# Critical Review of Basic Methods on DoA Estimation of EM Waves Impinging a Spherical Antenna Array
## 5 Numerical Experiments, Results, and Discussion
### Two Wideband Signals  
Four cases were simulated in the second example, where two signals arrive during the same time window. Likewise, as shown in [26], two wideband signals were simulated with one signal delayed by \(\mathbf{d}\) samples; i.e., the first and second delayed signals are \(s_{1}(t)=\delta(t-1024/f_{s})\) and \(s_{2}(t)=\delta(t-(1024+d)/f_{s})\), respectively. When the delay approaches zero, the signal covariance matrix is coherent, and when \(\mathbf{d}\) is increased, the covariance matrix of the signal is incoherent. The four delays considered here are \(d=0\), \(d=0.5\), \(d=1\), and \(d=8\) samples, and the covariance matrices of the signals are  
\[\widetilde{\mathbf{R}}_{ss}=E_{eq}\!\left[ \begin{array}{cc}1&1\\ 1&1\end{array} \right]\!,\,\widetilde{\mathbf{R}}_{ss}=E_{eq}\!\left[ \begin{array}[] {cc}1&0.85\\ 0.85&1\end{array} \right]\!,\,\widetilde{\mathbf{R}}_{ss}=E_{eq}\!\left[ \begin{array} []{cc}1&0.5\\ 0.5&1\end{array} \right]\!,\,\text{and}\,\widetilde{\mathbf{R}}_{ss}=\mathbf{E}_{eq}\! \left[ \begin{array}{cc}1&-0.04\\ -0.04&0.97\end{array} \right] \tag{65}\]  
respectively. The two signals were simulated as plane waves with DoAs \(\Omega_{1}=(90,0)^{\circ}\) and \(\Omega_{2}=(90,\phi_{2})^{\circ}\) and with harmonic order \(\text{N}_{h}=3\), and the second plane-wave direction was changed with \(2^{1+n_{k}\times 0.2}\), where \(n_{k}=0,\ 0.5,\ 1,\dots,7^{(\circ)}\). The value of SNR was then set at 20 dB. The SH coefficients in the time domain were computed as in the previous section and windowed by \(\text{N}_{w}=35\) samples of a lengthy rectangular window. The window was concentrated around the first signal at \(1024/f_{s}\). As a result, the covariance matrix estimate of the array was averaged with the time domain \(\text{N}_{w}=35\) samples, as given in Equation (34). Figure 3: Root-mean-square error over CRLB and 100 Monte Carlo samples: (**a**) elevation and (**b**) azimuth angle versus noise variance \(\sigma^{-2}\) for a single signal in \(\phi=\theta=60^{\circ}\). The azimuth angle has a larger CRLB and RMSE than the elevation angle. (**a**) \(\hat{\theta}\), elevation (**b**) \(\hat{\phi}\), azimuth. The RMSEs for CRLB and all of the techniques over 100 Monte Carlo samples against \(\phi_{2}\) when \(SNR=20\) dB and the second signal is delayed by d samples are shown in Figure 4. Firstly, it can be observed that PWD performs poorly in the estimation of DoA of multiple signals because its \(\phi_{2}\) increases as its RMSE increases. This can be interpreted as a bias in estimation: the PWD global maximum is found in between the true values, and the second maximum is \(180^{\circ}\) apart from the second one; the two signals generate a maximum in the spatial spectrum of PWD in between the true values of \(\phi_{1}\) and \(\phi_{2}\). Due to the large energy that this maximum has, RMSE is lower than CRLB, \(\phi_{2}\leq 3^{\circ}\) with \(d=0\). In addition, the next PWD spectrum maximum is generated by the steering vector's conjugate value that generated that global maximum. MUSIC has an identical profile to PWD, and its estimation is identically biased when \(d=0\). Similarly, MUSIC and PWD performance improves slightly when \(\phi_{2}>60^{\circ}\). Consequently, the segregation is longer than the Rayleigh resolution \(180^{\circ}/N_{h}=60^{\circ}\), and the techniques segregate the two signals. Nonetheless, estimates of PWD and MUSIC are also biased when \(\phi_{2}>60^{\circ}\). WSF and ML techniques result in smaller values than CRLB in the case of coherence when \(\phi_{2}\leq 3^{\circ}\), that is, at \(d=0\). This is because of the bias in estimation. All of the techniques have their global minimum within the true values and showcase compelling proof for a biased estimate, as in PWD. Hence, the information in second-order covariance matrix moments is highly coherent for an estimation that is not biased when \(\phi_{2}\leq 3^{\circ}\) and \(d=0\). Firstly, the steering vectors are identical when signals come from various directions that are close to each other.