## 4 Numerical Results  
In this section, we provide several simulation results to evaluate the performance of our proposed resource allocation scheme. Consider a single cell with a radius of 500 m where CUs and D2D pairs are uniformly and randomly distributed. Since the distance between D2D-Tx and D2D-Rx are relatively short, we assume all D2D pairs have the same transmitting distance. The main simulation parameters such as path loss models, shadowing standard deviation, subcarrier bandwidth, etc. are the same as these proposed in [24]. The related parameters used in the simulation experiments are summarized in Table 2. The tolerance \(\epsilon\) is set to \(10^{-5}\). We compare the proposed scheme with three other schemes. Scheme 1 is proposed in [9], where throughput of multiple D2D pairs is maximized by using the Hungarian method. Scheme 2 randomly selects the subcarrier for D2D pairs to reuse and performs power allocation optimization to maximize the total achievable rate of the D2D pairs. Scheme 1 and Scheme 2 can only be applied to the case that each D2D pair could reuse only one single cellular resource. Scheme 3 is developed in [17], where only one D2D pair has the authority to use all subcarriers. In Figure 2, we compare the spectrum efficiency of CUs and DUs versus the different transmit power budgets for D2D pairs, where \(K=20\), \(M=30\), \(P_{\rm thr}^{D}=20\) dBm, and \(R_{m,\rm min}^{C}=6\) bps\(/\)Hz. From the figure, the spectrum efficiency of D2D pairs increases dramatically with the D2D transmit power budget; on the contrary, since greater power of DUs brings more serious interference to CUs, the spectrum efficiency of CUs decreases with the increasing of D2D transmit power budget. When the D2D transmit power is small, for example, \(P_{\rm thr}^{D}\leq 10\) dBm, the movement of D2D transmit power has great influence on the spectrum efficiencies of DUs and CUs, which have great improved and reduced tendency, respectively. When the D2D transmit power budget is increased to some degree, it would not improve the spectrum efficiency of DUs effectively, the growth in spectrum efficiency of DUs tends to saturation. Moreover, we can see that as the distance of D2D decreases, the spectrum efficiency of DUs increases significantly, while the spectrum efficiency of CUs reduces slightly. The reason is that the smaller the D2D distance, the better the D2D channel conditions and the higher rate in D2D communications. Meanwhile, with the decrease of D2D distance (from 40 m to 20 m), the value range of D2D optimal transmit power is extended, which allows the D2D pairs to have more chance to reuse more subcarriers and causes more interference to CUs. While the D2D transmit power budget is large enough, the effect of interference caused by D2D transmit power on spectrum efficiency is more significant than that of D2D distance, and the curve becomes smooth. In addition, the increment of DUs in spectrum efficiency is greater than the decrement of CUs with the decrease of D2D distance, which implies that limiting the maximum communication distance of D2D can effectively improve the sum spectrum efficiency of the heterogeneous network. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Cellular layout & one isolated cellular cell \\ System area & 500 m \(\times\) 500 m \\ Maximum distance of D2D, \(r\) & 20, 30, 40 m \\ Uplink subcarrier bandwidth & 180 kHz \\ Noise spectral density & \(-\)174 dBm/Hz \\ Path loss model for cellular links & 128.1 + 37.6 log10 (d [km]) \\ Path loss model for D2D links & 148 + 40 log10 (d [km]) \\ Shadowing standard deviation & 10 dB for cellular links \\ Maximum Tx power of CU & 20 dBm \\ Maximum Tx power of D2D & \(-\)10 dBm\(\sim\)20 dBm \\ Minimum Rate of CU & 0 bps/Hz\(\sim\)6 bps/Hz \\ Number of CUs, \(M\) & 10\(\sim\)30 \\ Number of D2D pairs, \(K\) & 8\(\sim\)20 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Main Simulation Parameters.