## 3 Transmitter and Receiver Structure
### Precoder Design  
Assuming the time domain samples have been converted to the frequency domain through an \(N\)-point fast Fourier transform (FFT) and the CP has been discarded, the \(g\)-th PT-GSFIM symbol received by the user can be written as  
\[\mathbf{y}_{g}^{u}=\mathbf{H}_{g}^{u}\mathbf{x}_{g}+\mathbf{n}_{g}^{u}=\mathbf{ H}_{g}^{u}\mathbf{F}_{u}\mathbf{s}_{g}^{u}+\mathbf{H}_{g}^{u}\sum_{ \begin{array}{c} j=1\\ j\
eq u\end{array} }^{N_{u}}\mathbf{F}_{g}\mathbf{s}_{g}^{j}+\mathbf{n}_{g}^{u} \tag{3}\]  
Figure 3: Indoor office scenario layout. where \(\mathbf{y}_{g}^{u}\in\mathbb{C}^{N_{rx}N_{y}\times 1}\), \(\mathbf{n}_{g}^{u}\in\mathbb{C}^{N_{rx}N_{f}\times 1}\) represents the noise and \(\mathbf{H}_{g}^{u}\in\mathbb{C}^{N_{rx}N_{x}N_{x}N_{f}}\) is the frequency domain channel matrix between the BS and the user \(u\). Since we are considering an OFDM scheme, the channel matrix follows a block diagonal structure and can be written as  
\[\mathbf{H}_{g}^{u}=\text{blkdiag}\Big{\{}\mathbf{H}_{g}^{u,1},\cdots,\mathbf{ H}_{g}^{u,f},\cdots,\mathbf{H}_{g}^{u,N_{f}}\Big{\}} \tag{4}\]  
with \(\mathbf{H}_{g}^{u,f}\in\mathbb{C}^{N_{rx}\times N_{rx}}\), \(f=1,\cdots,N_{f}\). As seen on the right side of (3), the second term represents the interference between users. Using the approach of block diagonalization (BD) described in [38] for the design of the precoder matrices \(\mathbf{F}_{u}\), it is possible to eliminate all MUI from all the receivers. It is important to note that the signals conveyed on each subcarrier only generate interference on that subcarrier due to the block diagonal structure of \(\mathbf{H}_{g}^{u}\). This means that the precoder matrices \(\mathbf{F}_{u}\) for each subcarrier can be built independently and follow a block diagonal structure; i.e.,  
\[\mathbf{F}_{u}=\text{blkdiag}\Big{\{}\mathbf{F}_{u,1},\cdots,\mathbf{F}_{u,N_{ f}}\Big{\}} \tag{5}\]  
with \(\mathbf{F}_{u,f}\in\mathbb{C}^{N_{tx}\times N_{t}}\), \(f=1,\cdots,N_{f}\). Each of the component precoder matrices \(\mathbf{F}_{u,f}\) is generated so as to enforce \(\mathbf{H}_{g}^{v,f}\mathbf{F}_{u,f}=0\) for all \(v\
eq u\), canceling all MUI. Let us define the matrix that concatenates all the channel matrices between the base station and all users, except user \(u\), for the subcarrier \(f\) as  
\[\mathbf{\tilde{H}}_{g}^{u,f}=\bigg{[}\Big{(}\mathbf{H}_{g}^{1,f}\Big{)}^{T} \cdots\Big{(}\mathbf{H}_{g}^{u-1,f}\Big{)}^{T}\Big{(}\mathbf{H}_{g}^{u+1,f} \Big{)}^{T}\cdots\Big{(}\mathbf{H}_{g}^{N_{u,f}}\Big{)}^{T}\bigg{]}^{T} \tag{6}\]  
This matrix models the transmission of the signal targeted at the \(u\)-th user when it reaches all of the other receivers, which causes interference, thus corresponding to the interference channel matrix. The appropriate precoder matrix \(\mathbf{F}_{u,f}\) can be designed using an orthonormal basis of the null space of \(\tilde{\mathbf{H}}_{g}^{u,f}\), starting from the singular value decomposition (SVD) of \(\mathbf{H}_{g}^{u,f}\), which is given by  
\[\tilde{\mathbf{H}}_{g}^{u,f}=\mathbf{\tilde{U}}_{u,f}\tilde{\mathbf{\Lambda}}_{u,f}\bigg{[}\tilde{\mathbf{V}}_{u,f}^{(1)}\tilde{\mathbf{V}}_{u,f}^{(0)}\bigg{]} ^{H} \tag{7}\]  
where \(\mathbf{\tilde{U}}_{u,f}\) is the matrix with the left-singular vectors, \(\mathbf{\tilde{\Lambda}}_{u,f}\) is a rectangular diagonal matrix with the decreasing nonzero singular values, \(\tilde{\mathbf{V}}_{u,f}^{(1)}\) contains the right singular vectors corresponding to the nonzero singular values and \(\mathbf{\tilde{V}}_{u,f}^{(0)}\) contains the remainder right singular vectors. To guarantee zero MUI in each precoder matrix, each precoder can be set as \(\mathbf{F}_{u,f}=\tilde{\mathbf{V}}_{u,f}^{(0)}[,1:N_{s}]\). In this case, at each receiver, the resulting signal is reduced to  
\[\mathbf{y}_{g}^{u}=\tilde{\mathbf{H}}_{g}^{u}\mathbf{s}_{g}^{u}+\mathbf{n}_{ g}^{u} \tag{8}\]  
where \(\tilde{\mathbf{H}}_{g}^{u}=\mathbf{H}_{g}^{u}\mathbf{F}_{u}\) is the equivalent SU channel. It can be observed that there is no presence of MUI in the resulting signal.