# Multiuser Transmit Precoding Design for Dimming Compatible Visible Light Communications
## 2 System Model and Optical Power Constraints  
### Multiuser VLC Systems  
We consider a multiuser VLC system composed of \(N_{t}\) LED transmitters on the ceiling and \(N_{r}\) user terminals, each equipped with an photodetector (PD) on the receiver plane, as shown in Figure 1. Let \(\mathbf{s}=[s_{1},\cdots,s_{j}\cdots,s_{N_{t}}]^{T}\) denote the real finite-magnitude symbol vector of all users, where \(s_{j}\) represents the symbol transmitted to the \(j^{\text{th}}\) user. The previous works required that \(s_{j}\) be symmetrically bounded and have a zero mean, which restricts the modulation scheme to a very limited option, such as bipolar PAM, as Figure 2 depicts. This inherently limits their applications in VLC since some popular modulation schemes like PPM in Figure 2 cannot satisfy these strict requirements. In this paper, we consider a more general modulation scheme, with \(s_{j}\) bounded by:  
\[\Delta_{L,j}\leq s_{j}\leq\Delta_{H,j}, \tag{1}\]  
where \(\Delta_{L,j}\) and \(\Delta_{H,j}\) are known modulation-dependent values. Furthermore, let \(b_{j}\) and \(u_{j}\) respectively denote the mean and the average power of \(s_{j}\), i.e., \(b_{j}=\text{E}\{s_{j}\}\) and \(u_{j}=\text{E}\{s_{j}^{2}\}\).  
In order to mitigate the multiuser interference, TPC is employed by exploiting the channel state information (CSI). Then, a suitable DC bias is added for producing positive signals. Consequently, the signals transmitted by the LED arrays can be expressed as:  
\[\mathbf{x}=\mathbf{Ps}+\mathbf{d}=\sum_{j=1}^{N_{r}}\mathbf{p}_{j}s_{j}+ \mathbf{d}, \tag{2}\]  
Figure 1: Multiuser VLC systems.  
Figure 2: Pulse amplitude modulation (PAM) and pulse position modulation (PPM) waveforms.  
Here, \(\mathbf{P}\in\mathbb{R}^{N_{t}\times N_{t}}\) is the TPC matrix, \(\mathbf{p}_{j}\) is the \(j^{\text{th}}\) column of \(\mathbf{P}\), which denotes the TPC vector of the \(j^{\text{th}}\) user, and \(\mathbf{d}=[d_{1},\cdots,d_{N_{t}}]^{\text{T}}\) is the DC bias vector. Furthermore, the signal \(x_{i}\) transmitted by the \(i^{\text{th}}\) LED is given by:  
\[x_{i}=\sum_{j=1}^{N_{t}}p_{ij}s_{j}+d_{i}, \tag{3}\]  
where \(p_{ij}\) is the \((i,j)^{\text{th}}\) element of \(\mathbf{P}\).  
At the receiver, the light rays detected by the PD are converted into electrical signals. After removing the DC bias, the received signal of the \(k^{\text{th}}\) user is formulated as:  
\[y_{k}=\mathbf{h}_{k}^{\text{T}}\mathbf{P}\mathbf{s}+n_{k}=\mathbf{h}_{k}^{ \text{T}}\mathbf{p}_{k}s_{k}+\sum_{j\
eq k}\mathbf{h}_{k}^{\text{T}}\mathbf{p} _{j}s_{j}+n_{k}, \tag{4}\]  
where \(\mathbf{h}_{k}\) is the channel matrix spanning from the LED array to the \(k^{\text{th}}\) user and \(n_{k}\) is the additive white Gaussian noise (AWGN) having a variance of \(\sigma_{k}^{2}\). The received SINR of the \(k^{\text{th}}\) user is expressed as:  
\[\text{SINR}_{k} =\frac{\text{E}\left[(\mathbf{h}_{k}^{\text{T}}\mathbf{p}_{k}s_{k })^{2}\right]}{\text{E}\left[\left(\sum_{j\
eq k}\mathbf{h}_{k}^{\text{T}} \mathbf{p}_{j}s_{j}+n_{k}\right)^{2}\right]}\] \[=\frac{u_{k}(\mathbf{h}_{k}^{\text{T}}\mathbf{p}_{k})^{2}}{\sum \limits_{j\
eq k}\left[u_{j}\left(\mathbf{h}_{k}^{\text{T}}\mathbf{p}_{j} \right)^{2}+\underbrace{\sum_{i\
eq k,j}b_{i}b_{j}\mathbf{h}_{k}^{\text{T}} \mathbf{p}_{i}\mathbf{h}_{k}^{\text{T}}\mathbf{p}_{j}}_{\star}\right]+\sigma_{ k}^{2}}, \tag{5}\]  
where \(b_{j}\) and \(u_{j}\), respectively, denote the mean and the average power of \(s_{j}\), i.e., \(b_{j}=\text{E}\{s_{j}\}\) and \(u_{j}=\text{E}\{s_{j}^{2}\}\). The SINR is a ubiquitous metric used for evaluating the communication performance, which will also be adopted as the design criterion in the paper. Note that the term \(\ast\) exists due to the general modulation with non-zero mean. This is markedly different from the case of zero-mean modulation and makes the TPC design more intractable.