## 2 Preliminary Knowledge  
### OFDM with Index Modulation  
For an OFDM-IM system, information bit sequences \(X\) with length \(m\), are first split into \(G\) sub-blocks. Each sub-block \(\mathbf{X}_{g}\) contains \(p\) bits, thus \(m=pG\). Each sub-block is mapped to an OFDM sub-block of length \(n\), where \(N=nG\) is the number of available sub-carriers. OFDM-IM is not only performed by means of modulated symbols but also by the indices of sub-carriers in a modulated sub-block [4]. The illustration of an OFDM-IM system is given in Figure 1. For each sub-block, \(p_{1}=\left|\log_{2}C_{n}^{k}\right|\) bits are utilized to choose \(k\) active sub-carriers from \(n\) available sub-carriers, where \(\tilde{C}_{a}^{b}\) denotes the binomial coefficient. Here, the set of the indices of the \(k\) active sub-carriers in the \(g\)th sub-block is denoted as: \(\mathbf{I}_{g}=\{i_{g}(1),i_{g}(2),\ldots,i_{g}(k)\}\), where \(i_{g}(\gamma)\in\{1,2,\ldots,n\}\) for \(g=1,2,\ldots,G\) and \(\gamma=1,2,...,k\). The rest of the \(p_{2}=k\log_{2}M\)\((p_{1}+p_{2}=p)\) bits are modulated to \(M\)-ary constellations. Therefore, the \(g\)th set of modulated symbols can be denoted as \(\mathbf{S}_{g}=[S_{g}(1),S_{g}(2),\ldots,S_{g}(k)]\), where \(S_{g}(\gamma)\in S\) and \(S\) is the constellations. By considering \(I_{g}\) and \(\mathbf{S}_{g}\) for all \(G\) sub-blocks, an OFDM block \(\mathbf{X}\) can be written as:  
\[\mathbf{X}=[X(1),X(2),\ldots,X(N)]^{T} \tag{1}\]  
where \(N=Gn\), \(X(\alpha)\in\{0,S\}\), for \(\alpha=1,2,\ldots,N\). \(T\) denotes the transpose. The pattern for active sub-carrier selection can be completed by the look-up table method [4]. An example of such an active sub-carriers selection for an \(n=4\), \(k=2\) sub-block is given in Figure 2. It can be seen from Figure 2, that if the input information bit of the sub-block is \([110001]\), when \(M=2\), the first two bits \([11]\) are index bits, and the set of the indices is \(I=\{2,4\}\) according to the table. The remaining four bits \([0001]\) are mapped according to the constellation and carried by the second and fourth sub-carriers of the block. Therefore, OFDM-IM transmits four bits of information by utilizing only two sub-carriers, which improves the energy efficiency and inter-carrier interference (ICI) resilience. Therefore, by considering \(p_{1}\) and \(p_{2}\), we can conclude from the total bits that an OFDM-IM block is sent as:  
\[G(p_{1}+p_{2})=G(\left\lfloor\log_{2}C_{n}^{k}\right\rfloor+klog_{2}M) \tag{2}\]  
According to the transmission rule, the spectral efficiency of OFDM-IM is [21]:  
\[R_{ofdm-im}=\frac{(p_{1}+p_{2})}{n}=G(\frac{\left\lfloor\log_{2}C_{n}^{k} \right\rfloor+klog_{2}M}{N}) \tag{3}\]  
After, an inverse discrete Fourier transform (IDFT) block is applied to the baseband-equivalent OFDM-IM symbols, which can be written as:  
\[\mathbf{x}=\mathbf{F}^{H}\mathbf{X} \tag{4}\]  
where \(H\) represents the Hermitian transpose, \(\mathbf{F}\) denotes the unitary DFT matrix. Then, the cyclic prefix (CP) is appended, and parallel to serial (P/S) and digital to analog (D/A) conversions are followed. Figure 1: Sub-block grouping of an OFDM-IM system. Figure 2: Mapping rules of OFDM-IM when \(n=4\), \(k=2\). At the receiving side, after the removal of CP, a DFT operation needs to be implemented as the conventional OFDM. However, unlike the conventional OFDM, the OFDM-IM receiver needs to detect both the indices of active sub-carriers and the corresponding modulated symbols. For each sub-block, by considering a joint detection for the indices of active sub-carriers and the transmitted symbols carried on, the maximum likelihood (ML) detector for OFDM-IM is given by:  
\[\{\hat{S}_{g}(\gamma)\}_{\gamma=1}^{n}=\underset{\{S_{g}(\gamma)\}_{\gamma=1}^{n }}{\text{argmin}}\sum_{\gamma=1}^{n}|Y_{g}(\gamma)-H_{g}(\gamma)S_{g}(\gamma)|^ {2} \tag{5}\]  
where \(H_{g}(\gamma)\) denotes the channel frequency response (CFR) on the \(\gamma\)th sub-carrier of the \(g\)th sub-block.