## 7 Numerical Analysis  
### Performance Evaluation of DoA Estimation Techniques  
The performance of DoA estimation is assessed using a simulation environment in MATLAB and is measured in terms of the root mean square error (RMSE), as discussed in [1]. The expression for RMSE is as follows:  
\[\sigma(\theta)=\sqrt{E[(\hat{\theta})-(\theta)]^{2}}, \tag{25}\]  
where \(\hat{\theta}\) and \(\theta\) represent the estimated and actual values of the DoAs, respectively. Thus, the RMSE performance is measured in degrees, denoted as \(\sigma(\theta)\), plotted against the \(\gamma\), i.e., the signal-to-noise ratio in decibels [dB]. The simulation parameters for the signal model and the notations used for the uplink and downlink scenarios are specified in Table 1. Unless explicitly stated otherwise, the values specified in the table are used in this study. A flowchart illustrating the proposed model's complete analysis and working principle is presented in Figure 5. The flowchart serves as a visual aid of the overall process and enables clear comprehension of how the proposed model functions. The RMSE for the DAS and MUSIC methods were calculated to evaluate their performance under various channel conditions and diverse scenarios. The following channel conditions were considered: (i) AWGN only, (ii) AWGN and Rician fading, and (iii) spatially correlated channels with AWGN and Rician fading. The investigations were treated as separate cases, allowing for a comprehensive analysis of the method's performance under different conditions. Figure 5: Flowchart demonstrating the proposed model’s complete analysis and working principle. Figure 6: RMSE \(\sigma(\theta)\) performance of uplink geostationary and low Earth orbit satellites using DAS and MUSIC techniques for AWGN only (Case-i(a)). Note that, from Figure 7 onwards, all the outputs achieved are only for the downlink geometry under different cases. Therefore, the downlink scenario with AWGN for GEO and LEO satellites in Figure 7 (Case-i(b)) depicts the observations below:  
* The DAS and MUSIC methods are considered optimal when assuming AWGN (i.e., matched filtering provides the best results for a predefined noise distribution). * The RMSE performance of both DAS and MUSIC methods for the LEO satellite matches the \(\sigma_{\text{CR}}(\theta)\) within the range of (\(-10\) to \(50\)) [dB]. However, for the GEO satellite, the performance is close to \(\sigma_{\text{CR}}(\theta)\). * When the range of \(\gamma\) decreases below \(-10\) [dB], the DAS and MUSIC curves deviate beyond the Cramer-Rao lower bound (\(\sigma_{\text{CR}}(\theta)\)) for both GEO and LEO satellites. This deviation occurs because the estimation of the channel-phase random variable is limited to a range of (\(-\pi<\phi<\pi\)). Consequently, the standard deviation of the \(\phi\) analysis is constrained to this range and remains constant when the \(\gamma\) value drops to \(-20\) [dB]. In contrast, the \(\sigma_{\text{CR}}(\theta)\) applies to all values and can take any real numbers. Figure 8 (Case-ii) demonstrates the impact of AWGN and Rician fading, where the channel phase is modeled as a random variable. The performance of the DoA estimation methods for both GEO and LEO satellites is evaluated and denoted as \"DAS-RICE\" and \"MUSIC-RICE\". Notably, the RMSE statistics \(\sigma(\theta)\) exhibit a decline from \(0\) [dB] onwards for (Case-ii), indicating the adverse effects of a multipath environment on the system model. As a result, the performance of the DoA estimation methods deteriorates under these channel conditions. Figure 8: RMSE \(\sigma(\theta)\) performance of downlink geostationary and low Earth orbit satellites using DAS and MUSIC techniques for AWGN and Rician fading scenario (Case-ii). Figure 7: RMSE \(\sigma(\theta)\) performance of downlink geostationary and low Earth orbit satellites using DAS and MUSIC techniques for AWGN only (Case-i(b)).