# Critical Review of Basic Methods on DoA Estimation of EM Waves Impinging a Spherical Antenna Array
## 6 Conclusions  
DoA estimation of electromagnetic waves impinging on an SAA with a small number of snapshots was investigated. This paper studies DoA estimation of electromagnetic (EM) waves impinging on an SAA in short time windows, which finds applications in spacecraft and satellite communication. These techniques can avoid smoothing operations and cope with coherent signals. Multipath signal analysis in a short time window is of particular interest in this paper. Previously, ML techniques have been used within linear and planar antenna array configurations, but this paper presents them within the confines of an SAA configuration. Numerical computer simulations were conducted using the RMSE versus the square root of CRLB to evaluate the performance of MUSIC, PWD beamformer, stochastic and deterministic ML, and the subspace technique WSF (a large sample stochastic ML approximation) in direction estimation using an SAA. The results obtained indicate that the WSF and ML methods perform better than MUSIC and PWD in the coherent or partially correlated signals studied. They correctly estimated the DoA of various signals with adequate resolution, in contrast to PWD and MUSIC.  
Figure 7: GE performance evaluation of MUSIC, PWD, DML, SML, and WSF against SNR using the measured data in [40].  
Furthermore, deterministic ML or WSF performed better than stochastic ML because of lower convergence probability. For example, when \(=900\) MHz, \(f=2\) GHz, and \(f=4\) GHz, the estimation from all of the techniques was biased, as RMSE had an average of around \(2^{\circ}\) for all \(\phi_{2}\) values. At these frequencies, half of the separation angle was lower than CRLB for all the separation angles studied. Therefore, we expect RMSE and CRLB not to meet when there is high error variance. The poor effective SNR causes poor performance in these frequencies. The effective SNR in low frequencies is smaller than in higher frequencies because of the array geometry. Therefore, ML techniques are good candidates for DoA estimation of electromagnetic waves impinging on an SAA, while WSF is a better choice if the computational complexity factor is of great importance.  
In spite of the results presented in this study, there are a number of concerns that should be investigated in the future. For instance, we hope to further extend this work to the calibration and measurement of time delay in various applications, compare them to the state-of-the-art in terms of computational complexity, and evaluate their robustness to confounding factors that inevitably lead to alterations in practical SAA applications. Therefore, ML techniques, particularly DML and/or WSF, may spur positive development in SAA processing and its related applications, such as mobile communication systems, aerospace, and radar systems.  
Conceptualization, O.J.F. and T.S.; methodology, O.J.F.; software, O.J.F.; validation, O.J.F. and T.S.; formal analysis, O.J.F.; investigation, T.S.; resources, T.S.; writing--original draft preparation, O.J.F.; writing--review and editing, T.S.; supervision, T.S. All authors have read and agreed to the published version of the manuscript.  
The APC was funded by the University of Johannesburg, South Africa.  
Data availability StatementData are available upon request from authors.  
This research is supported in part by the University Research Committee (URC) of the University of Johannesburg, South Africa.  
The authors declare no conflict of interest.