# Critical Review of Basic Methods on DoA Estimation of EM Waves Impinging a Spherical Antenna Array
## 5 Numerical Experiments, Results, and Discussion
### Two Signals in One Frequency  
Here, the techniques' performance based on a single frequency is examined. The noise is whitened, as stated above, because of the applied frequency-domain processing [25]. That is, the steering vectors and whitened frequency-domain covariance matrix are employed against the time-domain forms (Equation (27)). Spatial aliasing should be considered when estimating DoA with an SAA in a single frequency. Typically, when \(kr>N_{h}\), spatial aliasing exists in high frequencies [36]. However, WSF and ML techniques are not affected by spatial aliasing, because \(\mathbf{B}(kr)\) does not have any zeros (i.e., continuous), and \(\mathbf{Y}(\mathbf{\Omega})\) and \(\mathbf{Y}(\mathbf{\Phi})\) are the same for all of the frequencies. When we conduct an analysis of signals in a single frequency, the resulting unavoidably coherent covariance matrix is  
\[\widetilde{\mathbf{R}}_{ss}(k)=||H_{\mathit{eq}}(k)||^{2}\left[ \begin{array}{cc} 1&1\\ 1&1\end{array} \right], \tag{66}\]  
and the covariance matrix estimate of the array becomes  
\[\mathbf{R}_{aa}(k)=\widetilde{\mathbf{q}}(k)\widetilde{\mathbf{q}}^{H}(k). \tag{67}\]  
In this simulation, the frequencies chosen are the resonant frequencies that fall within a band. It is worth mentioning that covariance matrices of the array are not averaged over the chosen band, but we examined the performance in single frequencies, as given in Equation (62). Wideband signals were simulated with \(d=0,\ N=2048,\ N_{h}=3\), and 20 dB SNR. The SH coefficients in the frequency domain were estimated by Equation (27) through the discrete Fourier transform (DFT), and frequency bins closest to frequencies \(f\) were adopted in this analysis. The resolution of the frequency is \(f_{s}/N\), and the bandwidth of each frequency analyzed is about 2 GHz due to windowing \"spectral leakage\" or aliasing. Figure 5 shows the simulation results for different frequencies. The techniques exhibit identical performance to that in the wideband scenario when \(=8\) GHz, \(f=16\) GHz, and \(f=32\) GHz, but \(f=16\) GHz performs best. The reason why CRLB \(>\) RMSE when \(f=16\) GHz, \(f=32\) GHz, \(f=40\) GHz, and \(f=75\) GHz is the same as in the aforementioned wideband scenario. This indicates that the difference between the covariance matrix of signals is coherent, and the steering vectors of signals impinging from that angle are small. MUSIC and PWD share the same problems in the single-frequency scenario as in the wideband scenario when \(d=0\). Furthermore, when \(=900\) MHz, \(f=2\) GHz, and \(f=4\) GHz, the estimations from all of the techniques are biased, as RMSE has an average of around \(2^{\circ}\) for all \(\phi_{2}\) values. At these frequencies, half of the separation angle is lower than CRLB for all 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. Figure 6 shows the simulation result for RMSE against SNR for MUSIC, PWD, DML, SML, and WSF. WSF shows better performance in the low-SNR regime. DML and SML show a high level of similarity to WSF. DML and WSF show greater performance than SML. This is because SML converges to a local minimum but does not exhibit convergence to the global minimum. MUSIC and PWD exhibit worse performance compared to other techniques, as in other cases. WSF and DML exhibit identical behavior. Hence, they are good choices in multiple signal analysis, especially in the single-frequency analysis. Furthermore, WSF is more computationally efficient for large U than DML. In the wideband scenario, there will be no wide gap in the performance of any of the techniques. Figure 5: _Cont_. Figure 5: _Cont._  
Generally, in this paper, the results obtained reveal that all of the techniques studied perform well for a single signal in the analysis window.