# Hybrid VLC-RF Channel Estimation for GFDM Wireless Sensor Networks Using Tree-Based Regressor
## 3 System Model
### Hybrid VLC-RF Channel  
(1) VLC Channel (LoS + AWGN)  
\[\mathbf{y}_{\text{VLC}}=h_{\text{vlc}}\cdot\mathbf{x}_{\text{CP}}+\mathbf{n}_ {\text{vlc}},\quad\mathbf{n}_{\text{vlc}}\sim\mathcal{N}(0,\sigma_{\text{vlc }}^{2}\mathbf{I}). \tag{7}\]  
The VLC channel is modeled as a pure line-of-sight (LoS) channel with additive white Gaussian noise (AWGN). The channel gain \(h_{\text{LoS}}\in\mathbb{R}_{+}\) represents the static optical gain, which depends on distance, LED angle, and receiver position. Since the environment is generally free of significant multipath scenarios, the VLC channel is deterministic and not faded. The noise \(\mathbf{n}_{\text{vlc}}\) accounts for thermal, shot, and ambient light noise at the photodetector, and it is assumed to be uncorrelated Gaussian noise across time.  
(2) RF Channel  
\[\mathbf{h}_{\text{RF}}\sim\mathcal{CN}(0,\sigma_{h}^{2}),\quad\mathbf{y}= \mathbf{h}_{\text{RF}}\cdot\mathbf{y}_{\text{VLC}}+\mathbf{n}_{\text{rf}}\,\quad\mathbf{n}_{\text{rf}}\sim\mathcal{CN}(0,\sigma_{\text{rf}}^{2} \mathbf{I}). \tag{8}\]  
The signal from the VLC stage is transmitted through an RF channel characterized by Rayleigh fading, which captures the effects of three-path propagation in the RF domain. This is modeled using a channel impulse response \(\mathbf{h}_{\text{RF}}\) as a complex Gaussian process. The delays in paths are defined as \(\tau_{l}=[0,\,0.1,\,0.2]\) in (\(\upmu\)s), and the corresponding power gains are \(G_{l}=[0,-3,-6]\) dB. The operation (.) between \(\mathbf{h}_{\text{RF}}\) and the VLC output introduces frequency selectivity and temporal dispersion, which is further impacted by the channel delay and power profiles. Additionally, additive white Gaussian noise (AWGN) \(\mathbf{n}_{\text{rf}}\) is introduced due to the receiver circuitry and thermal noise in the RF stage. Eventually, the received signal from the hybrid VLC-RF channel (the signal passes sequentially through both the VLC and RF stages) can be expressed as  
The VLC stage output is given by  
\[y_{\text{VLC}}=h_{\text{vlc}}\cdot\mathbf{x}_{\text{CP}}+n_{\text{vlc}}. \tag{9}\]  
Substituting \(y_{\text{VLC}}\) into the RF channel model yields  
\[y=h_{\text{RF}}\cdot(h_{\text{vlc}}\cdot\mathbf{x}_{\text{CP}}+n_{\text{vlc }})+n_{\text{rf}}, \tag{10}\]  
which simplifies to  
\[y=(h_{\text{RF}}\cdot h_{\text{vlc}})\cdot\mathbf{x}_{\text{CP}}+h_{\text{ RF}}\cdot n_{\text{vlc}}+n_{\text{rf}}. \tag{11}\]  
Thus, the equivalent hybrid channel model can be compactly expressed as  
\[y=h_{\text{eff}}\cdot\mathbf{x}_{\text{CP}}+n_{\text{eff}}, \tag{12}\]where  
\[h_{\text{eff}} =h_{\text{RF}}\cdot h_{\text{vcl}}, \tag{13}\] \[n_{\text{eff}} =h_{\text{RF}}\cdot n_{\text{vcl}}+n_{\text{rf}}. \tag{14}\]  
Here, \(h_{\text{vcl}}\) and \(h_{\text{RF}}\) represent the VLC and RF channel coefficients, respectively. The effective channel \(h_{\text{eff}}\) is given by the product of the two individual channels, corresponding to a sequential transmission through both. The total noise \(n_{\text{eff}}\) consists of the VLC noise term \(n_{\text{t,vclc}}\), which is scaled by the RF channel, and the additive noise \(n_{\text{t,rf}}\) from the RF stage.  
To estimate the effective hybrid channel \(\mathbf{h}_{\text{eff}}=h_{\text{vclc}}\cdot\mathbf{h}_{\text{RF}}\), we generated supervised learning pairs by simulating the hybrid channel's output \(\mathbf{y}\) for known transmitted signals \(\mathbf{x}_{\text{CP}}\) under varying SNR conditions. The known effective channel coefficients from te simulation serve as ground-truth labels. Our algorithms are trained to learn the non-linear mapping from the received signal \(\mathbf{y}\) to the estimated channel response \(\mathbf{\hat{h}}_{\text{eff}}\). This data-driven approach enables robust estimation, even under nonlinear and noisy hybrid VLC-RF conditions.