## 3 Proposed Scheme
### Problem Formulation  
To enhance the system performance, we minimized both co-channel interference and inter-cell interference. Thus, we can find the optimal resource sharing pair between cellular users and M-D2D groups. In the general case, to find the resource sharing partner for the M-D2D communications, we suppose in each cell, there are totally \(R_{T}\) uplink interference regions as shown in Figure 2b, constituting a set \(R_{U}\), where \(R_{U}=\{1,2,\,\ldots,\,R_{T}\}\). Within these regions, the D2D receivers are interfered with by cellular users, thus D2D communications cannot guarantee the system QoS [31]. Without loss of generality, in the proposed method, we assume one of the uplink interference regions, say uplink interference region-1, having the radius, \(U_{1}\). To avoid high interference from cellular users to D2D receivers, we should maintain an adequate distance between cellular users and M-D2D groups. Therefore, we assumed that the M-D2D groups in the cell outer-region can only reuse the resources of cellular users that exist in the cell inner-region. The reusing of cellular resource by the M-D2D groups in the same section of the cell is not allowed.  
**Proposition 1**.: _The radius, \(U_{1}\), of the uplink interference region-1 can be expressed as:_  
\[U_{1}=\frac{\gamma_{T}\frac{2}{\pi}\cdot l_{c^{in},B}\cdot l_{\frac{\pi}{8}}^{ edge}\cdot\frac{\pi^{edge}}{8}}{l_{\frac{\pi}{8}}^{edge}\cdot B} \tag{6}\]  
_where \(\gamma_{T}\)h is the threshold SINR._  
Proof.: The proof is presented in Appendix A.  
In cellular communication, the received power, \(P_{R}\), is the multiplication of the transmitted power, \(P_{T}\), and channel gain, \(L\), as:  
\[P_{R}=P_{T}\times L \tag{7}\]  
where \(L=l^{-\alpha}\times|H|^{2}\). The term, \(l\), is the channel distance and \(H\) is the channel coefficient. Therefore, we can express the SINR as:  
\[\gamma=\frac{P_{R}}{I+\sigma_{n}^{2}} \tag{8}\]  
where \(I\) is the interference in the network.  
To maintain reliable transmission, in the proposed method, we assume that the M-D2D groups are located in the cell outer region and cellular users are located in the cell inner region. According to Figure 2, we can express the SINR of the cellular user, \(c^{in}\), and the M-D2D group, \(g^{edge}\), as:  
\[\gamma_{c^{in}}=\ \frac{P_{B}\cdot l_{c^{in},B}^{-\alpha}\cdot\left|H_{c^{in},B} \right|^{2}}{P_{g^{edge}}\cdot l_{g^{edge},B}^{-\alpha}\cdot\left|H_{g^{edge},B}\right|^{2}+\ I_{c,B}+\ \sigma_{n}^{2}} \tag{9}\]and:  
\[\gamma_{g^{edge}}=\ \frac{P_{g^{edge}}\cdot\ l_{g^{edge}_{t},g^{edge}_{t}}^{- \alpha}\cdot\left|\ H_{g_{t}^{edge}_{t},g^{edge}_{t}}\right|^{2}}{P_{B}\cdot\ l_{c^{in},g^{edge}_{t}}^{- \alpha}\cdot\left|\ H_{c^{in},g^{edge}_{t}}\right|^{2}+\ I_{g,m}+\ \sigma_{n}^{2}} \tag{10}\]  
where \(I_{c,B}\) denotes co-channel interference from eNBs to the cellular user and \(I_{g,m}\) denotes co-channel interference from D2D transmitters in the upper layers of the network to M-D2D receivers. We can express \(I_{c,B}\) and \(I_{g,m}\) as:  
\[I_{c,B}=\sum_{n\
eq c,\ n=1}^{6}P_{B}\cdot l_{n,B}^{-\alpha}\cdot\left|H_{n,B} \right|^{2} \tag{11}\]  
and:  
\[I_{g,m}=\sum_{m=1}^{6}P_{m}\cdot l_{g,m}^{-\alpha}\cdot\left|H_{g,m}\right|^{2} \tag{12}\]