# Multicell Interference Management in Device to Device Underlay Cellular Networks
## 2 System Model  
We consider a multicell D2D underlay cellular network, where \(L\) neighbouring cells co-exist. The UEs and the eNBs are assumed to be equipped with a single omni-directional antenna. Each cell contains two types of users, cellular users (CUs) and D2D users (DUs), where their communication modes are assumed that have already been selected. The CUs communicate through the eNB of their serving cell, while the DUs communicate directly to each other in pairs, where the DUs of every D2D pair are adequately close to each other, thus forming a D2D link. The corresponding reference topology is shown in Figure 1.  
We assume that each cell has access to a fixed number of \(K\) RBs. Every cellular user of each cell uses one of those RBs which are orthogonal to each other. This means that there is no interference between cellular users of the same cell. The number of cellular users at each cell requesting a RB can be variable from cell to cell. Considering that the set of cellular users of cell \(i\) will be requesting and assigned a non-overlapping set of RBs, each cellular user of the same cell will occupy a different RB. Apart from the cellular users, D2D users reside in the network and each of them has to decide which RB to choose in order to transmit. The D2D users reuse one of the available \(K\) RBs at each cell. Multiple D2D users may share the same RB with one cellular user in each cell. As mentioned before, RBs reuse takes place between cellular users from different cells and between cellular and D2D users from the same or different cell. In principle, a RB may be occupied simultaneously from only one cellular user per eNB, and from multiple D2D users.  
In this multicell D2D underlay architecture, in order to treat cellular and D2D users in a homogeneous and holistic manner from RB allocation point of view, apart from the cellular cells i.e., eNBs, we consider the D2D receivers as virtual cells. Each D2D receiver can play the role of a (virtual) cell, where the serving users of this cell is exactly one i.e., the D2D transmitter of the respective D2D pair. From this point and on, when we refer to a cell, we may consider either the eNB of a cell or the D2D receiver of the respective D2D pair (that is we refer to either the physical \(L\) neighbouring cells or the considered virtual cells as defined above).  
Let \(E\) be the set of all the cells (including the virtual ones) and \(|E|\) be its cardinality that represents the number of the considered cells in the network. Let \(V_{i}\) be the set of users of cell \(i\) and let also \(K\) be the set of the available RBs at cell \(i\) which a user \(u\) can use. We define the SINR at the receiver of every user \(u\) of cell \(i\), either cellular or D2D, using RB \(k\), as follows:  
\[\gamma_{u}^{k}=\frac{G_{u}^{i,k}\cdot P_{u}^{k}}{\sum\limits_{j\
eq i}\sum \limits_{v\in V_{j}}G_{v}^{i,k}\cdot P_{v}^{k}+\sigma^{2}} \tag{1}\]  
where \(u\in V_{i}\), i.e., the set of users who belong to cell \(i\), \(G_{u}^{i,k}\) denotes the channel gain between user \(u\) of the cell \(i\) and the receiver-cell \(i\) at RB \(k\), \(P_{u}^{k}\) is the transmission power of user \(u\) at RB \(k\), \(V_{j}\) is the set of users who belong to cell \(j\), \(G_{v}^{i,k}\) is the channel gain between user \(v\) of the cell \(j\) and the receiver of cell \(i\) at RB \(k\), \(P_{v}^{k}\) is the transmission power of user \(v\) at RB \(k\) and \(\sigma^{2}\) refers to the thermal background noise power at the receiver of each link.  
Figure 1: System Model.