## 3 RSMA-Assisted Integrated Sensing and Communication Systems
### RSMA System Model  
RSMA has been studied in various forms depending on how user messages are split and common messages are combined for multiple users. In this paper, we focus on the key operations of one-layer downlink RSMA (see more comprehensive studies in [20; 21]).  
We consider the downlink RSMA system where the base station (BS) employs \(M\) transmit antennas and \(K\) users have a single antenna. By denoting the user set \(\mathcal{K}=\{1,2,\ldots,K\}\), as shown in Figure 3, the message of the \(k\)-th user (\(\forall k\in\mathcal{K}\)), \(W_{k}\), is split into a common part \(W_{c,k}\) and a private part \(W_{p,k}\). The common parts of all users, \(\{W_{c,1},W_{c,2},\ldots,W_{c,K}\}\), are combined into a single common message \(W_{c}\). The common message \(W_{c}\) and the \(K\) private messages \(\{W_{p,1},W_{p,2},\ldots,W_{p,K}\}\) are independently encoded into \(s_{c}\) and \(\{s_{1},s_{2},\ldots,s_{K}\}\), respectively. By defining \(\mathbf{p}_{c}\in\mathbb{C}^{N_{t}\times 1}\) and \(\mathbf{p}_{k}\in\mathbb{C}^{N_{t}\times 1}\) as the precoding vectors for the common and private streams, the transmitted signal \(x\in\mathbb{C}^{N_{t}\times 1}\) can be expressed as  
\[\mathbf{x}=\mathbf{p}_{c}s_{c}+\sum_{k\in\mathcal{K}}\mathbf{p}_{k}s_{k}. \tag{1}\]  
The overall transmitted symbol streams can be expressed as \(\mathbf{s}=[s_{1},s_{2},\ldots,s_{K},s_{c}]^{\mathrm{T}}\)\(\in\mathbb{C}^{(K+1)\times 1}\), following \(E\big{[}\mathbf{s}\mathbf{s}^{H}\big{]}=\mathbf{I}_{K+1}\). The transmit power constraint is expressed as \(P_{c}+\sum_{k\in\mathcal{K}}P_{k}\leq P\), where \(P_{c}=\|\mathbf{p}_{c}\|^{2}\) and \(P_{k}=\|\mathbf{p}_{k}\|^{2}\). The received signal at the \(k\)-th user can be expressed as  
\[y_{k}=\mathbf{h}_{k}^{H}\mathbf{x}+n_{k}=\mathbf{h}_{k}^{H}\mathbf{p}_{c}s_{c }+\mathbf{h}_{k}^{H}\sum_{k\in\mathcal{K}}\mathbf{p}_{k}s_{k}+n_{k} \tag{2}\]  
where \(\mathbf{h}_{k}\in\mathbb{C}^{N_{t}\times 1}\) is the channel vector between the transmitter and the \(k\)-th user, and \(n_{k}\sim\mathcal{CN}(0,\sigma_{k}^{2})\) is additive white gaussian noise (AWGN). Note that the first step on the receiver side is applying SIC to detect the common stream \(s_{c}\) and detect the common message part \(\tilde{W}_{c,k}\) by treating the private streams as noise. The signal-to-interference plus noise ratio (SINR) for the common stream at the \(k\)-th user can be expressed as  
\[\gamma_{c,k}=\frac{|\mathbf{h}_{k}^{H}\mathbf{p}_{c}|^{2}}{\sum_{j\in \mathcal{K}}|\mathbf{h}_{k}^{H}\mathbf{p}_{j}|^{2}+\sigma_{k}^{2}}. \tag{3}\]  
The common stream is reconstructed using the estimated \(\hat{W}_{c}\) and subtracted through the SIC operation. Then, the private message of the \(k\)-th user \(\tilde{W}_{p,k}\) can be detected by treating the other users' private streams as noise. Considering the perfect SIC operation, the SINR for the private stream at the \(k\)-th user can be expressed as  
\[\gamma_{p,k}=\frac{|\mathbf{h}_{k}^{H}\mathbf{p}_{k}|^{2}}{\sum_{j\in K,j\
eq k}| \mathbf{h}_{k}^{H}\mathbf{p}_{j}|^{2}+\sigma_{k}^{2}}. \tag{4}\]  
The maximum rate for the common stream is \(R_{c}=\min\{\log_{2}(1+\gamma_{c,1}),\ldots,\log_{2}(1+\gamma_{c,K})\}\), and the rate for the private stream is \(R_{p,k}=\log_{2}(1+\gamma_{p,k})\). Since the common stream contains the common messages intended for all users, we have \(R_{c}=\sum_{k\in K}C_{k}\), where \(C_{k}\) represents the portion of the common stream rate allocated to the \(k\)-th user. Therefore, the total achievable rate for the \(k\)-th user is \(R_{k,\text{tot}}=C_{k}+R_{p,k}\).