## 7 Numerical Analysis
### Performance Evaluation of DoA Estimation Techniques
Figure 9 illustrates the performance of DoA estimation techniques for GEO and LEO satellites under the scenario where spatial correlation is present in addition to AWGN and Rician fading. The figures denote the techniques as \"DAS-RICE-CORR\" and \"MUSIC-RICE-CORR\" for DAS and MUSIC methods, respectively. It can be observed that the performance of (Case-iii) shown in Figure 9 appears similar to that of (Case-ii) presented in Figure 8. For precise visualization of the graphs and to study the effect of spatial correlation in a multipath environment, the AWGN and MUSIC methods were excluded; the comparison between (Case-ii) and (Case-iii) is presented in Figure 10.  
Figure 10 illustrates that the performances of (Case-ii) and (Case-iii) are similar for both GEO and LEO satellites. However, (Case-iii) performs slightly better due to the inclusion of spatial correlation. The reason for this can be explained as follows: since the line-of-sight (LoS) angle is less random, the parameter estimation for it is the same for all paths. This leads to a better performance for a limited number of antennas (\(N=4\)), as seen in the \"LEO:DAS-RICE-CORR\" curve. Therefore, subsequent studies only focus on this curve for a more accurate comparison. Figure 11 presents the performance of the \"LEO:DAS-RICE-CORR\" case with a higher number of antennas (\(N=12\)), denoted as (Case-iv). The results reveal a noteworthy improve  
Figure 9: RMSE \(\sigma(\theta)\) performance of downlink geostationary and low Earth orbit satellites using the DAS and MUSIC techniques for spatially correlated channels with AWGN and Rician fading (Case-iii). ment in the RMSE \(\sigma(\theta)\) performance for both \"LEO:DAS-RICE-CORR\" and \(\sigma_{\text{CR}}(\theta)\) cases, which demonstrates the positive impact of a higher number of antennas on the system model. Certain parameters are often held constant to maintain consistency and ensure accurate results. In this work, three parameters are considered fixed unless otherwise specified: (i) \(\xi\) (i.e., samples) at 1000, (ii) \(K\) (i.e., K factor value) at 30 [dB] in the Rician fading channel and (iii) \(D_{0}\) (i.e., decorrelation distance) at the mobile station (MS) and base station (BS) as 0.5 \(\lambda\) and 13 \(\lambda\) for the prior investigations. However, the subsequent studies deviated from these fixed values in certain case scenarios to understand how these parameters affect the overall results. By varying these parameters, the study identified their individual impact and drew more robust conclusions from their analysis. Firstly, in Figure 12, two \(\xi\) values (i.e., 1000 and 2000) are evaluated as (Case-v) that studies the impact of change in sample values on RMSE \(\sigma(\theta)\) performance. A little deviation is observed on the \"LEO:DAS-RICE-CORR\" curve, while in the \(\sigma_{\text{CR}}(\theta)\) case, there is a good improvement for an increase in the \(\xi\) value. Similarly, in Figure 13, two \(K\) factor values (i.e., 30 and 100) [dB] are evaluated as (Case-vi) that study the impact of multipath on the system model. At a Rician fading channel with \(K=30\) [dB], the RMSE \(\sigma(\theta)\) consistently demonstrates poor performance  
Table 1.0 RMSE \(\sigma(\theta)\) performance of downlink low Earth orbit satellite using the DAS technique for change in antenna values in spatially correlated channels with AWGN and Rician fading (Case-iv). Table 1.1 RMSE \(\sigma(\theta)\) performance of downlink low Earth orbit satellite using the DAS technique for change in sample sizes in spatially correlated channels with AWGN and Rician fading (Case-v). due to the absence of a line-of-sight (LOS) path. However, for \(K=100\) [dB], the RMSE \(\sigma(\theta)\) performance is comparable to \(\sigma_{\text{CR}}(\theta)\), and the channel behavior resembles a Gaussian channel having a strong LOS path.