## 2 Signal Model  
Considering a spherical antenna array \(V\), the v\(th\) radiator is located at \(u_{v}=(R,\ \phi_{v})\), where \(\phi_{v}=(\theta_{v},\varphi_{v})\). R, \(\theta\), \(\varphi\) represent the radius, elevation, and azimuth. There exist \(I\) far field sources situated at \(\psi_{i}=(\theta_{v},\varphi_{v})\in\xi\) with \(\xi=\{(\theta,\varphi)|\theta\in(0,\pi],\varphi\in(-\pi,\pi]\}\). The signal impinging on the SAA \(\mathbf{G}(t)\) is  
\[\mathbf{G}(t)=\mathbf{A}(\overset{\rightarrow}{\psi})\mathbf{s}(t)+\mathbf{n}(t) \tag{1}\]  
where \(\mathbf{s}(t)\) and \(\mathbf{n}(t)\) denote the source and noise, respectively. \(\mathbf{A}(\overset{\rightarrow}{\psi})\) is the array manifold, \(\mathbf{A}(\overset{\rightarrow}{\psi})=[\mathbf{a}(\psi_{i}),\dots,\mathbf{a}(\psi_{I})] \in\mathbb{C}^{V\times I}\). If \(Y_{n}^{m}(\theta,\varphi)\) represents the SH of order \(n\) and degree \(m\), then  
\[ \begin{array}{c}Y_{n}^{m}(\theta,\varphi)=\sqrt{\frac{(2n+1)(n-m)!}{4\pi(n+ m)!}}P_{n}^{m}(cos\theta)e^{im\varphi}\\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\forall 0\leq n\leq N,\ 0\leq m\leq n \end{array}  \tag{2}\]  
\(P_{n}^{m}(cos\theta)\) denotes the connected Legendre polynomial, the v\(th\) element of \(\mathbf{a}(\psi_{i})\) is then formulated as the SH series decomposition by neglecting the high order \(b_{n}(\kappa R)\) when \(n>\kappa R\)[50]  
\[a_{v}(\psi_{i})=\sum_{n=0}^{N}\sum_{m=-n}^{n}b_{n}(\kappa R)[Y_{n}^{m}(\psi_{ i})]^{*}Y_{n}^{m}(\phi_{i}) \tag{3}\]  
where \(N\) is the maximum SH order, \(\kappa=\lambda/2\pi\) and \(\lambda\) denotes the wavelength, and the frequency dependent constituent \(b_{n}(\kappa R)\) is expressed as [50]  
\[b_{n}(\kappa R)= \begin{cases}4\pi i^{n}J_{n}(\kappa R)&\text{open sphere}\\ \qquad 4\pi i^{n}(J_{n}(\kappa R)-\frac{I_{n}(\kappa R)}{u_{n}(\kappa R)}&u_{n} (\kappa R))\end{cases} \quad\text{rigid sphere} \tag{4}\]  
\(u_{n}\) and \(J_{n}\) are the second kind of spherical Hankel function and the first kind of spherical Bessel function, correspondingly. \(\mathbf{J}_{n}^{\prime}\) and \(\mathbf{u}_{n}^{\prime}\) are the derivatives of \(J_{n}\) and \(u_{n}\), respectively. So, in matrix form, the SAA model is  
\[\mathbf{A}(\overset{\rightarrow}{\psi})=\mathbf{Y}(\overset{\rightarrow}{\phi})\bm {B}\mathbf{Y}^{H}(\overset{\rightarrow}{\psi}) \tag{5}\]  
where the \(v\)th original vector of \(\mathbf{Y}(\overset{\rightarrow}{\phi})\in\mathbb{C}^{V\times(N+1)^{2}}  
and \(\mathbf{B}\in\mathbb{C}^{(N+1)^{2}\times(N+1)^{2}}\) is  
\[\mathbf{B}=diag(b_{0}(\kappa R),\ldots,b_{N}(\kappa R)). \tag{7}\]  
Applying the SH transformation as  
\[\mathbf{P}(t)=\mathbf{Y}^{H}(\overset{\rightarrow}{\phi})\mathcal{H}\mathbf{G}(t)=\mathbf{B} \mathbf{Y}^{H}(\overset{\rightarrow}{\psi})\mathbf{s}(t)+\mathbf{c}(t) \tag{8}\]  
where \(\mathcal{H}=diag(\alpha_{1},\alpha_{2},\ldots\alpha_{v})\) satisfies [50; 51]  
\[\mathbf{Y}^{H}(\overset{\rightarrow}{\phi})\mathcal{H}\mathbf{Y}(\overset{ \rightarrow}{\phi})=\mathbf{I}. \tag{9}\]  
\(\mathbf{c}(t)\) represents the reshaped/transform added noise. Hence, the SH covariance matrix is expressed as  
\[\mathbf{R}=E\Big{\{}\mathbf{P}(t)\mathbf{P}^{H}(t)\Big{\}}=\mathbf{B}\mathbf{Y}^{H}(\overset{ \rightarrow}{\psi})\mathbf{R}_{\mathbf{s}}\mathbf{Y}(\overset{\rightarrow}{\psi})\mathbf{B} ^{H}+\mathbf{C} \tag{10}\]  
with \(\mathbf{C}=E\big{\{}\mathbf{c}(t)\mathbf{c}^{H}(t)\big{\}}\) and \(\mathbf{R}_{\mathbf{s}}=E\big{\{}\mathbf{s}(t)\mathbf{s}^{H}(t)\big{\}}\). We can remark that if \(E\big{\{}\mathbf{n}(t)\mathbf{n}^{H}(t)\big{\}}=\sigma^{2}\mathbf{I}\), where \(\sigma^{2}\) denotes the noise power. The position of the SAA elements \(\overset{\rightarrow}{\phi}\) satisfies the spherical t-design configuration in [50] (\(\mathcal{H}=\mathbf{I}\)), then \(\mathbf{C}=\sigma^{2}I\). Considering the v\(th\) column of \(\mathbf{R}\) in Equation (10) to be \(\mathbf{r}_{v}\) which can be regarded as one array measurement.