# Null Subcarrier Index Modulation in OFDM Systems for 6G and Beyond
## 4 Proposed NSC-OFDM-IM  
The OFDM symbol in the frequency domain is grouped into a number of sub-bands in Figure 3, while each has the same number of subcarrier indices. The size of the frequency sub-bands is determined as a power of two, which can also be equal to the order of the constellation mapper in the system. In the proposed scheme, one carrier is set to the inactive state within each sub-band group based on the incoming bit stream, and the others are used as the active subcarriers, which are modulated by the complex data after the conversion of the binary data at the output of the constellation mapper. The transmitter of the proposed NSC-OFDM-IM scheme is illustrated in Figure 4. The bit stream is first split into \(b_{k}\) main groups, and then, each \(b_{k}\) bit group is further divided into two subgroups with a different number of bits. The first subgroup contains the number of \(p_{0}\) bits as per the order of the index mapper, which is predefined as a power of two based on the size of the frequency domain sub-band groups. These \(p_{0}\) bits are used to modulate the zero subcarrier index within the OFDM frequency sub-band. The second bit subgroup is designed to have \(p_{1}\) bits to be used for the modulation of the active subcarriers in the complex subgroup, as in a conventional OFDM system. Figure 3: The subgroups in the frequency domain. The number of \(p_{0}\) and \(p_{1}\) bits, which are split from the \(b_{\kappa}\) main group, are then given by,  
\[ \begin{split} p_{0}&=log_{2}\widetilde{M}\\ p_{1}&=(\widetilde{M}-1)log_{2}M\end{split}  \tag{9}\]  
where \(\widetilde{M}\) denotes the size of complex sub-band group as a power of two, which is \(2^{p_{0}}\) in this case, and \(M\) is the modulation order by means of the constellation mapper. \(\widetilde{M}\) can also be called the null subcarrier index mapper order (IMO) since the number of \(p_{0}\) bits is conveyed to the inactive carrier index number. The number of bits in one of \(b_{\kappa}\) main bit groups is then defined as,  
\[ \begin{split} b_{1}&=p_{0}+p_{1}\\ &=log_{2}\widetilde{M}+(\widetilde{M}-1)log_{2}M\end{split}  \tag{10}\]  
where \(b_{1}\) indicates the first bit group of the main bit stream. The transmitted bits in one NSC-OFDM-IM symbol consist of all \(b_{\kappa}\) bit groups such that:  
\[b_{\kappa}=\{b_{1},b_{2},\ldots,b_{K}\} \tag{11}\]  
Then, the total number of \(m\) bits carried by the null subcarrier indices and active subcarrier phases in the NSC-OFDM-IM is given by:  
\[m=K(p_{0}+p_{1}) \tag{12}\]  
where \(K\) denotes the total number of \(b_{\kappa}\) main bit groups, which is also equal to the number of frequency domain complex sub-band groups. The subcarrier indices in the complex group are given by,  
\[X=\{X_{1},X_{2},\ldots,X_{\widetilde{M}}\} \tag{13}\]  
Figure 4: The proposed NSC-OFDM-IM transmitter. The index of the null subcarrier within the complex sub-band group is easily selected by the index mapper, which converts the binary \(p_{0}\) data to the decimal value. Based on the decimal value of the \(p_{0}\) bit sequence, the index mapper then finds the corresponding carrier location that is equal to one more than the decimal value of \(p_{0}\) bits as defined in Table 1. Upon finding the frequency location in the complex subgroup, the subcarrier in this group is set to an inactive state. In other words, there is no active carrier generated at this frequency, and hence, it will be a null subcarrier in the NSC-OFDM-IM symbol. The remaining subcarriers will be set as the active carriers and modulated by the complex dataset, which is generated by the constellation mapper, as in the classical OFDM systems. The corresponding NSC-OFDM-IM symbol in the frequency domain is then constructed by the complex subgroups. The frequency domain representation of the available bandwidth for the proposed scheme is given by:  
\[X=\{X(0),X(1),X(2),\ldots,X(N-1)\} \tag{14}\]  
where \(N\) denotes the size of the IFFT/FFT, which can also be called \(N_{FFT}\), in the system.