## 2 Proposed Integrated Mobile VLCP Scheme
### Proposed Scheme for MISO Configuration
Thus, for the \(n\)th VLCP data block, \(X_{i}^{n,l}(k)\) is the \(k\)th transmitted subcarrier symbol of the \(l\)th OFDM symbol at the \(i\)th LED, \(\mathbf{Y}^{n}(k)=[Y^{n,1}(k),\,Y^{n,2}(k),\,Y^{n,3}(k),\,Y^{n,A}(k)]^{\rm T}\), where \(Y^{n,l}(k)\) is the \(k\)th received subcarrier symbol of the \(l\)th OFDM symbol at PD1, \(\mathbf{H}^{n}(k)=[H_{1}^{n}(k),\,H_{2}^{n}(k),\,H_{3}^{n}(k),\,H_{4}^{n}(k)]^{ \rm T}\), where \(H_{i}^{n}(k)\) is the FDTF of the channel between the \(i\)th LED and PD1, and \(\mathbf{N}^{n}(k)\) is the noise matrix. Next, to recover the transmitted signal, we equalize the multipath channel. After zero-forcing equalization, the \(k\)th OFDM subcarrier symbol in the \(n\)th VLCP data block can be expressed as follows:  
\[\mathbf{\hat{S}}^{n}(k)= \begin{bmatrix}Y^{n,1}(k)\\ \overrightarrow{H}_{T1\_total}^{n}(k)\end{bmatrix} ,\frac{Y^{n,2}(k)}{ \overrightarrow{H}_{T2\_total}^{n}(k)},\frac{Y^{n,3}(k)}{\overrightarrow{H}_ {T3\_total}^{n}(k)},\frac{Y^{n,A}(k)}{\overrightarrow{H}_{T4\_total}^{n}(k)} \end{bmatrix}^{T}, \tag{3}\]where \(\hat{H}^{n}_{\textit{Tw\_total}}(k)\) (\(w=1,2,3,4\)) is the estimated FDTF of the channel between all LEDs activated in the \(w\)th time slot and PD1. According to Equation (3), before channel estimation, the receiver needs to have a priori knowledge of the CSI. Conventionally, training symbols at the beginning of each OFDM packet are employed for CSI estimation, assuming that the channel remains constant throughout the entire packet. However, in a mobile VLCP terminal, the channel characteristics vary over time. Consequently, the CSI estimated from training symbols may differ from that experienced by subsequent data symbols, leading to errors in CSI estimation and potentially affecting system performance. To effectively track the dynamic channel and achieve more accurate CSI, we implement the DFCE scheme in our system [20]. This scheme initially employs training symbols at the beginning of each OFDM packet to estimate the initial CSI. Subsequently, the estimated CSI is utilized to calculate \(\hat{\textbf{S}}^{n}(k)\) using Equation (3). Based on \(\hat{\textbf{S}}^{n}(k)\), the receiver can derive the decision value \(\hat{\textbf{E}}^{n}(k)\) to recover the original transmitted signal. The feedback of \(\hat{\textbf{E}}^{n}(k)\) is utilized to reconstruct the estimated transmitted signal matrix \(\hat{\textbf{X}}^{n}(k)\). Finally, \(\hat{\textbf{X}}^{n}(k)\) and \(\textbf{Y}^{n}(k)\) are used to update the CSI for subsequent channel estimation in the next VLCP data block, given by the following:  
\[\hat{\textbf{H}}^{n+1}(k)=\frac{1}{L}{\sum_{p=n-L+1}^{n}{\textbf{Y}^{p}(k)[ \hat{\textbf{X}}^{p}(k)]}^{-1}}, \tag{4}\]  
where \(L\) is the observation block length for VLCP data blocks. Equation (4) averages out noise by incorporating multiple blocks. This process is repeated to recover subsequent VLCP data blocks until the end of each OFDM packet. By doing so, we effectively track and mitigate channel variations between neighboring VLCP data blocks, thereby enhancing the channel estimation accuracy. It is worth mentioning that, based on the LED selection matrix in Equation (1), \(\hat{\textbf{X}}^{n}(k)\) is always invertible, ensuring the implementation of Equation (4). Meanwhile, benefiting from the enhanced channel estimation accuracy, the CSI, specifically \(\hat{H}^{n}_{\textit{Tw\_total}}(k)\), can further be utilized to estimate the location of the mobile receiver. Based on Equation (3) and the LED selection in Equation (1), the specific expansion of the FDTFs for the multipath channels can be derived as follows:  
Figure 2: Block diagram of the integrated mobile VLCP scheme for MISO configuration. \[