## 3 Proposed System
### Optimal Precoding Using Block Bi-Diagonalization  
The QAM-modulated signal of the indoor MIMO-VLC is given as input to the BBD for effectively mitigating the interferences during the communication over the indoor MIMO-VLC system. The desired and Inter-User Interference (IUI) users are paired together, and the computation order of the precoder is determined by user ordering. In the user ordering, the \(j+1\) user is fixed to the IUI user according to the \(j\)th desired user. From the channel matrix \(\mathbf{H}\), the \(j\) and \(j+1\) user's channel components are eliminated to define the \(\mathbf{H}_{bbd}^{j}\). Equation (12) expresses the SVD of \(\mathbf{H}_{bbd}^{j}\) identified in the proposed system. Subsequently, the simplified expressions for \(\mathbf{H}_{bbd}^{j}\) are given in Equations (13) and (14). \[\mathbf{H}_{bbd}^{j} =\left[\mathbf{H}_{1}^{t}\ldots\,\mathbf{H}_{j-1}^{t}\,\,\,\,\,\mathbf{H}_{j+2}^{ t}\ldots\,\mathbf{H}_{M}^{t}\right]^{T} \tag{12}\] \[=\mathbf{U}_{bbd}^{(j)}\mathbf{\Sigma}_{bbd}^{(j)}\mathbf{V}_{bbd}^{(j)H}\] (13) \[=\mathbf{U}_{bbd}^{(j)}\mathbf{\Sigma}_{bbd[s]}^{(j)}\mathbf{O}\left| \begin{array} []{c}\mathbf{V}_{bbd}^{(j)H}\\ \mathbf{V}_{bbd}^{(j)H}\end{array} \right. \tag{14}\]  
where, left and right singular vector matrices are denoted as \(\mathbf{U}_{bbd}^{(j)}\) and \(\mathbf{V}_{bbd}^{(j)}\), respectively, and the singular matrix is denoted as \(\mathbf{\Sigma}_{bbd}^{(j)}\). There are two different sub matrices used to specify the \(\mathbf{\Sigma}_{bbd}^{(j)}\). The two sub matrices are \(\left(M_{r}-M_{r,j},M_{r,j+1}\right)\) dimensional positive-definite square diagonal matrix \(\mathbf{\Sigma}_{bbd[s]}^{(j)}\) and zero matrix \(O\), and the unitary matrix of right singular vectors, \(\mathbf{V}_{bbd}^{(j)}\). Besides, the \(\mathbf{V}_{bbd}^{(j)}\) is separated into two sub matrices such as \(\mathbf{V}_{bbd}^{(j)}\) and \(\mathbf{V}_{bbd}^{(j)}\). The user space, except for \(j\)th and \((j+1)\)th user and zero matrix, are indicated in the submatrix \(\left(\mathbf{\Sigma}_{bbd[s]}^{(j)}\right)\), which is specified in Equation (14). Equation (15) shows the assumed channel matrix in the MIMO-VLC system. \[\mathbf{HV}_{bbd[n]}^{j}=\left[\ldots\mathbf{O}\left(\mathbf{H}_{j}\mathbf{V}_{bbd[n]}^{j} \right)^{t}\left(\mathbf{H}_{j+1}\mathbf{V}_{bbd[n]}^{j}\right)^{t}\mathbf{O}\ldots\right] ^{t} \tag{15}\]  
The suitable null steering for all users except the \(j\)th and \((j+1)\)th user is obtained by using the \(\mathbf{V}_{bbd[n]}^{j}\). For the \(j\)th user, the adequate beam space is generated by considering the channel component \(\left(\mathbf{HV}_{bbd[n]}^{j}\right)\), which is already mentioned in Equation (15). The eigenvector matrix \(\left(\mathbf{v}_{bbd[c]}^{j}\right)\) is assumed according to the singular values of \(\mathbf{H}_{j}\mathbf{V}_{bbd[n]}^{j}\) larger eigenvalues. The SNR for the \(j\)th user is enhanced by providing the required beamforming based on the multiplication between \(\mathbf{V}_{bbd[n]}^{j}\) and \(\mathbf{V}_{bbd[e]}^{j}\) from right side. Equation (16) gives the precoding matrix for the \(j\)th user, which is identified by using this BBD method. \[\mathbf{B}_{bbd,j}=\mathbf{V}_{bbd[n]}^{j}\mathbf{V}_{bbd[e]}^{j} \tag{16}\]  
Similarly, the precoding matrix \(\left(\mathbf{B}_{bbd}\right)\) for the remaining users is achieved by using the aforementioned procedure. The obtained precoding matrix is combined with the channel matrix of the transmitter to create an effective data transmission over the indoor MIMO-VLC system. The developed BBD precoding mitigates the 100-150 Hz of noise frequencies created by the thermal noise, shot noise and phase noise. The overall parameters used this indoor MIMO-VLC system are given in Table 1.  
In the MIMO-VLC system, the shot noise is generated by induced ambient light and transmitted optical signal. Additionally, the electronics noise of pre amplifier creates the thermal noise and phase noise is generated based on the random variations in the waveform phase.