## 2 System Model
### Precoding-Aided Transmitter-Side Generalized Space-Frequency Index Modulation (PT-GSFIM)  
We adopt an OFDM-based MU-MIMO as basis, where part of the information is mapped to conventional \(M\)-sized APM symbols and the remaining is encoded onto spatial and frequency indices. We assume a transmission block matrix where the bits are mapped in a simple way, with the encoding of the spatial indices executed independently from the frequency indices. In this case, simple look-up tables (LUTs) are inside each subblock: one for the spatial resources and another for the frequency resources. Although some SE is sacrificed when the mapping is done independently (the number of available space-frequency blocks that are indices is normally reduced), it simplifies the detection process as well the mapping/demapping process, as we will show further below. It is important to highlight that the approach based on a pair of LUTs is only adequate when a small number of both active antenna combinations and active subcarrier combinations exists. When this number is large, a different strategy can be employed, as described in [11; 34].  
Defining a PT-GSFIM symbol as \(\mathbf{s}_{g}^{u}\in\mathbb{C}^{N_{a}N_{f}\times 1}\), for the \(u\)-th user, it can be written as \(\mathbf{s}_{g}^{u}=\left[\left(\mathbf{s}_{g}^{u,1}\right)^{T}\cdots 0\cdots \left(\mathbf{s}_{g}^{u,N_{a_{f}}}\right)^{T}\right]^{T}\), which corresponds to the concatenation of (\(N_{f}\)-\(N_{a_{f}}\)) length-\(N_{s}\) vectors of zeros with \(N_{a_{f}}\)GSM symbol vectors \(\mathbf{s}_{g}^{u,i}\in\mathbb{C}^{N_{s}\times 1}\). These symbols are defined as \(\mathbf{s}_{g}^{u,i}=\left[\begin{array}{cccccccc}0&s_{g,1}^{u,i}&0&\cdots &0&s_{g,N_{a}}^{u,i}&0\\ \end{array}\right]^{T}\), where \(i=\left\{1,\cdots,N_{a_{f}}\right\}\), \(u=\left\{1,\cdots,N_{a_{f}}\right\}\) and \(g=\left\{1,\cdots,N_{GSFIM}\right\}\) (\(N_{GSFIM}\) is the number of PT-GSFIM symbols inside an OFDM block). Each of the \(N_{GSFIM}\) symbols is composed of all available antenna and subcarrier positions, where each is activated and deactivated according to their indices. The symbols \(s_{g,i}^{u,i}\), where \(j=\left\{1,\cdots,N_{a}\right\}\), are selected from an APM constellation. Since we are considering an MU downlink situation, where precoding is applied, the expression for the transmitted signal can be written as  
\[\mathbf{x}_{g}=\sum_{u=1}^{N_{u}}\mathbf{F}_{u}\mathbf{s}_{g}^{u}=\mathbf{F} \mathbf{s}_{g} \tag{1}\]  
where \(\mathbf{s}_{g}=\left[\left(\mathbf{s}_{g}^{1}\right)^{T}\cdots\left(\mathbf{ s}_{g}^{N_{u}}\right)^{T}\right]^{T}\) and \(\mathbf{F}=\left[\mathbf{F}_{1}\cdots\mathbf{F}_{N_{u}}\right]\), with \(\mathbf{F}_{u}\in\mathbb{C}^{N_{u}N_{f}\times N_{a}N_{f}}\) representing the precoder matrix. After the precoders and interleavers (which are explained further ahead), the frequency domain symbols are concatenated into \(N\)-sized blocks and converted to time domain through an inverse fast Fourier transform (IFFT). Before the transmission occurs, a CP with length greater than the delay spread of the channel is added to each block. It is important to observe that each PT-GSFIM symbol can carry a total of bits per user calculated through  
\[N_{bits}=N_{af}\Big{(}\left|\log_{2}\binom{N_{a}}{N_{a}}\right|+N_{a}\log_{2} M\Big{)}+\left|\log_{2}\binom{N_{f}}{N_{a_{f}}}\right| \tag{2}\]  
where \(N_{s}\) is the number of available spatial positions on each subcarrier which is constrained to \(N_{s}\leq N_{tr}/N_{u}\), and \(M\) is the size of the constellation adopted for the conventional modulated symbols.