# A Symmetric Matrix-Aided MIMO to Improve Reliability for Maritime Visible Light Communications
## 3 Symmetric Matrix-Aided MIMO-VLC System
### MIMO Multiuser Transmitter  
In the maritime MIMO-VLC system, from Figure 1, let \(N_{t}\) denote the number of transmitting apertures in the transmitter, these transmit apertures will transmit signals to \(K\) users. In this paper, let the user \(j\) as an example, the user \(j\) has \(N_{r,j}\) receiving apertures, the number of receiving apertures is \(N_{r}=\sum_{j=1}^{j=K}N_{r,j}\).  
Let \(\mathbf{L}_{j}\) represent the precoding matrix from the double singular value decomposition (SVD) operations. The objective of the double SVD is to generate the precoding matrix to help eliminate the multi-user interferences (MUls) using the subchannels decomposed by SVD orthogonal to each other. Then we use \(\mathbf{x}_{j}\) to denote the signal of the user \(j\) encoded by a symmetric matrix. After the signal \(\mathbf{x}_{j}\) has been precoded, the signal \(\widetilde{\mathbf{x}}_{j}\) is obtained by \(\widetilde{\mathbf{x}}_{j}=\mathbf{L}_{j}\mathbf{x}_{j}\).  
To derive the precoding matrix \(\mathbf{L}_{j}\), we first apply SVD to the \(((N_{r}-N_{r,j})\times N_{t})\) complementary channel matrix \(\mathbf{\widetilde{H}}_{j}\)[15]:  
\[\mathbf{\widetilde{H}}_{j}=\mathbf{\widetilde{U}}_{j}\mathbf{\widetilde{\Lambda }}_{j}[\mathbf{\widetilde{V}}_{j}^{(1)}|\mathbf{\widetilde{V}}_{j}^{(0)}]^{ \mathsf{T}}, \tag{7}\]  
where the \(((N_{r}-N_{r,j})\times(N_{r}-N_{r,j}))\) matrix \(\mathbf{\widetilde{U}}_{j}\) has the left singular vectors, and the \(((N_{r}-N_{r,j})\times N_{t})\) matrix \(\mathbf{\widetilde{\Lambda}}_{j}\) is the diagonal matrix which is formed by ordered singular values. Let \(\widetilde{P}_{j}\) denote the rank of the \(\mathbf{\widetilde{H}}_{j}\), then the \((N_{t}\times\widetilde{P}_{j})\) matrix \(\mathbf{\widetilde{V}}_{j}^{(1)}\) holds the first \(\widetilde{P}_{j}\) right singular vectors. In addition, the \((N_{t}\times(N_{t}-\widetilde{P}_{j}))\) matrix \(\mathbf{\widetilde{V}}_{j}^{(0)}\) forms the orthogonal basis for the null space of the \(\mathbf{\widetilde{H}}_{j}\). When the channel matrix has row full rank and the \(N_{t}=N_{r}\), \(N_{t}-\widetilde{P}_{j}\) is equal to the \(N_{r,j}\)[15]. Then we use the \(\mathbf{\widetilde{V}}_{j}^{(0)}\) obtained from Equation (7) to multiply the channel matrix \(\mathbf{H}_{j}\) to generate the equivalent channel matrix \(\mathbf{\overline{H}}_{j}\):  
\[\mathbf{\overline{H}}_{j}=\mathbf{H}_{j}\mathbf{\widetilde{V}}_{j}^{(0)}. \tag{8}\]  
Then we conduct the second SVD operation on Equation (8) [15]:  
\[\mathbf{\overline{H}}_{j}=\mathbf{H}_{j}\mathbf{\widetilde{V}}_{j}^{(0)}= \mathbf{U}_{j}\mathbf{\Lambda}_{j}\mathbf{V}_{j}^{\mathsf{T}}, \tag{9}\]  
where the \((N_{r,j}\times N_{r,j})\) matrix \(\mathbf{\Lambda}\) is a diagonal matrix that contains the singular values of the equivalent channel matrix \(\mathbf{\overline{H}}_{j}\) and the \((N_{r,j}\times N_{r,j})\) matrix \(\mathbf{U}_{j}\) is used in the receiver to decode. The \((N_{r,j}\times N_{r,j})\) matrix \(\mathbf{V}_{j}\) is the right singular vector when the equivalent channel matrix \(\mathbf{\overline{H}}_{j}\) is of the full rank. Moreover, we can notice that the precoding matrix \(\mathbf{L}_{j}=\mathbf{\widetilde{V}}_{j}^{(0)}\mathbf{V}_{j}\), which falls into the null space of the \(\mathbf{\widetilde{H}}_{j}\) to eliminate the multi-user interferences (MULs) [15].  
Then the transmitted vector \(\mathbf{s}\) can obtained as [26]  
\[\mathbf{s}=\sum_{j=1}^{K}(\mathbf{\widetilde{x}}_{j})+\mathbf{d}=\sum_{j=1}^{ K}(\mathbf{L}_{j}\mathbf{x}_{j})+\mathbf{d}, \tag{10}\]  
where \(\mathbf{d}\) is the direct current (DC) bias which is added to make sure the transmitted signal is non-negative [18]. Then the signals \(\mathbf{s}\) are fed into the transmit apertures.