# Critical Review of Basic Methods on DoA Estimation of EM Waves Impinging a Spherical Antenna Array
## 4 DoA Estimation
### Smoothing in Time Domain  
The formulations in [26] are followed in the processing of SH domain coefficients. The normalized spherical harmonic coefficients are achieved by multiplication of the left-hand side of Equation (17) by \(\mathbf{B}^{-1}(kr)\), which results in  
\[\mathbf{q}_{nm}(k)=\mathbf{Y}^{H}(\mathbf{\Omega})\mathbf{s}(k)+\mathbf{z}(k), \tag{27}\]  
where \(\mathbf{z}(k)=\mathbf{B}^{-1}\mathbf{Y}^{H}(\mathbf{\Phi})\mathbf{\Gamma}_{n}(k)\). The covariance matrix of the array in this scenario is expressed as  
\[\mathbf{R}_{aa}(k)=\mathbf{Y}^{H}(\mathbf{\Omega})\mathbf{R}_{ss}(k)\mathbf{Y}(\mathbf{\Omega})+\mathbf{R}_ {zz}(k) \tag{28}\]  
while the covariance matrix of the noise is  
\[\mathbf{R}_{zz}(k)=\sigma^{2}\mathbf{I}\Big{|}\mathbf{B}^{-1}(kr)\mathbf{Y}^{H}(\mathbf{\Phi})\mathbf{ \Gamma}\Big{|}^{2} \tag{29}\]  
which depends on \(k\), and \(||\cdot||\) is the Frobenius norm. The corresponding time-domain representation of \(\mathbf{q}_{nm}(k)\) is obtained through the inverse Fourier transform \(\mathcal{F}^{-1}\{\cdot\}\) as \(\mathbf{q}_{nm}(t)=\mathcal{F}^{-1}\{\mathbf{q}_{nm}(k)\}\). Nevertheless, multiplying by \(\mathbf{B}^{-1}\) results in unwanted noise amplification in some frequencies [26]. Hence, the same equalization is used in time-domain smoothing, as given in [26]:  
\[H_{eq}(k)=\left[\frac{\sigma^{2}}{\gamma}\sum_{n=0}^{N_{k}}\sum_{m=-n}^{n}|| \mathbf{B}^{-1}(kr)\mathbf{Y}(\mathbf{\Phi})\mathbf{\Gamma}||^{2}\right]^{-\frac{1}{2}} \tag{30}\]  
where \(\frac{\sigma^{2}}{\gamma}\) is the constant normalization factor that is the same for all frequencies. In the case of the antenna array considered in this work, the \(H_{eq}(k)\) magnitude response has the shape of a high-pass filter. As a result of equalization, the frequency-domain covariance matrix of the array is  
\[\widetilde{\mathbf{R}}_{aa}(k)=\mathbf{Y}^{H}(\Omega)\widetilde{\mathbf{R}}_{ss}(k)\mathbf{Y} (\mathbf{\Omega})+\widetilde{\mathbf{R}}_{zz}(k) \tag{31}\]  
where  
\[\widetilde{\mathbf{R}}_{zz}(k)=||H_{eq}(k)\,||^{2}\mathbf{R}_{zz}(k) \tag{32}\]  
\[\widetilde{\mathbf{R}}_{ss}(k)=R\Big{\{}s(k)s^{H}(k)=||H_{eq}(k)\,||^{2}\mathbf{R}_{ ss}(k) \tag{33}\]  
and it represents the equalized forms of variables possessing \(\widetilde{\cdot}\). In our estimation, it is assumed that the noise matrix in the time-domain form is spatially white and time-independent (i.e., \(\mathbf{R}_{zz}(t)=\widetilde{\mathbf{R}}_{zz}=\widetilde{\sigma}_{a}^{2}\mathrm{I}\), and \(\widetilde{\sigma}_{a}^{2}\) is the equalized variance).  
The covariance matrix estimate of the array for \(N\) snapshots (time instant \(t_{1},\dots,\ t_{N}\)) is  
\[\hat{\mathbf{R}}_{aa}=1/N\sum_{i=1}^{N}\widetilde{q}(t_{i})\widetilde{\mathbf{q}}^{H} (t_{i}), \tag{34}\]  
\(\widetilde{\mathbf{q}}_{nm}(t)=\mathcal{F}^{-1}\big{\{}H_{eq}(k)\widetilde{\mathbf{q} }_{nm}(k)\big{\}}\) denotes the equalized time-domain spherical harmonic coefficients.  
The performance in single-frequency bins is also of interest in this work. We introduce spatial whitening to the steering vectors and covariance matrix estimate in the frequency-domain estimate as in [26]. Therefore, other formulations in this section are provided for time-domain smoothing and are equal to the frequency-domain smoothing equivalent if each whitened form is a substitute for the steering vectors and the covariance matrix estimate.