## 2 System Model and Signal Design
### Bit and Subcarrier Allocation  
The main principle of OFDM-IM is to selectively activate some OFDM subcarriers according to the bit stream information and only carry out traditional constellation modulation on the activated subcarriers while the unactivated subcarriers are directly set to zero. After the receiver demodulates the OFDM signal, it can obtain the constellation information on the activated subcarrier, as well as obtain additional index information according to the index position of the activated subcarrier [41]. It is worth noting that the IM equivalently increases the subcarrier frequency spacing, resulting in better ICI resistance.  
The allocations of bits and subcarriers are as follows. Assume that the number of bits to be transmitted by the transmitter is \(N\) and the bits carried by each antenna are \(N_{1}=N/N_{1}\). In addition, to decrease mutual interference between subcarriers, we divide all subcarriers into \(I\) subblocks. It is assumed that each antenna has \(N_{c}\) available subcarriers, and then each subblock contains \(N_{c\_sub}=N_{c}/I\) subcarriers and \(N_{2}=N_{1}/I\) bits.  
If \(K\) activations are selected from each subblock, there are a total of \(C_{N_{c\_sub}}^{K}\) combinations of activated subcarriers, and \(C\) is the number of combinations. Therefore, each subblock can transmit \(p_{1}=\left\lfloor\log_{2}C_{N_{c\_sub}}^{K}\right\rfloor\) bits of index information, where \(\lfloor\cdot\rfloor\) means rounding down. Meanwhile, the indexes of \(K\) activated subcarriers are stored in matrix \(\mathbf{A}\), and the rest are saved in matrix \(\mathbf{B}\). In addition, the constellation modulation of \(M\) points is performed on the activated subcarriers so that the number of bits of modulation information in each subblock is \(p_{2}=K\log_{2}M\). Suppose that the number of OFDM symbols contained in each pulse signal is \(N_{s}\), and the modulation information on the \(k\)th subcarrier in the \(m\)th OFDM symbol is denoted as \(d_{M}(m,k),m=1,2,\cdots,N_{s}\); then, the data symbol on the \(k\)th subcarrier, \(m\)th OFDM symbol, \(i\)th subblock, and \(g\)th antenna can be formulated as  
\[d_{g}^{i}(m,k)=\left\{\begin{array}{cc}d_{M}(m,k)&k\in\mathbf{A}\\ 0&k\in\mathbf{B}\end{array}\right. \tag{1}\]  
Figure 2: Schematic diagram of MIMO-OFDM-IM integrated signal design in transmitter.  
The symbol matrix generated by the \(i\)th subblock of \(g\)th antennas can be written as  
\[\mathbf{d}_{g}^{i}=\left[\begin{array}{cccc}d_{g}^{i}(1,1)&d_{g}^{i}(1,2)&\cdots&d_ {g}^{i}(1,N_{c\_sub})\\ d_{g}^{i}(2,1)&d_{g}^{i}(2,2)&\cdots&d_{g}^{i}(2,N_{c\_sub})\\ \vdots&\vdots&\ddots&\vdots\\ d_{g}^{i}(N_{s},1)&d_{g}^{i}(N_{s},2)&\cdots&d_{g}^{i}(N_{s},N_{c\_sub})\\ \end{array}\right] \tag{2}\]  
where \(i=1,2,\cdots,I\) and \(g=1,2,\cdots,N_{t}\). Each subblock is processed in the same way as described above, and then all the subblocks are concatenated to obtain the OFDM-IM signal block for each antenna, and the obtained bit data matrix on the \(g\)th antenna can be represented as  
\[\mathbf{d}_{g}=\left[\mathbf{d}_{g^{\prime}}^{1}\mathbf{d}_{g^{\prime}}^{2}\cdots,\mathbf{d}_ {g}^{t}\right],g=1,2,\cdots,N_{t} \tag{3}\]  
Assume that the duration of OFDM symbols is \(T_{b}\). The complex envelope of a pulse signal transmitted on the \(g\)th antenna [15] can be written as  
\[s_{g}(t)=\sum_{m=0}^{N_{c}-1}\sum_{k=0}^{N_{c}-1}\mathbf{d}_{g}(m,k)\mathrm{e}^{j2 \pi k\Delta f(t-mT_{b})}\;\mathrm{rect}\bigg{(}\frac{t-mT_{b}}{T_{b}}\bigg{)} \tag{4}\]  
where \(\Delta f=1/T_{b}\) is the frequency interval of neighboring subcarriers, \(\mathbf{d}_{g}(m,k)\) is the element of the \(m\)th row and \(k\)th column of \(\mathbf{d}_{g}\), and \(\mathrm{rect}(\cdot)\) represents the rectangular window function defined on [0,1].