## 3 Numerical Analysis  
### JCAS STPA--A Design Example  
Following the JCAS aperture design pipeline in Figure 3, let us start with a uniform array with \(A_{x}\times A_{y}=30\times 30\) elements, with \(\lambda/2\) inter-element distance where \(\lambda\) is the wavelength at the center frequency of operation \(f_{c}=28\) GHz. First in a global scale, the uniformly and planarly placed antennas are tiled with dominoes based on the maximum entropy, as shown in Figure 5a. In order to show the capability of this large tiled array when scanning to \(\theta_{max}=30^{\circ}\) according to 5G base station requirements [14], its EBP, which corresponds to the scanning by linear beamforming, is shown in Figure 5b, where the maximum SLL is \(-13.24\) dB and the beamwidth is \(r_{b}=0.065\). Second, on a local scale, the sunflower array is used to thin these dominoes. In order to cover the whole aperture, the distance between two neighboring elements of the sunflower array is related to the parameter \(s=3.5\lambda\) (see Appendix A). The domino tiles that are in \(r_{c}\) distance from the elements of the sunflower are chosen for the subarrays, and the thinning rate depends on this \(r_{c}\). The simulation showed that \(r_{c}=1.15\lambda\) provides the best performance for low SLL and minimum required tiles, which results in 125 dominoes overall, as shown in Figure 6.  
This elimination of dominoes increases the maximum SLL (reaching \(-10.93\) dB), however, the locations of subarrays can also be optimized for improving the SLL via iterative convex optimization (see Appendix B). The parameters in the position optimization of subarrays for an EBP with \(\zeta=1.5\) are \(\tilde{r}_{b}=0.043\) for the main beamwidth and \(\mu=\lambda/25\) for the maximum of position steps. The angles \(0^{\circ}\leq\theta\leq 90^{\circ}\) and \(-180^{\circ}\leq\phi<180^{\circ}\) are swept with \(1^{\circ}\). The objective function is to minimize the maximum SLL in the EBP. As shown in Figure 6(a),b, the optimization reaches a maximum SLL of \(-13.95\) dB. For mutual coupling and implementation considerations, the distances of each tile in a subarray to the adjacent tiles in other subarrays are checked to be larger than \(\lambda\), which is shown in Figure 6(c). The EBP of the final geometry has an SLL of \(-13.5\) dB (due to the linear approximation in the optimization) while having a beamwidth of \(\tilde{r}_{b}=0.043\) as shown in Figure 6(d). This final array geometry in Figure 6(a) will be used for JCAS performance evaluation. The performance of this proposed STPA is analyzed in various scenarios in comparison to the classical CUPA and SPA. CUPA is a common choice for communication applications, whereas SPA is often utilized in sensing applications. For a fair comparison in regard to antenna element numbers, CUPA is constructed by a rectangular uniform array with \(12\times 12\) antennas; \(19\) antennas at the \(4\) corners, which are indicated by plus markers in Figure 7(a), are removed and \(125\) antennas remain which construct a uniform planar array that is almost symmetric in \(x\) and \(y\) axes as in Figure 7(a). Figure 7(d) shows the EBP of this CUPA with a maximum SLL of \(-15.6\) dB. The SPA design is based on the pattern of a sunflower array [42] with \(125\) antennas. Note that a maximum SLL for such SPA can even be \(-16.75\) dB when its aperture radius equals \(3.7\lambda\). However, we require the same aperture radius and beamwidth as that for STPA, therefore we apply an aperture radius of \(6.8\lambda\) as in  
Figure 5: **(a)** The tiled array with dominoes (the colors are only for differentiating the tiles), **(b)** its EBP with the blue circle with radius \(\tilde{r}_{b}=0.043\) denotes the \(3\) dB-beamwidth of the main beam in \((\hat{\mu},\hat{\sigma})\)-space. The beamwidth in \((u,v)\)-space is then \(r_{b}=\tilde{\zeta}r_{b}=0.065\). Figure 6: The subarrays chosen based on the sunflower array.