# Null Subcarrier Index Modulation in OFDM Systems for 6G and Beyond
## 4 Proposed NSC-OFDM-IM
### Spectral Efficiency  
The spectral efficiency (SE) is an important measurement parameter in OFDM systems, and it is defined as the symbol transmission rate in a unit band [34; 35]. For a conventional OFDM system, the SE can be expressed as,  
\[SE_{OFDM}=\frac{N_{used}(log_{2}M)}{N_{FFT}+N_{cp}} \tag{25}\]  
where \(M\) denotes the modulation order of the constellation mapper, \(N_{used}\) is the number of subcarriers used for data transmission, and \(N_{cp}\) is the length of the cyclic prefix.  
For the existing OFDM-IM and our proposed NSC-OFDM-IM schemes, the SE can be represented as,  
\[SE_{OFDM-IM}=\frac{K(k(log_{2}M)+\lfloor log_{2}(C(n,k))\rfloor)}{N_{FFT}+N_{cp}} \tag{26}\]  
\[SE_{NSC-OFDM-IM}=\frac{K(log_{2}\widetilde{M}+(\widetilde{M}-1)log_{2}M)}{N_{ FFT}+N_{cp}} \tag{27}\]  
where \(n\) and \(k\) denote the size of the complex subgroup and the number of active carriers in each subgroup for the OFDM-IM scheme, respectively. \(\widetilde{M}\) is the index modulation order (or subgroup size) for the NSC-OFDM-IM scheme. \(K\) is the number of complex subgroups in the frequency domain.  
The performance analysis in terms of the spectral efficiency of the proposed scheme is compared with the conventional OFDM and OFDM-IM schemes in Table 3.  
As can be seen from Table 3, based on the size of the complex subgroups and the constellation mapper order of the system, the proposed NSC-OFDM-IM scheme exhibits an improved SE over the conventional OFDM scheme while achieving the same SE as compared to the OFDM-IM scheme. The maximum spectral efficiencies (the green area in Table 3) are obtained for BSK modulation by using the subgroups with 4 and 8 subcarriers, for 4-QAM modulation by using the subgroups with 8 and 16 subcarriers, and for 8-PSK modulation by using the subgroups with 16 and 32 subcarriers. The proposed NSC-OFDM-IM scheme exhibits the same SE (the gray area in Table 3) as the conventional OFDM systems when the index modulation order \(\widetilde{M}\) equals the complex modulation order \(M\).  
The succession rate of obtaining the correct indices of the null subcarriers in the complex subgroups is given by:  
\[\eta=\frac{1}{N_{sym}}\sum_{\alpha=1}^{N_{sym}}\frac{K-\epsilon_{\alpha}}{K} \tag{28}\]  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & **Complex** & \multicolumn{3}{c}{**BPSK**} & \multicolumn{3}{c}{**4-QAM**} & \multicolumn{3}{c}{**8-PSK**} \\
**Waveforms** & **Group Size** & **Spectral** & **Extra** & **Spectral** & **Extra** & **Spectral** & **Extra** \\  & (\(\widetilde{M}\)) & **Efficiency** & **Bits** & **Efficiency** & **Bits** & **Efficiency** & **Bits** \\ \hline Classical & - & 0.80 & - & 1.60 & - & 2.40 & - \\ OFDM & & & & & & & \\ NSC-OFDM-IM & 2 & 0.80 & 0 & 1.2 & \(-\)1 & 1.60 & \(-\)2 \\ NSC-OFDM-IM & 4 & 1 & +1 & 1.60 & 0 & 2.20 & \(-\)1 \\ IN/OFDM-IM & & & & & & & \\ NSC-OFDM- & 8 & 1 & +2 & 1.70 & +1 & 2.40 & 0 \\ IN/OFDM-IM & & & & & & & \\ NSC-OFDM-IM & & & & & & & \\ IN/OFDM-IM & & & & & & & \\ NSC-OFDM-IM & & & & & & & \\ IM/OFDM-IM & & & & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 3: Spectral efficiencies of OFDM, OFDM-IM, and NSC-OFDM-IM (\(N_{FFT}=128\), \(N_{cp}=32\), \(n=\widetilde{M}\), and \(k=\widetilde{M}-1\)).  
where \(\eta\) denotes the succession rate of the null carrier indices per the SNR and \(\epsilon_{\alpha}\) is the number of failed indices.  
In the increasing SNR values, the null subcarriers in the subgroups are successfully captured, and the value of the succession rate \(\eta\) converges to one.