# Null Subcarrier Index Modulation in OFDM Systems for 6G and Beyond
## 7 Simulation Results
The spectral efficiencies of the proposed and the conventional waveforms by using the 4-QAM modulation are illustrated in Figure 12. In these simulations, we see that the spectral efficiency of the proposed scheme was also experimentally achieved by comparing to conventional OFDM and OFDM-IM systems. We also observed that the waveforms designed with the structure of complex subgroups that were configured by using eight and sixteen subcarriers theoretically had the same spectral efficiency as shown in Table 3. However, the performance of the waveforms was affected by the complex subgroup size, at low SNR values: the waveform configured with the complex subgroups that consisted of eight carrier locations exhibited better spectral efficiency than the other waveform configured with the subgroups that had sixteen carrier locations. In Figure 13, the conventional and the proposed NSC-OFDM-IM waveforms were simulated by using the 8-PSK complex modulation. The complex subgroups were configured with eight and sixteen subcarrier locations for index-modulated waveforms. Looking into the bit error rate performance, the performance of the index-modulated scheme was affected by the complex modulation order, and it expressed slightly better bit error rate performance than the conventional OFDM. Moreover, in terms of spectral efficiency, the system had better performance in the increasing SNR values, as shown in Figure 14.  
Figure 11: Succession rate of the null carrier indices for the proposed NSC-OFDM-IM system. Figure 12: Comparison of the proposed NSC-OFDM-IM with conventional OFDM/OFDM-IM in terms of spectral efficiency, where 4-QAM modulation was adopted. Figure 13: BER performance comparison between the proposed NSC-OFDM-IM and conventional OFDM-IM/OFDM, where 8-PSK modulation was adopted. In order to better illustrate the motivation for using the log-likelihood ratio mechanism for the null subcarrier index detection, we also simulated our proposed NSC-OFDM-IM waveform without the LLR design at the receiver, as illustrated in Figure 8:0 Instead, we designed the index detection mechanism just after the frequency domain equalization. The equalized spectrum was first divided into a number of K complex subgroups, and then, by taking the absolute values of the complex coefficients in each complex subgroup, we extracted the index of the minimum absolute value among them. The index number was assumed to belong to the null subcarrier location within the complex subgroup. Figure 15 depicts the BER performances of classical OFDM and the proposed NSC-OFDM-IM (without the LLR detector) waveforms using the BPSK complex modulation and the complex subgroup designs with four and eight subcarriers. In the simulations, we observed that the OFDM signal expressed better performance than the proposed scheme. This was due to the detection mechanism in NSC-OFDM-IM, which does not take into account the channel noise while detecting the null carrier index, hence causing a high number of detection failures at the null subcarrier positions. By comparing it to conventional OFDM, while both waveforms aim to obtain the complex modulated data on the active subcarriers, the index modulated waveform was also struggling to detect the null carrier indices in a noisy environment in this case. In summary, the performance of index modulation scheme was highly affected by the misdetection of null carrier indices. Figure 8:1 Comparison of the proposed NSC-OFDM-IM with conventional OFDM/OFDM-IM in terms of spectral efficiency, where 8-PSK modulation was adopted.