# Critical Review of Basic Methods on DoA Estimation of EM Waves Impinging a Spherical Antenna Array
## 3 Mathematical Formulations
### Spherical Harmonic Decomposition  
Using an SAA, the array covariance matrix and pressure are analyzed within the context of spherical harmonic decomposition (SHD). In this section, the presented formulation follows those given in [32; 33; 34; 26; 35]. The SHD model of the pressure signal \(p(k,r,\Omega)\) for an antenna array with harmonic order \(N_{h}\) and radius \(r\) is expressed using spherical Fourier transform (SFT) approximation and the corresponding inverse [26]:  
\[p_{nm}(k,r)=\sum\limits_{i=1}^{I}\alpha_{i}p(k,r,\Phi_{i})[Y_{n}^{m}(\Phi_{i}) ]^{*} \tag{14}\]  
and  
\[p(k,r,\Omega)=\sum\limits_{n=0}^{N_{h}}\sum\limits_{m=-n}^{n}p_{nm}(k,r)Y_{n} ^{m}(\Omega)\text{, respectively} \tag{15}\]  
where \(Y_{n}^{m}(\cdot)\) is the SH of degree \(m\) and order \(n\)[34; 35]:  
\[Y_{n}^{m}(\Omega)=\sqrt{\frac{2n+1}{4\pi}\frac{(n-m)!}{(n+m)!}}p_{n}^{m}\cos( cos\theta)e^{i\omega\phi} \tag{16}\]  
\(p_{nm}\) is the spherical harmonic domain coefficient, \(\Phi_{i}\) is the coordinates of the antenna, \(\Omega=(\theta,\phi)\) is the direction of steering where the inverse SFT is formulated, and \(\alpha_{i}\) is a sampling weight used to correct errors due to orthonormality. Moreover, \(p_{n}^{m}\) is associated with Legendre polynomials, while \(\left(\cdot\right)^{*}\) represents the complex conjugate. The sampling scheme defines the sampling weights, with the order defined by the number of elements (check [25] for details). Uniform sampling is assumed throughout this paper; therefore, the weight is reduced to \(\alpha_{i}=4\pi/I\). We also adopt harmonic coefficients \(\left(N_{h}+1\right)^{2}=16\), where the array harmonic order is \(N_{h}=3\), and radius \(r=8.4\ cm\), depending on the antenna array under consideration. In spherical arrays, whenever \(kr>N_{h}\), spatial sampling occurs [25]. The noiseless pressure in the spherical harmonic domain is expressed in matrix form as [26]  
\[\mathbf{P}_{nm}(kr)=\mathbf{Y}^{H}(\mathbf{\Phi})\mathbf{\Gamma}\mathbf{Y}(\mathbf{\Phi})\mathbf{B}(kr)\mathbf{Y }^{H}(\Omega)\mathbf{s}(k)=\mathbf{B}(kr)\mathbf{Y}^{H}(\Omega)\mathbf{s}(k), \tag{17}\]  
where \(\mathbf{\Gamma}=diag\{[\alpha_{1},\alpha_{2},\ldots,\alpha_{1}]\}\) is the sampling weights, \(\mathbf{Y}^{H}(\mathbf{\Phi})\mathbf{\Gamma}\mathbf{Y}(\mathbf{\Phi})=\mathbf{I}\), and \(\mathbf{\Phi}=[\Phi_{1},\ldots,\Phi_{I}]\) is the position of the element within the angular coordinates. The left-hand side of Equation (17) is the vectorized form of SH coefficients, i.e.,  
\[\mathbf{P}_{nm}(kr)=[p_{0,0},\ p_{1,-1},\ p_{1,0},\ p_{1,1},\...,\ pN_{h},\ N_{h} ]^{T}, \tag{18}\]  
\(k\) and \(r\) are not included for brevity's sake, and SH are presented in matrix form as  
\[\mathbf{Y}(\mathbf{\Omega})=\left[\begin{array}{cccc}Y_{0}^{0}(\Omega_{1}),&Y_{0}^{ 0}(\Omega_{2})&\cdots&Y_{0}^{0}(\Omega_{u}),\\ Y_{1}^{-1}(\Omega_{1}),&Y_{1}^{-1}(\Omega_{2})&\cdots&Y_{1}^{-1}(\Omega_{u}), \\ Y_{1}^{0}(\Omega_{1}),&Y_{1}^{0}(\Omega_{2})&\cdots&Y_{1}^{0}(\Omega_{u}),\\ Y_{1}^{1}(\Omega_{1}),&Y_{1}^{1}(\Omega_{2})&\cdots&Y_{1}^{1}(\Omega_{u}),\\ \vdots&\vdots&\vdots&\vdots\\ Y_{N_{h}}^{N_{h}}(\Omega_{1}),&Y_{N_{h}}^{N_{h}}(\Omega_{2})&\cdots&Y_{N_{h}}^ {N_{h}}(\Omega_{u}),\end{array}\right]^{T} \tag{19}\]  
Hence, antenna-array-dependent coefficients are described using a diagonal matrix:  
\[\mathbf{B}(kr)=diag\{[b_{0}(kr),\ b_{1}(kr),\ \ \ \ b_{2}(kr),b_{3}(kr),\ldots,b_{N}( kr)]\} \tag{20}\]  
where each array-dependent coefficient in the case of a rigid sphere is [19]  
\[b_{n}(kr)=4\pi i^{n}\left(j_{n}(kr)-\frac{j_{n}^{\prime}(kr)h_{n}(kr)}{h_{n^{ \prime}}(kr)}\right) \tag{21}\]  
\(h_{n}\), \(j_{n}\), \(h_{n^{\prime}}\), and \(j_{n^{\prime}}\) represent the Hankel and Bessel functions and the corresponding derivatives with respect to \(kr\), respectively.