## 3 RSMA-Assisted Integrated Sensing and Communication Systems
### RSMA-Assisted ISAC Architecture
The achievable private rate for the \(k\)-th user can be expressed as  
\[R_{p,k}=\text{log}_{2}\Bigg{(}1+\frac{|\mathbf{h}_{k}^{H}\mathbf{p}_{k}|^{2} }{\sum_{i\in\mathcal{K},i\
eq k}|\mathbf{h}_{k}^{H}\mathbf{p}_{i}|^{2}+\delta| \mathbf{h}_{k}^{H}\mathbf{p}_{R}|^{2}+\sigma_{k}^{2}}\Bigg{)}. \tag{8}\]  
Then, the total achievable rate for the \(k\)-th user is \(R_{p,k}+C_{k}\), where \(C_{k}\) is the \(k\)-th user's common rate portion. Since the common stream is decoded by all users, the sum of the common rate portions \(\sum_{k\in\mathcal{K}}C_{k}\) should be less or equal than \(\min_{k}R_{c,k}\). For the reflected signal, BS receives the \(N_{r}\times 1\) echo signal at time index \(t\) as follows  
Figure 4: RSMA-assisted ISAC systems. \[\mathbf{y}_{r}[t]=\alpha e^{j2\pi\mathcal{F}_{D}tT_{\mathrm{sym}}}\mathbf{b}( \theta)\mathbf{a}^{T}(\theta)\mathbf{x}[t]+\mathbf{n}_{r}[t] \tag{9}\]  
where \(\alpha\) is the complex reflection coefficient related to the radar cross-section (RCS). \(\mathcal{F}_{D}=\frac{2\pi f_{c}}{c}\) is the Doppler frequency, where \(v\) is the velocity of the radar target. \(f_{c}\) and \(c\) are the carrier frequency and the speed of light, respectively. \(T_{\mathrm{sym}}\) denotes the symbol period. \(\theta\) denotes the radar target's direction of departure (DoD) as well as the direction of arrival (DoA) of the target, which are the same in the monostatic system. \(\mathbf{a}(\theta)\in\mathbb{C}^{N_{l}\times 1}\) and \(\mathbf{b}(\theta)\in\mathbb{C}^{N_{r}\times 1}\) are the transmit and receive steering vectors, respectively. \(\mathbf{n}_{r}[t]\) is AWGN. For the joint design of communication and radar functions, communication performance metrics and radar performance metrics should be jointly optimized. Specifically, the minimum fairness rate [22], energy efficiency [23], and weighted sum rate [14] can be considered as the communication performance metrics. As the radar sensing metrics, the beampattern matching [24] and Cramer-Rao bound (CRB) [25] are considered. For the beampattern matching, the mean square error (MSE) for the beampattern can be exploited as \(\mathrm{MSE}=\sum_{m=1}^{M}\left|P_{d}(\theta_{m})-\mathbf{a}^{H}(\theta_{m} )\mathbf{R}_{\mathbf{x}}\mathbf{a}(\theta_{m})\right|^{2}\) where \(\theta_{m}\) is the \(m\)-th azimuth angle grid among all \(M\) grids, \(P_{d}(\theta_{m})\) is the desired beam pattern level at \(\theta_{m}\). \(\mathbf{a}(\theta_{m})=[1,e^{j2\pi\Delta\sin(\theta_{m})},\ldots,e^{j2\pi(N_{ l}-1)\Delta\sin(\theta_{m})}]^{T}\in\mathbb{C}^{N_{l}\times 1}\) is the transmit steering vector and \(\Delta\) is the normalized distance between adjacent array elements. \(\mathbf{R}_{\mathbf{x}}\) is the covariance matrix of transmit waveforms. A smaller MSE results in a higher signal-to-noise ratio (SNR) at the receiver, leading to improved target detection probability and estimation accuracy [14]. In addition, CRB can be exploited for target estimation. CRB is a lower bound on the variance of an unbiased estimator, which can be obtained by calculating the inverse of the Fisher information matrix (FIM), i.e., CRB = \(\mathbf{F}^{-1}\). The matrix \(\mathbf{F}\) contains the estimation of the angular direction, the complex reflection coefficient, and the Doppler frequency of the radar echo signal [26]. Since RSMA-assisted ISAC systems offer flexibility in both message construction and resource allocation, RSMA-assisted ISAC systems outperform ISAC systems based on SDMA and NOMA. In [17], RSMA-assisted ISAC systems show better communication-sensing performance trade-offs, higher energy efficiency, and robust interference management compared to both SDMA-based ISAC and NOMA-based ISAC schemes. In [27], RSMA-assisted ISAC systems demonstrate better spectral efficiency compared to NOMA-based ISAC in IoT environments where CSI is imperfect or interference cancellation is limited.