# Null Subcarrier Index Modulation in OFDM Systems for 6G and Beyond
## 7 Simulation Results  
In this section, we present the experimental results of the proposed NSC-OFDM-IM along with conventional OFDM and existing OFDM-IM. The performance of the proposed method and the conventional systems was evaluated in terms of spectral efficiency and bit error rate. The parameters in Table 6 were used for the computer simulations. The system was designed with 128-point subcarriers, which were used for the complex and index modulations, in the frequency domain. In all waveforms, the length of the cyclic prefix was chosen as 32 to prevent ISI. As a wireless channel environment, the flat fading Rayleigh multipath channel was used in the presence of additive white Gaussian noise. The complex data were generated by the BPSK, 4-QAM, and 8-PSK constellation mappers, and they were used to modulate the active subcarriers in all waveforms. Since the receiver has perfect channel state information (CSI), the system does not need a pilot symbol insertion in the transmitted signals for the channel estimation process. All the subcarriers are active in the conventional OFDM signal. In the proposed NSC-OFDM-IM and existing OFDM-IM schemes, 128 carrier locations are divided into \(K\) complex subgroups with a size of \(\bar{M}\), which is also equal to \(n\). The complex subgroups were designed by using \(4,8,\) and 16 subcarrier locations for the computer simulations. Along the frequency axis, the number of complex subgroups \(K\) were selected as \(32,16,\) and \(8\), corresponding to the \(\bar{M}\) of \(4,8,\) and \(16\) subcarriers, respectively. In the proposed method, in each complex subgroup, one carrier was set to inactive to convey the \(p_{0}\) bits to the index of this null subcarrier. The total number of bits conveyed by the indices of the null subcarriers was obtained by \(K(log_{2}\bar{M})\). \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Parameter Value** \\ \hline FFT Size & 128 \\ Number of used subcarrier locations & 128 \\ Cyclic Prefix Length & 32 \\ Channel type & Rayleigh fading + AWGN \\ Number of channel taps & 20 \\ Complex subgroup sizes & 4, 8, 16 \\ Complex modulations & BPSK, 4-QAM, 8-PSK \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The simulation parameters. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Sorting Algorithms** & \multicolumn{3}{c}{**Time Complexity**} & \multicolumn{1}{c}{**Space**} \\  & **Best Case** & **Average Case** & **Worst Case** & **Complexity** \\ \hline Quick & \(\mathcal{O}(n\log n)\) & \(\mathcal{O}(n\log n)\) & \(\mathcal{O}(n^{2})\) & \(\mathcal{O}(n\log n)\) \\ Merge & \(\mathcal{O}(n\log n)\) & \(\mathcal{O}(n\log n)\) & \(\mathcal{O}(n\log n)\) & \(\mathcal{O}(n)\) \\ Heap & \(\mathcal{O}(n\log n)\) & \(\mathcal{O}(n\log n)\) & \(\mathcal{O}(n\log n)\) & \(\mathcal{O}(1)\) \\ Insertion & \(\mathcal{O}(n)\) & \(\mathcal{O}(n^{2})\) & \(\mathcal{O}(n^{2})\) & \(\mathcal{O}(1)\) \\ Bubble & \(\mathcal{O}(n)\) & \(\mathcal{O}(n^{2})\) & \(\mathcal{O}(n^{2})\) & \(\mathcal{O}(1)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Computational complexities of commonly used sorting algorithms. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**OFDM-IM**} & \multicolumn{2}{c}{**NSC-OFDM-IM**} \\
**Operations** & **Time** & **Space** & **Time** & **Space** \\  & **Complexity** & **Complexity** & **Complexity** & **Complexity** \\ \hline Combinadic Unranking (Tx) & \(\mathcal{O}(n)\) & \(\mathcal{O}(nk)\) & \(-\) & \(-\) \\ Combinadic Ranking (Rx) & \(\mathcal{O}(n)\) & \(\mathcal{O}(k^{2})\) & \(-\) & \(-\) \\ Sort LLR values (Quick-Rx) & \(\mathcal{O}(n\log n)\) & \(\mathcal{O}(n\log n)\) & \(-\) & \(-\) \\ Find min value index & \(-\) & \(-\) & \(\mathcal{O}(n)\) & \(\mathcal{O}(1)\) \\ (Equation (21)) & \(-\) & \(-\) & \(\mathcal{O}(n)\) & \(\mathcal{O}(1)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Complexity comparisons between NSC-OFDM-IM and OFDM-IM schemes.