# Precoded Generalized Spatial Modulation for Downlink MIMO Transmissions in Beyond 5G Networks
## 3 Transmitter and Receiver Structure
### Transmitter Design  
Channel state information at the transmitter (CSIT) will be used to pre-process the symbols through a linear precoder \(\mathbf{F}=[\mathbf{F}_{0},\ldots,\mathbf{F}_{N_{a}-1}]\), where \(\mathbf{F}_{k}\in\mathbb{C}^{N_{a}\times N_{s}}\). Considering that the transmitted signal propagates through a flat fading channel, the baseband signal received by user k can be written as  
\[\mathbf{y}_{k}=\mathbf{H}_{k}\mathbf{x}_{k}+\mathbf{H}_{k}\sum_{\begin{array}[ ]{c}j=0\\ j\
eq k\end{array}}^{N_{a}-1}\mathbf{x}_{j}+\mathbf{n}_{k}. \tag{3}\]  
where  
\[\mathbf{x}=\sum_{k=0}^{N_{a}-1}\mathbf{x}_{k}=\sum_{k=0}^{N_{a}-1}\mathbf{F}_ {k}\mathbf{s}_{k}. \tag{4}\]  
In this expression, \(\mathbf{H}_{k}\in\mathbb{C}^{N_{n}\times N_{tr}}\) corresponds to the channel matrix for the link between the BS and user \(k\) and \(\mathbf{n}_{k}\in\mathbb{C}^{N_{rx}\times 1}\) is the noise vector with samples taken according to a zero-mean circularly symmetric Gaussian distribution with covariance \(2\sigma^{2}\mathbf{I}_{N_{rx}}\). The first term in (4) is related to the desired signal and the second one is the interference caused by the other users' signals. Moreover, the multiuser interference can be eliminated by using a BD method as proposed in [9]. Following this approach, the equivalent overall channel matrix \(\mathbf{HF}\), with \(\mathbf{H}=\left[\mathbf{H}_{0}{}^{T},\ldots,\mathbf{H}_{N_{a}-1}{}^{T}\right] ^{T}\), will become block diagonal. A simple BD precoder without any power loading optimization is assumed in this paper, with each precoder matrix \(\mathbf{F}_{k}\) designed so as to enforce that \(\mathbf{H}_{i}\mathbf{F}_{k}=0\) for all \(i\
eq k\). This particular condition can be satisfied using vectors selected from the null space of matrix \(\mathbf{\hat{H}}_{k}\), which is defined as  
\[\mathbf{\hat{H}}_{k}=\big{[}\mathbf{H}_{0}{}^{T},\ldots,\mathbf{H}_{k-1}{}^{ T},\mathbf{H}_{k+1}{}^{T},\ldots,\mathbf{H}_{N_{a}-1}{}^{T}\big{]}^{T}. \tag{5}\]  
\(\mathbf{\hat{H}}_{k}\) corresponds to the concatenation of the channel matrices between the BS and all users except user \(k\). An orthonormal basis for the null space of \(\mathbf{H}_{k}\) can be found by computing its singular value decomposition (SVD) as  
\[\mathbf{\hat{H}}_{k}=\mathbf{\hat{U}}_{k}\mathbf{\hat{\Lambda}}_{k}\big{[} \mathbf{\hat{V}}_{k}{}^{(1)}\mathbf{\hat{V}}_{k}{}^{(0)}\big{]}^{H}, \tag{6}\]  
where \(\mathbf{\hat{U}}_{k}\) is the matrix with the left-singular vectors and \(\mathbf{\hat{\Lambda}}_{k}\) is a rectangular diagonal matrix containing the nonzero singular values. \(\mathbf{\hat{V}}_{k}{}^{(1)}\) and \(\mathbf{\hat{V}}_{k}{}^{(0)}\) contain the right singular vectors corresponding to the nonzero singular values and the null singular values, respectively. \(\mathbf{F}_{k}\) is obtained from \(\mathbf{\hat{V}}_{k}{}^{(0)}\) by selecting its first \(N_{s}\) columns. In this case, the signal arriving at each receiver reduces to  
\[\mathbf{y}_{k}=\mathbf{\hat{H}}_{k}\mathbf{s}_{k}+\mathbf{n}_{k}, \tag{7}\]where \(\hat{\mathbf{H}}_{k}=\mathbf{H}_{k}\mathbf{F}_{k}\) is the equivalent single user channel seen by the receiver.