## 2 Augmentation of Li-Fi Wireless DMT Transmission Capacity Through Sparse Coded Mask Optimization
### Sparse Coded Mask Modeling for DMT Signal
First, the waveform of the DMT signal with the CP, \(X(m)\) is divided into \(S\) segments, where the length of a single segment, \(x_{i}(m)\) is \(\big{(}2N+N_{cp}\big{)}/S\big{)}\) as shown in Equation (3). Next, they are transformed into \(S\) DMT patches, \(P_{i}(m)\) using a one-dimensional (1-D) convolution filter (kernel size, \(K=16\), stride \(=16\)) as shown in Equation (4):  
\[X(m)=\sum_{i=1}^{S}x_{i}(m) \tag{3}\]  
\[P_{i}(m)=\sum_{k=0}^{K-1}h_{k}\cdot x_{i}(m-k),\ i=1,2,\ldots,S \tag{4}\]  
where \(P_{i}(m)\) is the i-th extracted feature patch, \(x_{i}(m)\) is the i-th DMT signal segment, \(S\) represents the total number of patches, and the kernel \(h_{k}\) is applied uniformly across all segments. The summation iterates over the length of the kernel \(K\), ensuring localized feature extraction. This convolution operation allows the network to learn meaningful representations of the input signal by capturing its temporal dependencies. Figure 1: Masking and reconstruction process of DMT signal using the proposed sparse coded masked modeling technique based on masked autoencoder (MAE). The DMT signal is segmented, masked, transmitted via Li-Fi link, and reconstructed at the receiver. In our proposed technique, the 1-D convolution filter is applied to process the discrete multi-tone (DMT) signal in order to obtain a feature representation before the masking operation. The primary purpose of this step is to transform each segment of the DMT signal into a feature space suitable for masked autoencoder (MAE)-based processing. The 1-D convolution filter is designed as a learnable kernel during the training phase. It is initialized randomly and optimized through backpropagation to extract the most informative features from the input DMT signal. all DMT signal segments are processed using the same 1-D digital convolution filter. The rationale behind using the same filter for all segments is to maintain consistency in feature extraction, ensuring that the learned representations are generalized across the entire signal. This shared filtering process enhances the robustness of the feature extraction while preserving the structural characteristics of the input DMT waveform. Also, the positional encoding patch is represented as shown in Equation (5). \[E_{position}(n,m)= \begin{cases}\sin\left(\frac{n}{10000^{\frac{m}{m_{model}}}} \right),&m=2j\\ \cos\left(\frac{n}{10000^{\frac{m}{m_{model}}}}\right),&m=2j+1\end{cases}  \tag{5}\]  
where \(n\in\{1,\ldots,k\},j\in\{1,\ldots,\lfloor L_{model}/2\rfloor\}\), \(L_{model}\) is the feature dimension after encoding. \(n\) is the position index of the DMT signal segment in the sequence, \(m\) is the feature dimension index within the model, \(L_{model}\) is the total feature dimension, \(j\) is an integer index. This formulation is inspired by sinusoidal positional encoding, commonly used in transformer-based models. It ensures that different frequency components are assigned to each position in a way that helps the model learn relative positions between different DMT signal segments. To be more specific, the position index of the DMT signal, \(n\) represents the absolute position of a given DMT signal segment within the entire sequence. This ensures that each segment of the input signal is uniquely encoded with a position-dependent value. Since DMT signals are time-series data, maintaining this position information is crucial for preserving sequential dependencies. The feature dimension index, \(m\) corresponds to the dimension of the feature vector in the model. The model typically transforms the original DMT signal into a high-dimensional feature space, where each feature dimension is indexed by \(m\). The equation differentiates even-indexed and odd-indexed feature dimensions, ensuring orthogonality between different components of the encoding. The use of sine and cosine functions in Equation (5) serves the purpose of injecting position-dependent variations into the input features, which allows the model to differentiate between different time steps in a continuous manner.