## 3 Formulation of a Concept of Radar Antennas for CA Systems
### Design of a Prototype Implementing the Candidate Architecture  
For the sake of simplicity and ease of manufacture, we opted for an implementation based on a circular array of sectorial apertures. Starting from a baseline model in CST Studio\({}^{\text{\textregistered}}\)--a well-established, State-of-the-Art full-wave electromagnetic simulator [51]--we utilised CST's optimisation capabilities to obtain the final design shown in Figure 1. This model was constructed using a material with the same electrical conductivity as AlSi10 Mg, which was also used in the prototype developed for the experimental proof of concept described in Section 4. Our final design consists of a circular array with two rows of eight sector waveguide apertures, each fed through 50 \(\Omega\) coaxial lines. The diameter of the circular plates is 90 mm, with a distance of 10 mm between them. To improve isolation between the feeding ports, we added 10 mm high flanges to both the upper and lower plates. All elements have a width of 1.5 mm, selected based on the manufacturing method we chose for the prototype (details are provided in Section 4). We incorporated a central 15 mm diameter metallic post for mechanical robustness, and the feeding points are positioned on the symmetry plane of the sectors, 21 mm from the centre of the circular plates. We determined this distance through an optimisation process aimed at achieving return losses below 10 dB and isolation between adjacent feeding points greater than 40 dB. Although these values are not specified in existing standards, we set them as design goals based on our experience in the field. This configuration generates eight fixed monopulse beams, each directed at an azimuth angle of \(\varphi_{B,n}=n\pi/4\) radians, where \(0\leq n\leq 7\), thus ensuring full 360\({}^{\circ}\) coverage. The elevation angle is fixed at \(\theta_{B}=\pi/2\) radians. Each monopulse pattern is formed by feeding a cluster of four sector antennas, as illustrated in Figure 2.  
Figure 2(a) depicts the classical monopulse architecture implemented using circuit components. The circuits employed to combine the signals received from the four antennas are 180\({}^{\circ}\) hybrid couplers, characterised by the scattering matrix shown in Equation (2), referenced to ports 1-4 as depicted in Figure 2(b). Note that labels I-IV in Figure 2(b) refer to the corresponding sector in Figure 2.  
Figure 1: Final model of the radar antenna in CST. Figure 2: Representative view of the excited cavities. \[S=\frac{1}{\sqrt{2}} \begin{bmatrix}1&1&0&0\\ 1&-1&0&0\\ 0&0&1&1\\ 0&0&1&-1\end{bmatrix}  \tag{2}\]  
Assuming that all ports are terminated with the circuit's characteristic impedance \(Z_{0}\), it follows that \(b_{1}=b_{2}=0\) and \(a_{3}=a_{4}=0\), while \(a_{1}\) and \(a_{2}\) represent the input signals. Consequently, the output signals at ports 3 and 4 can be expressed as:  
\[ \begin{bmatrix}0\\ 0\\ b_{3}\\ b_{4}\end{bmatrix} =\frac{1}{\sqrt{2}} \begin{bmatrix}1&1&0&0\\ 1&-1&0&0\\ 0&0&1&1\\ 0&0&1&-1\end{bmatrix}\begin{bmatrix}a_{1}\\ a_{2}\\ 0\\ 0\end{bmatrix} , \tag{3}\]  
This leads to \(b_{3}=(a_{1}+a_{2})/\sqrt{2}\) and \(b_{4}=(a_{1}-a_{2})/\sqrt{2}\). Therefore, according to the diagram in Figure 3a, the \(\Sigma\) (sum) channel is obtained by coherently adding the signals received from all four sectors (i.e., \(\mathrm{I}+\mathrm{II}+\mathrm{III}+\mathrm{IV}\)). The \(\Delta_{\varphi}\) (azimuth difference) channel is formed by adding the signals from sectors I and IV, and subtracting those from sectors II and III (i.e., \(\mathrm{I}+\mathrm{IV}-\mathrm{II}-\mathrm{III}\)). Similarly, the \(\Delta_{\theta}\) (elevation difference) channel is derived by adding the signals from sectors I and II, and subtracting those from sectors III and IV (i.e., \(\mathrm{I}+\mathrm{II}-\mathrm{III}-\mathrm{IV}\)).