## 2 System Model MIMO-OFDM-IM  
This article considers a MIMO-OFDM-IM system equipped with \(N_{t}\) transmitting antennas, \(N_{t}\) receiving antennas, and a transmission of \(N\) samples during a block period of \(T_{s}\). The block diagram of the MIMO-OFDM-IM transmitter is depicted in Figure 1. Each system frame is composed of \(mN_{t}\) data bits. These bits are divided into \(N_{t}\) groups for each of the transmitting antennas, each of which contains \(m\) bits that will be modulated by the OFDM-IM technique for transmission by the \(N_{t}\) transmitting antennas. The \(m\) bits are further divided into \(N_{B}\) groups, each with \(b\) bits; hence, \(m=N_{B}b\). The OFDM-IM symbol consists of \(N=N_{d}+N_{p}+N_{g}\) subcarriers, where \(N_{d},N_{p},N_{g}\) are the carriers assigned to data, pilots, and guards, respectively. IM is performed by dividing the \(N_{d}\) subcarriers into \(N_{B}=N_{d}/N_{C}\) sub-blocks, where \(N_{C}\) is the number of subcarriers per sub-block. Carrier index modulation (IM) transmits a different symbol inspired by the technique known as QSM and considers the \(N_{C}\) subcarriers within a sub-block as transmitting antennas. This allows for the transmission of \(b=2\log_{2}(N_{C})+\log_{2}(M)\) bits for each sub-block of OFDM-IM symbol, where \(M\) represents the size of the constellation \(\Omega\) used with cardinality equal to \(|\Omega|=2^{M}\). OFDM-IM separates the \(b\) bits of each sub-block into two parts for different purposes: the first part with \(b_{1}=2\log_{2}(N_{C})\) bits is used to select \(K\) active subcarriers, denoted by the following set:  
\[\mathcal{I}_{t}^{\delta}=\{i_{t}^{\delta}(0),\cdots,i_{t}^{\delta}(K-1)\}, \tag{1}\]  
Figure 1: MIMO-OFDM-IM transmitter proposed. where \(g=\{0,\ldots,N_{B}-1\}\), \(i_{I}^{S}(k)\in\{0,\cdots,K-1\}\) for \(k=\{0,\cdots,N_{c}-1\}\), and the active elements of \(\mathcal{I}_{t}^{S}\) are assigned using a spatial constellation among the \(N_{c}\) subcarriers. The second part with \(b_{2}=\log_{2}(M)\) bits is mapped to a symbol \(\mathcal{D}_{t}^{S}\) using an \(M\)-ary modulation. \(\left\{\mathfrak{R}(\mathcal{D}_{t}^{S}(n))\right\}_{n\in\mathcal{I}_{t}^{S}}\) and \(\left\{\mathfrak{I}\left(\mathcal{D}_{t}^{S}(n)\right)\right\}_{n\in\mathcal{ I}_{t}^{S}}\) are transmitted by the same or different subcarriers indicated by \(\mathcal{I}_{t}^{S}\), where \(\mathcal{D}_{t}^{S}\) is extracted from a complex alphabet, and we assume that \(E\{\left|\mathcal{D}_{t}^{S}\right|^{2}\}=1\) for signal normalization. Therefore, the OFDM-IM symbol element of the \(g\)-th sub-block in the \(t\)-th transmitting antenna can be expressed as \(\mathbf{s}_{t}^{g}=[s_{t}^{g}(0)\quad s_{t}^{g}(1)\quad\cdots\quad s_{t}^{g}( N_{c}-1)]^{T}\), where:  
\[s_{t}^{g}(n)=\left\{ \begin{array}{ll}\mathfrak{R}(\mathcal{D}_{t}^{S}(n)),&n \in\mathcal{I}_{t}^{S}\\ 3\left(\mathcal{D}_{t}^{S}(n)\right),&n\in\mathcal{I}_{t}^{S}\\ 0,&\text{otherwise}\end{array} \right. \tag{2}\]  
Therefore, each sub-block of the OFDM-IM symbol contains one or two active carriers whose positions carry information through the subcarrier indices. Table 1 shows the carrier index assignment for a single sub-block with \(N_{c}=2\) using 4-QAM constellation. For example, in Table 1, for the input sequence \([0\quad 0\quad 0\quad 0]\), the real and imaginary parts of the symbol (\(\mathcal{D}=+1+j\)) are transmitted by the first carrier of the sub-block, while the second carrier remains inactive. A detailed explanation of the QSM scheme can be found in [25]. Each OFDM-IM symbol contains \(G\) OFDM-IM sub-blocks in each transmitter branch, denoted by \(\mathbf{s}_{t}\). Each OFDM-IM block is transformed to the time domain using the normalized inverse discrete Fourier transform (IDFT) matrix of size \(N\times N\) for transmission.