# Multiuser Transmit Precoding Design for Dimming Compatible Visible Light Communications
## 2 System Model and Optical Power Constraints
### Optical Constraints  
In VLC, LEDs are the main source of non-linearity. Due to the LED non-linearity, the signal transmitted by the \(i^{\text{th}}\) LED is constrained to a limited linear dynamic range, i.e., \(d_{L,i}\leq x_{i}\leq d_{H,i}\), where \(d_{L,i}\) and \(d_{H,i}\) denote the minimum and maximum drive current permitted by the \(i^{\text{th}}\) LED, respectively [22]. Next, we will derive the specific optical constraints imposed on the design to ensure that the signals transmitted by LEDs operate within their limited linear dynamic range. Let us first define \(\Delta_{j}=\frac{\Delta_{H,i}-\Delta_{L,i}}{2}\) and \(c_{j}=\frac{\Delta_{H,i}+\Delta_{L,i}}{2}\). Since we have:  
\[|s_{j}-c_{j}|\leq\Delta_{j}, \tag{6}\]  
where \(|\cdot|\) denotes the absolute value operator, it follows that:  
\[|\sum_{j=1}^{N_{t}}p_{ij}(s_{j}-c_{j})|\leq\sum_{j=1}^{N_{t}}|p_{ij}||s_{j}-c _{j}|\leq\sum_{j=1}^{N_{t}}|p_{ij}|\Delta_{j}. \tag{7}\]  
Equivalently, it yields:  
\[-\sum_{j=1}^{N_{t}}|p_{ij}|\Delta_{j}\leq\sum_{j=1}^{N_{t}}p_{ij}(s_{j}-c_{j} )\leq\sum_{j=1}^{N_{t}}|p_{ij}|\Delta_{j}. \tag{8}\]By rearranging the above inequalities and introducing \(d_{i}\) in the inequalities, we have:  
\[-\sum_{j=1}^{N_{r}}|p_{ij}|\Delta_{j}+\sum_{j=1}^{N_{r}}p_{ij}c_{j}+d_{i}\leq\sum _{j=1}^{N_{r}}p_{ij}s_{j}+d_{i}\leq\sum_{j=1}^{N_{r}}|p_{ij}|\Delta_{j}+\sum_{j =1}^{N_{r}}p_{ij}c_{j}+d_{i}. \tag{9}\]  
Based on the expression of \(x_{i}\) in (3), we can formulate the dynamic range of \(x_{i}\) in the following matrix form:  
\[-\text{abs}\{\textbf{e}_{i}^{T}\textbf{P}\}\textbf{\Delta}+\textbf{e}_{i}^{T} \textbf{P}\textbf{c}+d_{i}\leq x_{i}\leq\text{abs}\{\textbf{e}_{i}^{T}\textbf{ P}\}\textbf{\Delta}+\textbf{e}_{i}^{T}\textbf{P}\textbf{c}+d_{i}, \tag{10}\]  
where the above formulation stems from the fact that:  
\[\left\{\begin{array}{l}\text{abs}\{\textbf{e}_{i}^{T}\textbf{P}\}\textbf{ \Delta}=\sum_{j=1}^{N_{r}}|p_{ij}|\Delta_{j},\\ \textbf{e}_{i}^{T}\textbf{P}\textbf{c}=\sum_{j=1}^{N_{r}}p_{ij}c_{j},\\ \textbf{e}_{i}=\left[\textbf{0}_{1\times(i-1)}\quad 1\quad\textbf{0}_{1 \times(N_{r}-i-1)}\right]^{T}.\end{array}\right. \tag{11}\]  
Here, \(\textbf{e}_{i}\) denotes the \(i^{\text{th}}\) standard basis vector for the \(N_{r}\)-dimensional space, \((\cdot)^{T}\) represents the transpose operator, \(\text{abs}\{\cdot\}\) is the element-wise absolute value operator, \(\textbf{0}_{1\times n}\) is an all-zero row-vector of size \(n\), \(\textbf{\Delta}=[\Delta_{1},\cdots,\Delta_{N_{r}}]^{T}\), and \(\textbf{c}=[c_{1},\cdots,c_{N_{r}}]^{T}\). To ensure that \(d_{L,i}\leq x_{i}\leq d_{H,i}\), \(\textbf{p}_{i}\) and \(d_{i}\) should satisfy:  
\[\left\{\begin{array}{l}-\text{abs}\{\textbf{e}_{i}^{T}\textbf{P}\}\textbf{ \Delta}+\textbf{e}_{i}^{T}\textbf{P}\textbf{c}+d_{i}\geq d_{L,i},\\ \text{abs}\{\textbf{e}_{i}^{T}\textbf{P}\}\textbf{\Delta}+\textbf{e}_{i}^{T} \textbf{P}\textbf{c}+d_{i}\leq d_{H,i}.\end{array}\right. \tag{12}\]  
Consequently, in order to guarantee that all LEDs operate within their limited linear dynamic range, the following optical constraints are imposed:  
\[\left\{\begin{array}{l}\text{abs}\{\textbf{P}\}\textbf{\Delta}-\textbf{P} \textbf{c}\leq\textbf{d}-\textbf{d}_{L}\\ \text{abs}\{\textbf{P}\}\textbf{\Delta}+\textbf{P}\textbf{c}\leq\textbf{d}_{H }-\textbf{d}\end{array}\right., \tag{13}\]  
where \(\textbf{d}_{L}=[d_{L,1},\cdots,d_{L,N_{l}}]^{T}\) and \(\textbf{d}_{H}=[d_{H,1},\cdots,d_{H,N_{l}}]^{T}\).