## 3 Formulation of a Concept of Radar Antennas for CA Systems
### Design of a Prototype Implementing the Candidate Architecture
The azimuth and elevation off-boresight angles (\(\varphi_{OB}\) and \(\theta_{OB}\), respectively) are then determined by comparing the phase of the corresponding \(\Delta\) channel with that of the \(\Sigma\) channel [29]:  
\[\varphi_{OB}=2\mathrm{atan}\bigg{(}\frac{\Delta_{\varphi}}{\Sigma}\bigg{)} \tag{4}\]  
\[\theta_{OB}=2\mathrm{atan}\bigg{(}\frac{\Delta_{\theta}}{\Sigma}\bigg{)} \tag{5}\]  
Figure 3: (**a**) Block diagram of a 3D monopulse receiver; (**b**) labelling of ports of the 180\({}^{\circ}\) hybrid couplers used to illustrate the concept of a monopulse receiver. The corresponding azimuthal boresight angle (\(\varphi_{B,n}\)) is determined by the sectors through which the signals are received, while the elevation boresight is fixed at \(\theta_{B}=\pi/2\) in all cases. The angular position of the targets is, therefore, computed as:  
\[\varphi=\varphi_{B,n}+2\text{atan}\bigg{(}\frac{\Delta_{\varphi}}{\Sigma}\bigg{)} \tag{6}\]  
\[\theta=\frac{\pi}{2}+2\text{atan}\bigg{(}\frac{\Delta_{\theta}}{\Sigma}\bigg{)} \tag{7}\]  
Figure 4 shows the scattering parameters of the antenna presented in Figure 2, obtained using CST, with port labelling consistent with that defined in Figure 3. The results demonstrate that both the return losses and isolation are well below the design targets at the radar operating frequency (24 GHz). These results confirm that the design criteria have been satisfied, with a comfortable margin to accommodate potential manufacturing effects. Figure 5a shows the 3D \(\Sigma\) radiation pattern in dBi, Figure 5b shows the horizontal 2D cut, and Figure 6 shows the simulated radiation patterns for the \(\Sigma\) and \(\Delta_{\varphi}\) channels on the horizontal plane (\(\theta=90^{\circ}\)) for a boresight angle of \(\varphi_{B,n}=45^{\circ}\). Figures 7 and 8 illustrate the corresponding \(\Sigma\) and \(\Delta_{\theta}\) patterns on the vertical plane for the same boresight direction. Figure 5b's 2D cut in the horizontal plane exhibits a ripple of \(\pm 1.5\) dB, which is typical for electrically large antennas, exceeding \(5.5\lambda\) in the horizontal plane for the specified boresight angle in our case. This antenna size is necessary to achieve the desired gain. Similar behaviour has been reported in the literature (see, for instance, [46]). Accordingly, the \(\Sigma\) radiation pattern exhibits a gain of 13.1 dBi, with \(-3\) dB beamwidths of \(\boldsymbol{\varphi}_{-3\mathbf{dB}}=84^{\circ}\) in the horizontal plane and \(\boldsymbol{\theta}_{-3\mathbf{dB}}=95^{\circ}\) in the vertical plane. Regarding the different patterns, the most relevant parameter is the depth of the null, which reaches 58 dB in the \(\boldsymbol{\Delta_{\varphi}}\) pattern and 26 dB in the \(\boldsymbol{\Delta_{\theta}}\) pattern. Figure 8 corresponds to a boresight angle of \(\boldsymbol{\varphi}_{B}=45^{\circ}\). As expected from the antenna's symmetry, we obtained similar patterns for the other seven boresight angles. When combined, these patterns provide a full \(360^{\circ}\) azimuthal monopulse coverage, as illustrated in Figure 9.  
Figure 4: S parameters of the antenna prototype obtained by simulation using CST. Figure 5: (**a**) 3D \(\Sigma\) radiation pattern in dBi; (**b**) 2D cut in the horizontal plane \(\Sigma(\varphi,\theta=90^{\circ})\).