## 3 Numerical Analysis
### STPA Performances in JCAS Scenarios and Benchmarking
#### 3.2.2 Sensing Performance in Scenario 1: Non-Overlapping Communication and Sensing Beams  
One of the key performance metrics for sensing is the angular resolution in the scanning mode, which is studied here for the STPA in comparison to CUPA and SPA. STPA and SPA have a smaller beamwidth than CUPA and presumably perform better at target detection [10]. Besides, the quality of SLLs is different for STPA and SPA, which influences the detection results. We adopt a Monte Carlo simulation to study these effects. The scenario in Figure 4a is considered, where UE and the targets are distributed in the FoV of JCAS apertures. Suppose that UE is located in the same range and angle as before and the targets are in a square region at \(z\)-axis, and generate \(N_{\text{targets}}\) point targets at \(z_{\text{targets}}=70\) m, as shown in Figure 12. The targets have the same velocities of \(v_{\text{targets}}=20\) m/s and their \(x\) and \(y\) coordinates are sampled uniformly from \(x\in[x_{min},x_{max}]\) and \(y\in[y_{min},y_{max}]\) where \(x_{max}=-x_{min}=y_{max}=-y_{min}=z_{\text{targets}}\times\tan\theta_{max}\). Note that if a target coincides with the angle of UE, it is neglected. Suppose \(\mathbf{A}^{p}=[a_{1}^{p},...,a_{N_{\text{targets}}}^{p}]\) is a matrix containing the Cartesian coordinates of \(p^{\text{th}}\) set of targets, where \(a_{n_{t}}^{p}=(x_{n_{t}}^{p},y_{n_{t}}^{p})\) for \(n_{t}^{\text{th}}\) target. The radiation gain of JCAS tiles and auxiliary antennas, and the noise figure of the receiver, are assumed to be unit. Then, the two-way channel gain in (5) is \(b_{l}=\frac{\lambda 2}{(4\pi)^{3/2}\tau_{l}^{2}}\), where \(r_{l}\) is the range of \(l\)th target. The carrier frequency is \(f_{c}=28\) GHz and the symbol duration equals \(T_{s}=51.2\) us. The coherent processing duration for Doppler estimation is \(N_{d}T_{s}\) and a communication packet includes \(N_{d}N_{e}\) symbols. Here, we intend to compare these apertures in the worst scenario, namely all targets are located at the same range and velocity bins but different cross-range bins. Therefore, to expedite the Monte Carlo simulations,  
Figure 11: The spectral efficiency for the designed CUPA, SPA, and STPA when there is no overlapping of communication and sensing beams. SNR denotes the SNR level in the LoS path. simply \(N_{sc}=8\) and \(N_{d}=8\). The spatial sampling is adjustable according to the purpose of scanning, here \(N_{x}=N_{y}=60\) provides sufficient resolutions, where \(N_{x}\) and \(N_{y}\) are the numbers of bins in the \(x\)-axis and \(y\)-axis, respectively. We applied a conversion from Cartesian to spherical coordinate on the points in FoV and then utilized (8) to generate synthetic data, and then employed (9) for range-Doppler and angle estimation of targets. The images in \(x\)-\(y\) plane are generated, and cell averaging 2D constant false alarm rate (CA-CFAR) detection is applied with a guard band size of \([2,2]\) and a training band size of \([5,5]\). As CA-CFAR determines its threshold based on a measured SNR in training cells to guarantee a constant probability of false-alarm \(P_{FA}\), a variation of SNR in simulations is therefore pointless. Instead, we study a variation in \(P_{FA}=\{10^{-3},10^{-2},10^{-1}\}\). Depending on the level of threshold resulting from different \(P_{FA}\)s, the number of detection points can vary. The density-based spatial clustering of applications with noise (DBSCAN) is applied to cluster the detected points, where a threshold for a neighborhood search radius of 0.05 and a minimum number of neighbors of 1 are considered. Then the number of clusters is the number of detected targets \(N_{\text{targets}}^{\text{detected}}\) and the centroid of each cluster is supposed to be the coordinate of the detected target. The association of the estimated and true targets is based on the \(k\)-nearest neighbors (\(k\)-NN) algorithm, where a single nearest neighbor to a true target is determined by the Minkowski distance of second.