# Efficient Modulation Scheme for Intermediate Relay-Aided IoT Networks
## 4 Numerical Results  
In our simulation, all Rician fading channels are assumed to have the same Rician parameter \(K\). Additionally, we assume that during the second phase, the phases of the LOS components of channels \(h_{1}\) and \(h_{2}\) can be aligned at the source and relay, respectively. Consequently, we define \(\mu_{h_{0}}=\mu_{h_{1}}=\sqrt{K/(K+1)}\), \(\mu_{h_{2}}=a\sqrt{K/(K+1)}\), \(\sigma_{h_{0}}^{2}=\sigma_{h_{1}}^{2}=1/(K+1)\) and \(\sigma_{h_{2}}^{2}=|a|^{2}/(K+1)\), where \(a\) denotes the ratio between the amplitude of the LOS component of the relay to destination and the source to destination link. The noise power is parameterized with \(E_{b}/N_{0}\) of the source to destination link. First, we provide an example of quadratic assignment problem-optimized modulation diversity for 16-QAM. For \(E_{b}/N_{0}=2\)dB, \(K=10\) and \(a=1\), the remapping schemes \(\lambda_{1}\) and \(\lambda_{2}\) are depicted in Figure 2. Here, \"S-HARQ\" denotes the simple HARQ retransmission, where \(\lambda_{1}=\lambda_{2}=\lambda_{0}\), \"Heuristic\" indicates the conventional heuristic approach where \(\lambda_{1}=\lambda_{2}=\lambda_{S}\) and \"Proposed\" represents the proposed modulation diversity solution. The results support the use of heuristics discussed in Section 3.2 as both \(\lambda_{1}\) and \(\lambda_{2}\) are essentially very close to \(\lambda_{S}\). However, we note that in general \(\lambda_{1}\
eq\lambda_{2}\). Next, we compare the empirical BER for three mapping schemes. The empirical BER is evaluated by applying the ML demodulator in (2) and over \(M=10^{7}\) randomly generated channels, noise instances and data symbols. In Figure 3, there is a considerable performance gain for the proposed modulation diversity scheme versus S-HARQ. Moreover, the gain of the proposed scheme over \(\lambda_{1}=\lambda_{2}=\lambda_{S}\) indicates that different rearrangements at the source and relay during the cooperative transmission can further reduce BER and boost performance. However, as \(K\) decreases, heuristic modulation diversity becomes a good approximate solution with a diminishing performance gap from the proposed scheme. Finally, we test the coded BER performance and the robustness of the modulation diversity design with a typical LDPC coded system. Although our proposed scheme depends on noise power and statistical CSI, in this simulation we only use modulation diversity schemes designed for the specific channel condition of \(E_{b}/N_{0}=-2 dB\) under the additive white Gaussian noise (AWGN) channel. We test the modulation diversity design on Rician channels of \(K=10\) and \(a=1\), and various \(E_{b}/N_{0}\) to demonstrate its robustness. The coded BER results for 16-QAM and 32-QAM are plotted in Figure 4a,b, respectively. We can see that the proposed solution provides additional gains  
Figure 2: Illustration of constellations \(\lambda_{1}\) and \(\lambda_{2}\). over the heuristic approach and S-HARQ, even for mismatched design conditions. Additionally, we notice that the proposed solution provides a larger performance gain for 32-QAM than for 16-QAM constellation. Figure 5 shows the performance of the proposed technique against conventional S-HARQ and heuristic approaches. It can be noted that the proposed approach outperforms the other approaches. Note that \"S-HARQ\" here represents the simple HARQ retransmission, \"Heuristic\" indicates the conventional heuristic approach where \(\lambda_{1}=\lambda_{2}=\lambda_{S}\), \"Proposed (Bound)\" indicates the upper performance bound of the proposed solution and \"Proposed\" represents the exact proposed modulation diversity solution. Note that the proposed solution is upper bounded by the dashed curve and the distance between the two curves reduces for higher values of signal power. Furthermore, for smaller values of signal power, the S-HARQ approach performs better, and at larger values the proposed bounded technique appears more suitable.