## 2 System Model and Problem Formulation
### System Model
Moreover, we assume that, if RIS \(n,n\in\mathcal{N}\) associates with user \(n\), it adjusts its phase shift so that the reflected link, via itself, is aligned with the corresponding direct link of user \(n\). To streamline the practical design of BS selections, we introduce binary variables \(\gamma_{m,n},m\in\mathcal{M},n\in\mathcal{N}\) to represent whether BS \(m\) is selected by user \(n\) (\(\gamma_{m,n}=1\)) or not (\(\gamma_{m,n}=0\)). We assume that each BS \(m\) serves at most one user, while each user can be served by multiple BSs. Additionally, we ensure that there is at least one BS in \(\mathcal{M}\) selected by each user \(n\) to provide service coverage. Thus, we have  
\[\sum_{n\in\mathcal{N}}\gamma_{m,n}\leq 1,\ \forall m\in \mathcal{M}, \tag{1}\] \[\sum_{m\in\mathcal{M}}\gamma_{m,n}\geq 1,\ \forall n\in\mathcal{N}. \tag{2}\]  
Figure 2: The system model of the considered scenario in the RIS-CF network. For coherent joint transmission, we assume, in this paper, the synchronization of all BSs associated with user \(n\), such that user \(n\) could receive the exact same symbols from those BSs. Specifically, we denote the received symbol of each user \(n\) served by BS \(m\) as \(x_{m,n}=w_{m,n}s_{n}\), where \(s_{n}\) and \(w_{m,n}\) represent the transmission by each user \(n\) and the precoding matrix, respectively. The power of \(s_{n}\) is normalized, i.e., \(\mathbb{E}\{|s_{n}|^{2}\}=1\). It is assumed, in this paper, that intended signals with predetermined phase shifts are reflected by each RIS \(r\) purposefully for its associated user \(n\), while the unintended signals are scattered at random. Thus, each user \(n\) receives the desired signal from its associated RIS \(n\) as well as that scattered by other non-associated RISs [31]. To account for the significant double path loss in the RIS-aided cascaded channel, we exclude any signals that are reflected or scattered more than once by the RIS. The channels from BSs to user \(n\) consist of direct links from BSs as well as reflected and scattered links via RISs. Since user \(n\) is served by multiple BSs simultaneously, each user \(n\) could receive the effective signal composed of the aggregated signals originating from its selected BSs. Thus, the desired signal at user \(n\)'s receiver can be expressed as  
\[y_{n}=\sum_{m\in\mathcal{M}}\gamma_{m,n}H_{m,n}x_{m,n}, \tag{3}\]  
where \(H_{m,n}=h_{m,n}+f_{n,n}^{H}\mathbf{\Phi}_{n}g_{m,n}+\sum_{r\in\mathcal{N},r\
eq n }f_{r,n}^{H}\mathbf{\Phi}_{r}g_{m,r}\). If BS \(m\) sends information signals to user \(n\) in the downlink communication, \(\gamma_{m,n}\) is set to 1. In other words, we can consider the BSs selected by user \(n\) as a single multi-antenna BS that employs precoding weights \(w_{m,n}\)'s. Consequently, each reflecting element on RIS \(n\) adjusts its reflection phase to align the cascaded BS-RIS-user channel with the direct BS-user channel [6]. To achieve this, the \(l\)-th phase shift of RIS \(n\) needs to be configured as \(\theta_{n,l}=\angle\sum\limits_{m\in\mathcal{M}}\gamma_{m,n}h_{m,n}w_{m,n}- \angle\sum\limits_{m\in\mathcal{M}}\gamma_{m,n}f_{n,n}^{H}g_{m,n}w_{m,n}\). Thus, \(H_{m,n}\) in (3) can be rewritten as  
\[H_{m,n}=h_{m,n}+\sum_{l\in\mathcal{L}}|g_{m,n,l}||f_{n,l}|+\sum_{r\in\mathcal{ N},r\
eq n}f_{r,n}^{H}\mathbf{\Phi}_{r}g_{m,r}. \tag{4}\]  
Meanwhile, regardless of the desired signal from its own selected BSs, each user \(n\) would also receive both the direct interference transmitted by other BSs and the interference reflected by RISs. Thus, each user \(n\) would receive the following total interference:  
\[I_{n}=\sum_{j\in\mathcal{N},j\
eq n}\sum_{m\in\mathcal{M}}\gamma_{m,j}\Bigg{(} h_{m,n}+\sum_{r\in\mathcal{N}}f_{r,n}^{H}\mathbf{\Phi}_{r}g_{m,r}\Bigg{)}x_{m,j}. \tag{5}\]  
In light of (3) and (5), we could further obtain the average SINR of each user \(n\) as  
\[\widetilde{\lambda}_{n}=\mathbb{E}\bigg{[}\frac{|y_{n}|^{2}}{\sigma^{2}+|I_{ n}|^{2}}\bigg{]}, \tag{6}\]  
where \(\sigma^{2}\) is the Gaussian noise power.