# Critical Review of Basic Methods on DoA Estimation of EM Waves Impinging a Spherical Antenna Array
## 3 Mathematical Formulations
### Spherical Antenna Array (SAA) with Spherical Harmonic Decomposition (SHD)  
The application of theoretical discrete spherical harmonic transformation (SHT) to the receptions of a real antenna array as in Equation (15) leads to serious loss of accuracy in the transformed coefficients, as compared to the theoretical result of the antenna array of equal specifications to the actual one [11; 12; 25]. This is because of the differences between the real antenna array and the theoretical antenna array model, such as calibration errors between antenna elements and antenna mutual coupling effects and positioning error. Conversely, the SHT performance approximates the theoretical counterpart using real array calibration measurements in an anechoic chamber. The SHT of Equation (15) is expressed as an array of encoding filters, and the equalization with inverse modal weights \(\mathbf{B}^{-1}\) can be optimally obtained within the context of least-squares. This is the same as the challenge faced when trying to obtain the spherical steering vector that is optimal w.r.t. the measured properties of the array. In this work, such a measurement-dependent steering vector is adapted. Section 5.3 presents the SAA measurement procedure under consideration. Let \(\hat{\mathbf{Y}}(\mathbf{\Omega})\) represent the measurement steering vector. In an ideal scenario, by approximation, the steering vector is equal to the theoretical steering vector for a plane wave in the transform domain  
\[\hat{\mathbf{Y}}^{H}(\Omega)\approx\mathbf{Y}^{H}(\Omega) \tag{22}\]Equation (17) is the response of a theoretically transformed antenna array, and after equalization, it can be incorporated into the components  
\[\hat{\mathbf{Y}}^{H}(\Omega)=\left(\mathbf{B}^{-1}\mathbf{Y}^{H}(\mathbf{\Phi})\mathbf{\Gamma}_{\phi} \right)\left(\mathbf{Y}(\mathbf{\Phi})\mathbf{B}\mathbf{Y}^{H}(\mathbf{\Omega})\right)=\mathbf{W}_{enc} \mathbf{h}(\mathbf{\Omega}) \tag{23}\]  
where \(\mathbf{h}(\mathbf{\Omega})\) represents the array response vector to the \(\mathbf{\Omega}\) direction or the steering vector, and \(\mathbf{W}_{enc}\) is the encoding matrix that implements the equalization and SHT. Ideally, \(\mathbf{B}^{-1}\mathbf{Y}^{H}(\mathbf{\Phi})\mathbf{\Gamma}_{\phi}\mathbf{Y}(\mathbf{\Phi})\mathbf{B}=\mathbf{I}\), and Equation (22) holds correctly. In a practical situation, both the interpolated array response for an arbitrary \(\mathbf{\Omega}\) and encoding filters can be calculated from response measurements in \(\mathbf{L}\) directions \(\mathbf{\
abla}=[\mathbf{\Psi}_{1},\dots,\mathbf{\Psi}_{L}]\) arranged uniformly in a grid. We can then compute the encoding filters using the solution of regularized weighted least-squares from measurements [34; 35; 36]:  
\[\hat{\mathbf{W}}_{enc}(k)=\mathbf{Y}^{H}(\mathbf{\
abla})\mathbf{\Gamma}_{\psi}\hat{\mathbf{H}}^{ H}(k)\left(\hat{\mathbf{H}}(k)\mathbf{\Gamma}_{\psi}\hat{\mathbf{H}}^{H}(k)+\beta(k)\mathbf{I} \right)^{-1} \tag{24}\]  
where \(\mathbf{\Gamma}_{\psi}\) is the correct weight for the measurement grid, and \(\hat{\mathbf{H}}(k)=[\hat{\mathbf{h}}_{1},\dots,\hat{\mathbf{h}}_{L}]\) is the set of array-measured snapshots at frequency \(k\) in the direction \(\mathbf{\
abla}\), while \(\beta\) is a set of regularization parameters. In this study, we computed the weights of \(\mathbf{\Gamma}_{\psi}\) via the areas of spherical Voronoi cells of the measurement grid. In order to acquire the steering vector \(\mathbf{h}(\mathbf{\Omega})\) of the antenna array with high accuracy in any direction, spherical interpolation was conducted through the expansion of the steering vector as a function of its measured spherical harmonic coefficients, as the SH matrices \(\mathbf{\Upsilon}\) are computed up to an order driven by the measurement number. It is discovered that an accurate truncation order \(N_{tr}\) is accurately described by \(\left(N_{tr}+1\right)^{2}=L/4\) for a regular grid.