## 4 Numerical Results
Figure 3 shows the sum spectrum efficiency of the heterogeneous network with different number of CUs, where \(K=8\), \(r=30\) m, \(P^{D}_{\rm thr}=20\) dBm, and \(R^{C}_{m,\rm min}=6\) bps/Hz. Note that we assume the number of CUs equals the number of subcarriers, since each subcarrier is occupied by a CU. We observe that the sum spectrum efficiency goes up with the number of CUs increasing in all compared schemes, which indicates that more DUs contribute to a higher system sum spectrum efficiency. From the figure, our proposed scheme, i.e., the scheme of reusing multiple subcarriers, performs much better than the other three schemes. In particular, when the number of CUs is relatively small, for example, \(M=10\), the performance of the proposed scheme is close to Scheme 1. With the increasing number of CUs, the absolute performance gap expands. The relative performance gap when there are 30 CUs is about 19%. This is due to the fact that each D2D pair can use more subcarriers as the number of CUs increases in the scenario of reusing multiple CUs' subcarriers. Obviously, Scheme 3 has the lowest sum spectrum efficiency. This is due to the fact that only one D2D pair gets involved in resource sharing; as a result, there may be many cellular subcarriers that may not be reused, leading to inefficiency in resource utilization. Figure 3: The sum spectrum efficiency versus the number of CUs \(M\). Figure 2: The spectrum efficiencies of the D2D pairs and CUs versus the transmit power budget. The sum spectrum efficiency of the heterogeneous network with different D2D transmit power budget is illustrated in Figure 4, where \(K=20\), \(M=30\), \(r=30\) m, and \(R_{m,\text{min}}^{C}=6\)\(\text{bps}/\text{Hz}\). Intuitively, the sum spectrum efficiency rises with the increase of D2D transmit power budget. Since the total power can be allocated to multiple subcarriers of multiple D2D pairs, our proposed sharing scheme exhibits the best performance and the largest increases in sum spectrum efficiency. In addition, the sum spectrum efficiency increases rapidly in the lower D2D transmission power case. When \(P_{\text{thr}}^{D}\geq 10\) dBm, the growth curve becomes smooth, which implies that the sum spectrum efficiency starts to saturate and the D2D transmit power does not have a major impact on the performance improvement. Figure 5 shows the sum spectrum efficiency with different rate requirement of the CUs, where \(K=20\), \(M=30\), \(r=30\) m, and \(P_{\text{thr}}^{D}=20\) dBm. Similarly, it is obvious that the proposed scheme performs much better than the other three. Moreover, with the increasing rate requirements of CUs, the sum spectrum efficiency of the heterogeneous network decreases slightly. With higher rate requirements, more transmission powers for CUs are needed to meet the required QoS, and thus more D2D pairs that share the same subcarriers with these CUs are restricted, resulting in that only the D2D pairs which generate low interference to the eNB can reuse the subcarriers. Figure 6 plots the sum spectrum efficiency of the heterogeneous network with different D2D pairs, where \(M=30\), \(r=30\) m, \(P_{\text{thr}}^{D}=20\) dBm, and \(R_{m,\text{min}}^{C}=6\) bps/Hz. We observe that with the increase of D2D pairs, the proposed scheme shows a trend of rapid growth in sum spectrum efficiency and outperforms other schemes, whereas the performance of Scheme 3 remains nearly unchanged. Since the performance of D2D communication on spectrum efficiency is far better than that of traditional cellular communication, multiple D2D pairs reusing multiple subcarriers can achieve a much higher throughput than reusing only single subcarrier (Scheme 1 and Scheme 2), or a single D2D pair reusing multiple subcarriers (Scheme 3). We also compare our proposed sharing scheme with Scheme 1 and Scheme 2 on the performance of average D2D spectrum efficiency. From Figure 7, It is obviously that the average D2D spectrum efficiency decreases with the increasing number of D2D pairs.