## 3 Numerical Analysis
### STPA Performances in JCAS Scenarios and Benchmarking
#### 3.2.2 Sensing Performance in Scenario 1: Non-Overlapping Communication and Sensing Beams
Before introducing the quantitative metrics for comparison of sensing by these apertures, an example of a sensing scenario is provided here. The detection of 8 randomly generated targets at \(SNR=10\) dB and \(P_{FA}=10^{-2}\) are simulated. The normalized images and qualitative performance are shown in Figure 13a,b for CUPA, where only 5 targets are detected and 3 are missed. In Figure 13c,d for SPA, 7 targets are estimated and 1 target is missed. Finally, in Figure 13e,f for STPA, 6 targets are estimated and 2 are missed. Therefore, we adopt two quantitative performance metrics; namely _detection loss rate_ (DLR) and _root-mean-square error_ (RMSE). Due to different beamwidths, the maximum detectable targets \(N_{\text{targets}}^{\text{max}}\) for these apertures are different, and considering a symmetrical FoV of \(2\theta_{\text{max}}\) in \(x\) and \(y\) axes as in Figure 12b, it is obtained by \(N_{\text{targets}}^{\text{max}}=\lfloor(2\theta_{\text{max}})/(\theta_{3 \text{dB}}/2)\rfloor^{2}\). Considering the radiation patterns in Figure 8d-f, the 3 dB-beamwidths \(\theta_{3\text{dB}}\) for CUPA, SPA, and STPA are \(24^{\circ},10^{\circ}\) and \(10^{\circ}\), respectively, therefore \(N_{\text{targets}}^{\text{max,CUPA}}=25\) and \(N_{\text{targets}}^{\text{max,SPA}}=N_{\text{targets}}^{\text{max,STPA}}=144\). Knowing \(0<N_{\text{targets}}^{\text{detected,p}}<N_{\text{targets}}^{\text{max}}\) for \(p^{\text{th}}\) set of targets, we define a normalized detection (ND) as \(ND^{p}=N_{\text{targets}}^{\text{detected,p}}/N_{\text{targets}}^{\text{max}}\) which, as a random variable, follows a Beta distribution with an average value of \(\mu_{ND}\). DLR basically accounts for the discrepancy in the number of detected \(N_{\text{targets}}^{\text{detected}}\) and true targets \(N_{\text{targets}}\), and is defined as:  
\[DLR=\frac{|\mu_{ND}N_{\text{targets}}^{\text{max}}-N_{\text{targets}}|}{|N_ {\text{targets}}^{\text{max}}-N_{\text{targets}}|} \tag{15}\]  
Figure 12: The investigation domain in FoV for scenario 1. Ideally, \(DLR\) equals zero, however, it is often a positive number, particularly when \(N_{\text{targets}}\) increases or the angular resolution of the aperture is limited, therefore \(0<DLR<1\), and it is the first metric for comparison. The RMSE of true and estimated targets is defined as:  
\[\text{RMSE}(\hat{\mathbf{A}}^{p},\mathbf{A}^{p})=\sqrt{\frac{1}{N_{\text{ targets}}}\sum_{n_{t}=1}^{N_{\text{targets}}}|\hat{a}_{n_{t}}^{p}-a_{n_{t}}^{p}|^{2}} \tag{16}\]  
Figure 13: Normalized images, associated, missed, and true targets for (**a,b**) CUPA, (**c,d**) SPA, and (**e,f**) STPA. where \(\hat{\mathbf{A}}^{p}\) denotes the estimated coordinates of \(p^{\text{th}}\) set of targets. The RMSE as a random variable can be modeled by Rayleigh distribution, with an average value of \(\mu_{RMSE}\), which is the second metric to compare the performance of these apertures. The process of generating targets and calculating DLR and RMSE is repeated for \(P=200\) times at \(SNR=10\) dB. \(DLR\) and \(\mu_{RMSE}\) are calculated for these apertures in \(P_{FA}=\{10^{-3},10^{-2},10^{-1}\}\) and \(N_{\text{targets}}=\{4,8,12\}\). Figure 14a-c demonstrate that \(DLR\) for CUPA is larger than SPA and STPA in any \(P_{FA}\) and \(N_{\text{targets}}\). STPA also has higher \(DLR\) than SPA. The effect of the increase in \(N_{\text{targets}}\) can be observed in all apertures where \(DLR\) increases accordingly. The increase in \(P_{FA}\) results in the decrease in \(DLR\) for all apertures. Figure 15a-c shows the \(\mu_{RMSE}\) for these apertures, where the performances, although similar, are still comparable, as in \(\mu_{RMSE}^{\text{CUPA}}>\mu_{RMSE}^{\text{STPA}}>\mu_{RMSE}^{\text{SPA}}\) for all \(P_{FA}\)s and \(N_{\text{targets}}\)s. Although the angular resolutions of SPA and STPA are smaller than CUPA, the differences in \(\mu_{RMSE}\) are small due to the possible side lobes, which can create ghost targets and result in errors in the location of targets.