## 2 Proposed STPA Configuration for JCAS, Its Design Pipeline, and Performance
### JCAS Scenarios and Performance Metrics
#### 2.3.1 JCAS Scenario 1: Non-Overlapping Communication and Sensing Beam Directions  
Let us consider the scenario in Figure 4a. A geometric model for both communication and sensing is assumed to include \(L\) multipath signals with the angle-of-departure (AoD), \((\theta_{T,l},\phi_{T,l})\) and angle-of-arrival (AoA), \((\theta_{R,l},\phi_{R,l})\). The time-varying channel [33] can be expressed as:  
\[\mathbf{H}=\sum_{l=1}^{L}b_{l}\delta(t-\tau_{l})e^{l2\pi f_{Dj}t}\mathbf{a}_{T }(\theta_{T,l},\phi_{T,l})\mathbf{a}_{R}^{T}(\theta_{R,l},\phi_{R,l}) \tag{5}\]  
Figure 4: The scenarios for JCAS in downlink communication, (**a**) without overlapping of communication and sensing beams, (**b**) with overlapping of communication and sensing beams, when the UE blocks the sensing link, and (**c**) forthcoming overlapping of communication and sensing beams, when the sensing target can block the communication link. where \(b_{l}\) is the complex value modeling signal attenuation, \(\tau_{l}\) is the propagation delay, and \(f_{DJ}\) is the Doppler frequency. \(\mathbf{a}_{T}\) and \(\mathbf{a}_{R}\) are the steering vectors for transmit and receive arrays in the directions of multipath AoD and AoA, respectively. The channel (5) might also include the LoS path for communication. Note that \(\mathbf{H}\in\mathbb{C}^{N_{JCAS}\times N_{R}}\), where \(N_{JCAS}\) denotes the number of radiating elements (either tiles or single-antennas) in the JCAS aperture, and \(N_{R}\) is the number of antennas in the receive aperture of the UE for the communication channel or the auxiliary aperture in BS for monostatic sensing. The multibeam weighting vector at the JCAS aperture is [24]:  
\[\mathbf{w}_{JCAS}=\sqrt{\rho}\mathbf{w}_{c}+\sqrt{(1-\rho)}\mathbf{w}_{s} \tag{6}\]  
where \(0\leq\rho\leq 1\) is the power allocation factor between beams, which equals 0.5 in this paper. \(\mathbf{w}_{c}\in\mathbb{C}^{N_{JCAS}\times 1}\) is the weighting vector for communication and is calculated by the right singular vector of the channel corresponding to the maximum singular value of \(\mathbf{H}\) with \(\|\mathbf{w}_{c}\|_{F}^{2}=1\)[34]. \(\mathbf{w}_{s}\in\mathbb{C}^{N_{JCAS}\times 1}\) denotes the weighting vector for sensing to a desired direction \(\Delta_{s}=(\theta_{s},\phi_{s})\) and calculated by \(\mathbf{w}_{s}=\mathbf{a}_{T}^{*}(\theta_{s},\phi_{s})/\|\mathbf{a}_{T}(\theta _{s},\phi_{s})\|_{F}\). **Communication Performance**: We use spectral efficiency as the performance metric for communication. With Gaussian symbols transmitted in a mmWave channel, the spectral efficiency is [34]:  
\[R=\text{log}_{2}\text{det}\Big{(}1+\frac{\mathbf{H}^{*}\mathbf{w}_{JCAS}^{*} \mathbf{w}_{JCAS}\mathbf{H}}{\sigma_{n}^{2}}\Big{)} \tag{7}\]  
where \(\sigma_{n}^{2}\) denotes the variance of a zero-mean Gaussian noise. **Sensing Performance**: The main figure of merit of sensing in a JCAS scenario is the detection of targets as well as the angular-delay-Doppler estimates of targets. While the waveform configuration, including the subcarriers and bandwidth as well as symbol duration and the number of coherent processing symbols, determine the delay/range and Doppler estimates, respectively, the array configuration determines the accuracy and ambiguity of AoA/AoD estimates. Assume that \(N_{d}\) OFDM symbols are required for a coherent processing interval for Doppler estimation. \(N_{e}\) and \(N_{a}\) are the number of elevation and azimuth angles, respectively, and \(N_{e}\times N_{a}\) angles form a full sensing coverage. We suppose that self-interference is already canceled by a combination of passive [35] and active methods [26; 36], therefore the received signal at the auxiliary Rx is simplified [24] to:  
\[ \begin{split}&\tilde{y}_{n,n_{e},q}=\tilde{h}_{n,n_{e},q}\tilde{s}_{ n}+\tilde{z}_{n},\\ &\tilde{h}_{n,n_{e},q}=\sum_{l=1}^{L}b_{l}\mathbf{w}_{JCAS}^{T} \mathbf{a}_{T}(\theta_{T,l},\phi_{T,l})e^{-j2\pi n\tau_{l}f_{0}}e^{j2\pi f_{DJ }(qT_{s}+(n_{a}-1)T_{f})},\end{split}