# Critical Review of Basic Methods on DoA Estimation of EM Waves Impinging a Spherical Antenna Array
## 3 Mathematical Formulations
### Analysis of the Space Domain  
The position of the antenna's \(i\)th element is presented in the 3-D Cartesian coordinate form as  
\[\mathbf{m}_{i}=r_{i}[\sin(\theta_{i})\cos(\phi_{i}),\sin(\theta_{i})\sin(\phi_{i} ),\cos(\theta_{i})]^{T} \tag{3}\]  
where \(\Phi_{i}=(\theta_{i},\phi_{i})\) and \(r_{i}\) are the general spherical coordinates, azimuth \(\phi_{i}\in[-180,\ 180)^{\circ}\), radius \(r_{i}\in[0,\infty)\), and elevation \(\theta_{i}\in[0,\ 180]^{\circ}\). Each signal event \(\mathbf{u}\) arriving at the antenna  
Figure 2: SAA with 64 elements as modeled with CST: (**a**) side view and (**b**) top view [7]. has propagated a distance \(d_{u}\) from the source to the antenna with a DoA \(\Omega_{u}=(\theta_{u},\phi_{u})\) w.r.t. the array origin and can be expressed in Cartesian coordinates as  
\[\mathbf{r}_{u}=d_{u}[\sin(\theta_{u})\cos(\phi_{u}),\sin(\theta_{u})\sin(\phi_{u}), \cos(\theta_{u})]^{T} \tag{4}\]  
Each signal is considered a plane wave affected by the free space phenomena mentioned above. A wideband pressure signal \(p_{i}\) in the \(i\)th antenna for a particular signal source \(s\) is depicted in the frequency domain as the addition of all snapshots or EM signals \(u=~{}1,\ldots,~{}U\):  
\[p_{i}(k)=\sum_{u=1}^{U}\dot{h}_{iu}(k)\dot{s}_{u}(k)+n_{i}(k)~{}~{}\in\mathbb{C} \tag{5}\]  
where \(\dot{h}_{iu}(k)\) is the frequency response of a snapshot (an event consists of multiple waves), while \(n_{i}(k)\) is a noise variable, which is assumed to be identically distributed and independent for each antenna element. In addition, wave number \(k=\omega/c=2\pi f/c\). The signal source \(\dot{s}_{u}(k)\) characterizes the frequency response of the source, which is originally transmitted in a few directions, arriving at the receiving antenna array from direction \(\mathbf{r}_{u}\). For a snapshot with multiple signals, the frequency response depends on the time delay \(\tau_{iu}(k)\), which defines the travel time of the signal to the antenna and the complex amplitude of each event \(c_{iu}\), i.e.,  
\[\dot{h}_{iu}(k)=c_{iu}(k)\exp(-i\tau_{iu}(k)) \tag{6}\]  
For a plane-wave model to be applied, it is assumed that both signals and sources are in the far field w.r.t. the receiving array. Thus, in line with the plane-wave model, the complex amplitude is expressed as  
\[c_{iu}(k)=\dot{c}_{iu}(k)\bar{c_{u}}(k)~{}~{}\forall~{}~{}i \tag{7}\]  
where \(\dot{c}_{iu}\) is the response of receiving antenna \(i\) in the \(\Omega_{u}\) direction and is assumed to be known a priori, and \(\overline{c_{u}}\) denotes the complex amplitude of the plane wave \(u\), which factors all signal phenomena experienced before reaching the receiving array. In addition, due to the assumption of a short time window analysis and a homogeneous medium, the pathway distance \(d_{u}\) is therefore neglected in the directional analysis. As a result, the time delay of the plane wave can be represented with respect to the origin (that is, the mid-point of the sphere in the SAA) as  
\[\tau_{iu}(k)=k\mathbf{\tau}_{u}^{T}\mathbf{m}_{i} \tag{8}\]  
where \(\mathbf{\tilde{r}}_{u}=\left[\sin(\theta_{u})\cos(\phi_{u}),\sin(\theta_{u})\sin( \phi_{u}),\cos(\theta_{u})\right]^{T}\) is the normal to the wavefront. The SAA response and corresponding delay term is presented as  
\[\dot{h}_{iu}(k)=\dot{c}_{iu}(k)\exp\Bigl{\{}-i\mathbf{k}\bar{\mathbf{\tau}}_{u}^{T}\bm {m}_{i}\Bigr{\}} \tag{9}\]  
and referred to as the steering vector. For instance, \(\dot{c}_{iu}=1\) for an ideal antenna array. The source response and amplitude of the plane wave are given as the product  
\[s_{u}(k)=\overline{c}_{u}(k)\dot{s}_{u}(k) \tag{10}\]and regarded as the actual signal. For brevity, the array measurements are presented in matrix form. The impulse responses of array elements \(i=1,\ldots I\), i.e., their inputs, are a vector [31]:  
\[