## 2 System Model
### Communication Signal Model  
In V2V communication, the source vehicle embeds its position, velocity, and beam direction information into the communication signal. The target vehicle demodulates the communication signal to obtain these parameters, compensates for the Doppler shift and delay, and constructs a beamformer to maximize the communication SINR. In time slot \(n\), the received signal of target vehicle \(k^{\prime}\) can be expressed as  
\[\begin{split} c_{k^{\prime},n}=&\zeta\sqrt{p_{k^{ \prime},n}}\kappa_{k^{\prime},n}\mathbf{w}^{H}(\tilde{\theta}_{k^{\prime},n}) \mathbf{u}(\theta_{k^{\prime},n})\mathbf{a}^{H}(\theta_{k^{\prime},n})\mathbf{ f}(\tilde{\theta}_{k^{\prime},n})s_{k^{\prime},n}\\ &+\mathbf{w}^{H}(\tilde{\theta}_{k^{\prime},n})\left[\sum_{m=1}  
where \(\mathbf{w}^{H}\big{(}\theta_{k^{\prime},n}\big{)}\) represents the communication receive beamforming vector; \(\mathbf{u}\) denotes the communication receive steering vector; \(\kappa_{k^{\prime},n}\) represents the path loss of the signal from source vehicle \(k\) to target vehicle \(k^{\prime}\). When \(i=k\) and \(j=k^{\prime}\), the beam does not cause self-interference; hence, \(\delta_{i,j}=0\); otherwise, \(\delta_{i,j}=1;z_{c}\) represents complex additive white Gaussian noise with zero mean and variance \(\sigma_{c}^{2}\). The communication SINR at target vehicle \(k^{\prime}\) can be expressed as  
\[\omega_{k^{\prime},n}^{c}=\frac{\zeta^{2}p_{k^{\prime},n}|\kappa_{k^{\prime}, n}\mathbf{w}^{H}\big{(}\theta_{k^{\prime},n}\big{)}\mathbf{u}\big{(}\theta_{k^{ \prime},n}\big{)}\mathbf{a}^{H}\big{(}\theta_{k^{\prime},n}\big{)}\mathbf{f} \big{|}^{2}}{\sigma_{c}^{2}+I_{I2V}+I_{I2V}+I_{Echo}}. \tag{5}\]  
Similar to the sensing SINR, the communication SINR is also directly related to the transmit power, beam direction, and frequency band allocation. According to the above formula, the achievable communication rate between source vehicle \(k\) and target vehicle \(k^{\prime}\) can be expressed as  
\[R_{k^{\prime},n}^{c}=\log_{2}\Big{(}1+\omega_{k^{\prime},n}^{c}\Big{)}. \tag{6}\]  
In I2V communication, since the RSU uses orthogonal frequency bands to communicate with each vehicle, the received signal of target vehicle \(m\) can be expressed as  
\[\begin{split} c_{m,n}=&\zeta\sqrt{p_{m,n}}\kappa_{ m,n}\mathbf{w}^{H}(\bar{\theta}_{m,n})\mathbf{u}(\theta_{m,n})\mathbf{a}^{H}( \theta_{m,n})\mathbf{f}(\bar{\theta}_{m,n})q_{m,n}\\ &+\mathbf{w}^{H}(\bar{\theta}_{m,n})\sum_{i=1}^{K}\sum_{j=1}^{K ^{\prime}}\psi_{i,m}\xi\sqrt{p_{j,n}}\mathbf{H}_{i,m}\mathbf{f}_{j}s_{j,n}+z_ {c},\end{split} \tag{7}\]  
where \(\kappa_{m,n}\) represents the path loss of the signal from the RSU to vehicle \(m\). Since the RSU knows the motion parameters of all I2V vehicles, precise beam alignment can be achieved, i.e., \(\bar{\theta}_{m,n}=\theta_{m,n}\). The received signal of target vehicle \(m\) can be rewritten as  
\[\begin{split} c_{m,n}=&\zeta\sqrt{p_{m,n}}\kappa_{ m,n}q_{m,n}\\ &+\mathbf{w}^{H}(\bar{\theta}_{m,n})\sum_{i=1}^{K}\sum_{j=1}^{K^{ \prime}}\psi_{i,m}\xi\sqrt{p_{j,n}}\mathbf{H}_{i,m}\mathbf{f}_{j}s_{j,n}+z_{c}.\end{split} \tag{8}\]  
According to the above formula, the total achievable rate for I2V can be expressed as  
\[R_{n}^{R}=\sum_{m=1}^{M}\log_{2}\Bigg{(}1+\frac{\zeta^{2}p_{m,n}|\kappa_{m,n} |^{2}}{\sigma_{c}^{2}+I_{V2V}}\Bigg{)}. \tag{9}\]  
It can be seen that \(R_{n}^{R}\) is related not only to the interference power of V2V but also directly to the transmit power of I2V. Therefore, joint power allocation for I2V and V2V is crucial for enhancing the overall sensing-communication performance of the network. In the following section, we introduce the I2V and V2V beam allocation problem aimed at maximizing overall sensing-communication performance and present the corresponding optimization algorithm.