## 3 System Model and Problem Formulation
### Problem Formulation
#### 3.2.2 Interference Management  
In the network-assisted D2D communications, first the eNB collects CSI and required SINR information of all the users. Then, eNB calculates the transmit powers of CUs and DPs. In this subsection, we formulate the SINR problem which aims at minimizing the sum interference. The SINR is defined as the ratio of the power of a certain desired signal to the sum of interference power (from all other interfering devices) and noise power [26]. The uplink interference scenarios for a multicell cellular network are shown in Figure 3. Taking the cell 1 as target cell, in which the D2D transmitter \(m_{t}\) communicates with D2D receiver \(m_{r}\) located in edge zone by reusing subcarrier resource \(k\) of the CU \(n\) located in center zone. During data transmission between the D2D transmitter \(m_{t}\) and receiver \(m_{r}\), the possible uplink interference introduced in the network are, namely co-channel interference from D2D transmitter \(m_{t}\) to eNB (\(I_{B}^{m_{t}}\)), co-channel interference from CU \(n\) to D2D receiver \(m_{r}\) (\(I_{m_{t}}^{n}\)), interference from the D2D transmitter \(m_{t}^{\prime}\) of neighboring cell 2 to eNB of cell 1 (\(I_{B}^{m_{t}^{\prime}}\)), interference from the CU \(n^{\prime}\) of neighboring cell 2 to D2D receiver \(m_{r}\) of cell 1 (\(I_{m_{r}}^{n^{\prime}}\)), and mutual interference between D2D receiver \(m_{r}\) and neighboring cell D2D transmitter \(m_{t}^{\prime}\) (\(I_{m_{t}}^{m^{\prime}}\)). Therefore, the SINRs of CU \(n\) and DP \(m\) on subcarrier \(k\) are:  
\[\gamma_{n}^{k}=\frac{p_{n}^{k}\cdot I_{n,B}^{-\alpha}.|h_{n,B}|^{2}}{I_{B}^{m_{ t}}+I_{B}^{m_{t}^{\prime}}+\sigma_{n}^{2}},\forall n\in C,\ m\in D,k\in X \tag{13}\]  
and  
\[\gamma_{m}^{k}=\frac{\delta_{n,n^{\prime}}^{k}\cdot P_{m_{t}}^{k}\cdot I_{m_{t }}^{k}\cdot I_{m_{t},m_{r}}.|h_{m_{t},m_{r}}|^{2}}{I_{m_{r}}^{n}+I_{m_{r}}^{n^ {\prime}}+I_{m_{r}}^{m_{r}^{\prime}}+\sigma_{n}^{2}},\forall n^{\prime}\in C, \ m_{t},m_{r},\ m_{t}^{\prime}\in D,k\in X, \tag{14}\]  
respectively, where \(P_{n}^{k}\) and \(P_{m_{t}}^{k}\) represent the transmit power of CU \(n\) and D2D transmitter \(m_{t}\) for channel \(k\), respectively, and \(\sigma_{n}\) is the noise power. Figure 3: Demonstration of the interference between two neighboring cells, when a D2D communications is overlapped on a cellular resource. Thus, we define the interference terms as follows:  
\[I_{B}^{m_{t}}=\sum_{m=1}^{M}\sum_{n=1}^{N}\delta_{m,n}^{k}\cdot P_{m_{t}}^{k} \cdot l_{m_{t},B}^{-\alpha}\cdot|h_{m_{t},B}|^{2}, \tag{15}\]  
\[I_{B}^{m_{t}^{\prime}}=\sum_{ \begin{array}{c}m=1\\ m=1\\ m_{t}^{\prime}\
eq m_{t}\end{array} }^{M}P_{m_{t}^{\prime}}^{k}\cdot l_{m_{t}^{ \prime},B}^{-\alpha}\cdot|h_{m_{t}^{\prime},B}|^{2}, \tag{16}\]  
\[I_{m_{t}}^{n}=\sum_{m=1}^{M}\sum_{n=1}^{N}P_{n}^{k}\cdot l_{n_{t}m_{t}}^{- \alpha}\cdot|h_{m_{t},m_{t}}|^{2}, \tag{17}\]  
\[I_{m_{t}}^{n^{\prime}}=\sum_{m=1}^{M}\sum_{n=1}^{N}P_{n^{\prime}}^{k}\cdot l_{ n^{\prime},m_{t}}^{-\alpha}\cdot|h_{n^{\prime},m_{t}}|^{2}, \tag{18}\]  
\[I_{m_{t}}^{m_{t}^{\prime}}=\sum_{m=1}^{M}\sum_{ \begin{array}{c}m=1\\ m_{t}^{\prime}\
eq m_{t}\end{array} }^{N}P_{m_{t}^{\prime}}^{k}\cdot l_{m_{t}^{ \prime},m_{t}}^{-\alpha}\cdot|h_{m_{t}^{\prime},m_{t}}|^{2}, \tag{19}\]  
where \(l_{m_{t},B}\) and \(l_{m_{t}^{\prime},B}\) denote the distance from the eNB to D2D transmitter \(m_{t}\) and D2D transmitter \(m_{t}^{\prime}\), respectively, \(l_{n,m_{t}}\), \(l_{n^{\prime},m_{t}}\), and \(l_{m_{t}^{\prime},m_{t}}\) denote the distance from D2D receiver \(m_{r}\) to CU \(n\), CU \(n^{\prime}\), and D2D transmitter \(m_{t}^{\prime}\), respectively.