# Adaptive DCS-SOMP for Localization Parameter Estimation in 5G Networks
## 4 Proposed Method
### Adaptive Search in DCS-SOMP Approach for 3D Parameter Estimation
We selected arbitrary actual values for two paths as described in Table 2. We set \(q_{az}=18\) and \(q_{el}=18\) when not using the adaptive search, i.e., DCS-SOMP, as this is the maximum value our simulator could process. When using the adaptive search, we set \(q_{az}=6\) and \(q_{el}=6\). Additionally, we defined the parameters as follows: \(f_{c}=28\) GHz, \(B=100\) MHz, \(N_{t}=64\), \(N_{r}=64\), \(N=10\), and \(M_{t}=20\). To establish a maximum number of iterations, we determined the difference between the estimated value in iteration \(k\) and the estimated value in iteration \(k-1\), and as long as this difference is greater than the threshold (\(\varrho=10^{-6}\)), the next iteration will be performed. Figures 4a-6 display the comparison between DCS-SOMP and adaptive DCS-SOMP for 2D-AoD, 2D-AoA, and ToA. The results correspond to a random run for Path 1. It's evident that, for all cases, the adaptive DCS-SOMP significantly enhances the estimation of the respective angle as early as the second iteration, gradually converging towards the actual value with each subsequent iteration. We analyzed the complexity of the methods in terms of execution time and number of mathematical operations. Following the analysis provided in [18; 19], the complexity of the SOMP algorithm is \(O(LN_{t}q_{c})\), where \(q_{c}=q_{az}q_{el}\) denotes all possible combinations for the candidates for azimuth (\(q_{az}\)) and elevation angles (\(q_{el}\)). In the DCS-SOMP algorithm, we perform the Kronecker product (tensor product for matrices) to estimate 2D-AoD and 2D-AoA parameters, resulting in \(O(LN_{t}(q_{c})^{2})\). In our adaptive DCS-SOMP approach, we fix the number of candidates and we update the values for the candidates at each iteration. Thus, the complexity of the adaptive DCS-SOMP is \(O(LN_{t}q_{c}q_{it})\), where \(q_{it}\) is the number of \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Path** & \(\mathbf{\phi^{az}}\) **(rad)** & \(\mathbf{\phi^{el}}\) **(rad)** & \(\mathbf{\theta^{az}}\) **(rad)** & \(\mathbf{\theta^{el}}\) **(rad)** & \(\mathbf{\tau}\) **(us)** \\ \hline
1 & 1.33 & 0.45 & 3.50 & 0.90 & 0.0615 \\
2 & 2.80 & 1.15 & 5.20 & 1.45 & 0.0767 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Actual Values Used in Simulations. Figure 4: Comparison of (**a**) Azimuth AoD estimation and (**b**) Elevation AoD estimation results using DCS-SOMP and adaptive DCS-SOMP. The outcomes pertain to Path 1. In the DCS-SOMP case, \(q_{az}\) and \(q_{el}\) are both set to 18, whereas in the adaptive DCS-SOMP scenario, \(q_{az}\) and \(q_{el}\) are both set to 6.

iterations. The main mathematical operations are due to (7) and (8). We determine the total number of mathematical operations, \(q_{op}\), as follows:

\[q_{op}=(q_{op}^{w}+q_{op}^{h})(q_{il}L+1), \tag{24}\]

where \(q_{op}^{w}\) is the number of mathematical operations to construct the sensing matrix, (7), determined as follows:

\[q_{op}^{w}=N(q_{c}M_{t}N_{t}+q_{c}(M_{t}N_{t}-1)+M_{t}N_{t}(q_{c})^{2}), \tag{25}\]

and \(q_{op}^{h}\) is the number of mathematical operations to determine the max correlation, (8), determined as follows:

\[q_{op}^{h}=(2N_{t}M_{t})N(q_{c})^{2}. \tag{26}\]

Table 3 presents the execution times and the number of mathematical operations for each method. For the adaptive DCS-SOMP method, we utilized 14 iterations. These values represent the total duration required by the method to estimate all five parameters for both paths. Our observations indicate that the adaptive method is notably faster and involves fewer mathematical operations compared to the traditional method. Figure 5: Comparison of (**a**) Azimuth AoA estimation and (**b**) Elevation AoA estimation results using DCS-SOMP and adaptive DCS-SOMP. The outcomes pertain to Path 1. In the DCS-SOMP case, \(q_{az}\) and \(q_{el}\) are both set to 18, whereas in the adaptive DCS-SOMP scenario, \(q_{az}\) and \(q_{el}\) are both set to 6. Figure 6: Comparison of ToA estimation results using DCS-SOMP and adaptive DCS-SOMP. The outcomes pertain to Path 1.