# Null Subcarrier Index Modulation in OFDM Systems for 6G and Beyond
## 3 Index Modulation for OFDM Systems  
The transmission of information through indices was originally carried out by using the spatial modulation (SM) technique [22; 23]. The SM method was then utilized in OFDM systems by allowing it to transmit additional data bits by modulating the indices of the active subcarriers [24]. However, in this method, the number of active subcarriers changes between the consecutive OFDM symbols. Thus, in order to make a suitable design of the active subcarriers and enhance the performance of the previous approaches, a new technique called OFDM-IM was introduced in [7]. The primary concept of OFDM-IM is to use the active subcarrier indices for additional data transmission besides modulating the same active subcarriers with complex data [25; 26]. The basic OFDM-IM transmitter diagram is given in Figure 2.  
The incoming bit stream is divided into a number of \(K\) groups, each containing a predetermined number of \(q\) bits according to the complex modulation order used in the system and the number of used \(k\) active subcarriers in the frequency domain complex subgroups. The total number of complex subgroups in the frequency domain is also equal to \(K\) such that \(K=N/n\), where \(N\) is the number of total subcarrier locations in the frequency axis, regardless of whether they are active or inactive for the complex and index modulations, and \(n\) denotes the size of the complex subgroups. Then, each group of bits is divided into two separate subgroups within itself, containing \(d_{0}\) and \(d_{1}\) bits. In other words, one subgroup with \(d_{0}\) bits is used to modulate the indices of the active subcarriers, while the other \(d_{1}\) bits are converted into complex data via the constellation mapper and used to modulate the active subcarriers as in the classical OFDM method. The number of \(d_{0}\) bits is determined by the size of the complex subgroup and the number of active subcarriers within the complex subgroup. In other words, the number of bits that come with the \(d_{0}\) bit sequence is determined by the combinational selection number \((C(n,k))\), where all possible selections of \(k\) active subcarriers in the subgroup are obtained [27; 28]. The \(d_{0}\) bits, which are used for selecting and modulating the indices of the active subcarriers, can be represented as:  
\[d_{0}=\lfloor log_{2}(C(n,k))\rfloor \tag{5}\]  
The \(d_{1}\) bits for the number of \(k\) activated subcarriers are determined according to the order of the constellation mapper. The number of \(d_{1}\) bits is obtained as follows:  
\[d_{1}=klog_{2}(M) \tag{6}\]  
so the number of \(q\) bits per subgroup, which are conveyed via the indices of the active subcarriers and mapped to the complex data for the modulation of the phases of the active subcarriers, can be represented as:  
\[q=d_{0}+d_{1} \tag{7}\]  
Figure 2: OFDM-IM system transmitter. then the total number of bits transmitted over one symbol is given by:  
\[ \begin{split} m&=K(d_{0}+d_{1})\\ &=\frac{N}{n}((d_{0}+d_{1}))\end{split}  \tag{8}\]  
Each complex subgroup is designed to have the number of \(n\) subcarrier locations; \(q\) is the number of total bits to be transmitted by the subcarriers in one complex subgroup. The index selector at the transmitter uses an unranking combination algorithm [29] to be able to produce the corresponding \(k\)-size subset vector from the \((^{n}_{k})\) combinations. By using the vector elements, the index selector specifies the number of active subcarriers in the complex subgroup, where \(k\) of \(n\) subcarriers are activated [30]. Hence, the active subcarriers are the source of the information in the subgroup and transmit the data via their indices and phases [31]. At the receiver, the signal is analyzed in the frequency domain to make the detection of the transmitted bits. The data are extracted from the indices and phases of the active subcarriers. In order to obtain the indices of active subcarriers, the maximum likelihood (ML) or LLR detection methods are employed for the OFDM-IM system [32; 33].