# Energy Efficient Underlaid D2D Communication for 5G Applications
## 2 System Model  
Figure 1 shows the whole network underlaid with D2D communication. The D2D users and the cellular users share the same bandwidth in the uplink direction. Here, the base station is used to allocate the resources to the whole network underlaid with D2D communication [24]. At present, the existing works were based on a single-cell scenario, but in this paper, we tried to work on multiple bands which means that at a specific time it transfers the data to various devices. The total bandwidth of the ith band is Wi and it is divided into M sub-bands. The bandwidth of each sub-band is Wi, i = 1, 2, 3,..., M, having these parameters the channel fading may vary and the power transmission is adjustable and it also improves energy efficiency [24; 25]. Both cellular and D2D are divided into sub-bands so they can use these sub-bands with the density \(\lambda\)c,i, \(\lambda\)d,i. For each, there is a different density and if we use more bandwidth for both users, the multiple users are allowed to transmit and the density also increases. The transmission power allocation for the ith band for cellular users is C\({}_{\text{p,i}}\) and the total transmission power C\({}_{\text{P}}\). \[\sum_{i=1}^{k}C_{\text{p,i}}=C_{\text{P}} \tag{1}\]  
Similarly, the transmission power for the ith band for D2D users is D\({}_{\text{p,i}}\) and the total transmission power for D2D is D\({}_{\text{P}}\). \[\sum_{i=1}^{k}D_{\text{p,i}}=D_{\text{P}} \tag{2}\]  
We considered the channel model as a channel path loss and small-scale channel fading, which represents the Rayleigh fading. To observe the performance of cellular and D2D communication of the whole network, we focused on the typical receiver without loss, the typical receiver for D2D communication, and the typical base station for the cellular network [26]. The received power for both devices obtained by considering small and large-scale fading is expressed as  
\[R_{\text{P}}=T_{\text{P}}\delta R^{-\alpha} \tag{3}\]  
where T\({}_{\text{P}}\) represents power transmission, \(\delta\) indicates Rayleigh fading, R is for the distance between transmitter and receiver, R\({}_{\text{P}}\) represents the power received for cellular and D2D users and \(\alpha\) represents the path loss exponent. To obtain a better quality of communication, the probabilities should be less than the threshold values. \[1-Pr\left(\text{SIRc,i}\geq\text{Tc,i}\right)\leq\theta c,i, \tag{4}\]  
\[1-Pr\left(\text{SIRd,i}\geq\text{Td,i}\right)\leq\theta d,i, \tag{5}\]  
where \(\theta c\),i and \(\theta d\),i are the probabilities for cellular transmission and device to device communication for the ith band. But the power transmission for D2D can't hold interference  
Figure 1: The system model of underlaid with D2D communication. for very long, in which case the network is complicated to coordinate. In that situation, the base station reduces the number of D2D users to access until the probabilities become small. The power in the ith band should not be less than zero and it should not be greater than the upper bound and it is represented as \(\mathrm{D_{p,i,up}}\). Then we get  
\[0\leq\mathrm{D_{p,i}}\leq\mathrm{D_{p,i,up}} \tag{6}\]  
The optimal resource allocation is mainly used to maximize energy efficiency; finally, we obtain the maximum EE. Any optimization issue where the objective or any of the limitations are non-convex is referred to as a non-convex problem. Such a problem may have several possible zones, each with several locally optimal points. \(\mathrm{D_{p,i,inf}}\) and \(\mathrm{D_{p,i,sup}}\) are used to represent the lower limit and upper limit of the feasible area of \(\mathrm{D_{p,i}}\). We have \(\mathrm{D_{p,i,inf}}=\max\left\{0,\mathrm{D_{p,i,low}}\right\}\) and \(\mathrm{D_{p,i,sup}}=\min\left\{\mathrm{D_{p,i,up}},\mathrm{D_{p,i,high}}\right\}\).