## 3 Theoretical Performance Analysis on Sidelink JCS System  
Multiple vehicles simultaneously transmit data packets via full-duplex broadcasts and use echo signals to detect and locate passive targets in the surrounding environment, obtaining sensing information such as target distance, relative speed, and the signal-to-noise ratio (SNR). In this scenario, multiple vehicles send sidelink signals, such as a cooperative awareness message (CAM) or collective perception message (CPM) [41]. It is initially assumed that all vehicles are equipped with in-band full-duplex transceivers with perfect SIC, enabling the use of echoes to detect channel quality [42]. Assume that during time slot \(t\in\mathcal{T}\), vehicle \(i\in\mathcal{I}_{t}\) generates a data packet and begins broadcasting the OFDM symbols \(s_{m}(t)\). In this context, the symbol for the \(n\)-th subcarrier of the \(m\)-th symbol is denoted as \(x_{n,m}\), with each symbol power being \(P_{n,m}\). The symbol duration is defined as \(T_{sym}=T+T_{cp}\), where \(T_{cp}\) represents the cyclic prefix duration and \(T=1/\Delta f\), where \(\Delta f\) is subcarrier space. The representation in the complex baseband is as follows:  
\[s_{m}(t)=\sum_{n=0}^{N-1}\sqrt{P_{n,m}}x_{n,m}e^{\beta 2\tau n\Delta ft} \operatorname{rect}\left(\frac{t-mT_{\text{sym}}}{T_{\text{sym}}}\right) \tag{1}\]  
The sidelink JCS data packet consists of \(N\) subcarriers and \(M\) OFDM symbols. Assume that all the symbols \(x_{n,m}\) in the packet form an \(N\times M\) matrix \(\mathbb{X}=\{x_{n,m}\}\in\mathbb{C}^{N\times M}\), where each column represents an OFDM symbol and each row represents a subcarrier. Assume that the power of each symbol on each subcarrier is the same and normalized, i.e., \(P_{n,m}=P_{avg}\). Therefore, for all \(N\) and \(M\), the transmission power of each vehicle is given by \(P_{T}=N\cdot P_{avg}\). Sidelink data packets are not only used for communication transmission but also for echo sensing. Assume the target vehicle is \(d\) away from the transmitting vehicle and the relative velocity is \(v\). The complex baseband representation of the received echo signal for the \(n\)-th subcarrier of the \(m\)-th symbol is given by:  
\[y_{m,n}\approx\alpha\ e^{\beta 2\tau mT_{\text{sym}}=v}e^{-\beta 2\tau n\Delta f \tau}x_{m,n} \tag{2}\]  
Among them, \(\alpha\) is the channel coefficient. By receiving the echoes at the transmitting receiver, estimates of \(v\) and \(\tau\) are obtained as \(\hat{v}\) and \(\hat{\tau}\). The parameters \(v\) and \(\tau\) represent the Doppler shift and delay of the echo signal, respectively. The distance estimate \(\hat{d}\) is calculated from \(\hat{\tau}\) as:  
\[\hat{d}=\frac{\hat{\tau}}{2c} \tag{3}\]  
where \(c\) represents the speed of light. The carrier frequency is \(f_{x}\). The relative speed estimate \(\hat{v}\) is calculated from \(\hat{v}\) as:  
\[\hat{v}=\frac{\hat{v}c}{2f_{x}} \tag{4}\]\(P_{R}\) is the received power of the full duplex, given by:  
\[P_{R}=P_{T}\frac{G^{2}c^{2}\sigma}{(4\pi)^{3}f_{\xi}^{2}d^{4}} \tag{5}\]  
Among them, \(P_{T}\) represents the transmit power, and \(G\) represents the transmit/receive antenna gain. The noise power is represented as \(P_{n}=k_{B}T_{0}FW\), where \(k_{B}\) is the Boltzmann constant, \(T_{0}=290\) is the reference temperature, \(F\) is the noise figure of the full-duplex radar receiver, and the bandwidth of the sidelink transmission signal is \(W=N\Delta f\). In practice, during the transmission slot, interference occurs when other vehicles transmit using the same frequency resources, which affects radar sensing performance. The calculation formula is:  
\[I_{t,i}=\sum_{k\in\mathcal{I}_{t},k\
eq i}\eta_{t,ki}\frac{h_{t,ki}P_{\text{T} }}{L(d_{t,ki})} \tag{6}\]  
In (6), \(L(d_{t,ki})\) represents the path loss at distance \(d_{t,ki}\) between the interfering vehicle \(k\) and the transmitting vehicle \(i\) at time slot \(t\), and the parameter \(h_{t,ki}\) represents the large-scale fading at time slot \(t\).