive approach to enhance the parameters estimated in the two aforementioned stages: (1) sensing matrix construction and (2) application of the DCS-SOMP approach for 3D parameter estimation. Our method dynamically adapts the search space for each angle to be estimated, i.e., 2D-AoD and 2D-AoA. To illustrate how dynamic adaptation is performed, Figure 3 demonstrates the improvements made in each iteration using our adaptive approach for the azimuth AoD angle. In summary, in iteration \(k\), the search interval (\(\mathbf{\hat{\phi}}^{(k)}\)) is dynamically adapted using the index (\(t_{l}^{az}\)) of the previous search interval (\(\mathbf{\hat{\phi}}^{(k-1)}\)). New parameters are estimated in each iteration. The process continues until a predetermined number of iterations (\(K\)) is reached or a pre-established threshold (\(\epsilon\)) is exceeded. We adaptively calculate the candidates for AoD azimuth angle (\(\mathbf{\hat{\phi}}^{(k)}\)) with \(q_{az}\) values that are uniformly spaced within a new interval \([\hat{\phi}_{start}^{az},\hat{\phi}_{end}^{az}]\), where  
\[\hat{\phi}_{start}^{az}= \begin{cases}\mathbf{\hat{\phi}}_{t_{l}^{az}-1}^{(k-1)}, \text{ if }t_{l}^{az}>1\\ 0,\text{ if }t_{l}^{az}=1\end{cases} \text{ and }\quad\hat{\phi}_{end}^{az}= \begin{cases}\mathbf{\hat{\phi}}_{t_{l}^{az}+1}^{(k-1)},\text{ if }t_{l}^{az}<q_{az}\\ q_{az},\text{ if }t_{l}^{az}=q_{az}\end{cases} . \tag{17}\]  
Similarly, we adaptively calculate the candidates for AoD elevation angle (\(\mathbf{\hat{\phi}}^{(k)}\)) with \(q_{el}\) values that are uniformly spaced within a new interval \([\hat{\phi}_{start}^{el},\hat{\phi}_{end}^{el}]\), where  
\[\hat{\phi}_{start}^{el}= \begin{cases}\mathbf{\hat{\phi}}_{t_{l}^{al}-1}^{(k-1)}, \text{ if }t_{l}^{el}>1\\ 0,\text{ if }t_{l}^{el}=1\end{cases} \text{ and }\quad\hat{\phi}_{end}^{el}= \begin{cases}\mathbf{\hat{\phi}}_{t_{l}^{al}+1}^{(k-1)},\text{ if }t_{l}^{el}<q_{el}\\ q_{el},\text{ if }t_{l}^{el}=q_{el}\end{cases} . \tag{18}\]  
For the AoA azimuth angle, we adaptively calculate the candidates (\(\mathbf{\hat{\theta}}^{(k)}\)) with \(q_{az}\) values that are uniformly spaced within a new interval \([\hat{\theta}_{start}^{az},\hat{\phi}_{end}^{az}]\), where  
\[\hat{\theta}_{start}^{az}= \begin{cases}\mathbf{\hat{\theta}}_{\gamma_{l}^{az}-1}^{ (k-1)},\text{ if }\gamma_{l}^{az}>1\\ 0,\text{ if }\gamma_{l}^{az}=1\end{cases} \text{ and }\quad\hat{\theta}_{end}^{ az}= \begin{cases}\mathbf{\hat{\theta}}_{\gamma_{l}^{az}+1}^{(k-1)},\text{ if }\gamma_{l}^{az}<q_{az}\\ q_{az},\text{ if }\gamma_{l}^{az}=q_{az}\end{cases} . \tag{19}\]  
Figure 3: Adaptive Procedure for Selecting Eligible Candidates for Azimuth AoD. Similarly, we adaptively calculate the candidates for AoA elevation angle (\(\dot{\mathbf{\theta}}^{(k)}\)) with \(q_{el}\) values that are uniformly spaced within a new interval \([\dot{\theta}^{cl}_{start},\ddot{\mathbf{\theta}}^{cl}_{end}]\), where  
\[\ddot{\mathbf{\theta}}^{cl}_{start}= \begin{cases}\dot{\mathbf{\theta}}^{(k-1)}_{\gamma^{ cl}_{l}-1^{\prime}}\text{, if }\gamma^{cl}_{l}>1\\ 0\text{, if }\gamma^{cl}_{l}=1\end{cases} \text{ and }\quad\ddot{\mathbf{\theta}}^{cl}_{end}= \begin{cases}\dot{\mathbf{\theta}}^{(k-1)}_{\gamma^{cl}_{l}+1^{\prime}}\text{, if }\gamma^{cl}_{l}<q_{el}\\ q_{el}\text{, if }\gamma^{cl}_{l}=q_{el}\end{cases} .