# Hybrid VLC-RF Channel Estimation for GFDM Wireless Sensor Networks Using Tree-Based Regressor
## 3 System Model  
Throughout this model, we describe the end-to-end transmission and reception operations of a BPSK-based GFDM system deployed over a hybrid VLC-RF channel. The transmitter processes the binary data stream through channel coding using the BPSK modulation in GFDM, as well as cyclic prefixing. The signal then travels through a two-stage channel, where the first stage is a LoS-based VLC channel with AWGN, and the second stage is an RF Rayleigh fading channel. The receiver inverts the transmission chain using matched filtering, demapping, and decoding (see Figure 2). In a GFDM environment, let the number of subcarriers be \(K\), the number of subsymbols be \(M\), and the total GFDM block length be \(N=KM\). The total number of bits transmitted per block is \(N_{b}\). Figure 2: Block diagram of GFDM. A. _Transmitter_  
(1) Binary Source  
\[\mathbf{b}=[b_{0},b_{1},\ldots,b_{N_{b}-1}],\quad b_{i}\in\{0,1\}. \tag{1}\]  
This vector represents the raw binary information that the system aims to transmit. It is generated by the source and typically contains the payload data. Each element \(b_{i}\) represents a single bit and forms the basis for modulation. The length \(N_{b}\) depends on the block configuration and the channel coding parameters are used subsequently. (2) Channel Encoder  
\[\mathbf{b}_{c}=\mathcal{E}(\mathbf{b}),\quad N_{c}=\frac{N_{b}}{r}. \tag{2}\]  
To model the system's bit-level processing behavior, the original bit vector \(\mathbf{b}\) is mapped to a transformed bit sequence \(\mathbf{b}_{c}\) through the encoding function \(\mathcal{E}(\cdot)\). This captures any expansion or modification that may occur before modulation, such as bit padding. The transformation increases the bit vector size from \(N_{b}\) to \(N_{c}\), where \(r\in(0,1]\) represents a generic expansion factor. This step allows our system model to accommodate scenarios where additional bits are introduced intentionally or by system design before transmission. (3) BPSK Mapping  
\[d_{i}=2b_{c,i}-1\quad\Rightarrow\quad\mathbf{d}=[d_{0},d_{1},\ldots,d_{N-1}]^{ \top}. \tag{3}\]  
Each encoded bit is mapped to a BPSK symbol: bit 0 becomes \(-1\), and bit 1 becomes \(+1\). This mapping results in a real-valued symbol vector \(\mathbf{d}\in\{-1,+1\}^{N}\). These symbols are later shaped and modulated by GFDM. BPSK was chosen for its robustness and simplicity, especially under low SNR and fading conditions, which, in turn, are a suitable choice for hybrid VLC-RF systems, where reliability is critical. (4) GFDM Modulation  
\[g_{k,m}[n]=g[(n-mK)\text{ mod }N]\cdot e^{-j\frac{2\pi\mathrm{bit}}{K}}. \tag{4}\]  
The function \(g_{k,m}[n]\) represents the time-frequency localized pulse shaping function applied to each data symbol \(d_{k,m}\). It is a circularly time-shifted version of a filter \(g[n]\) modulated in frequency by the subcarrier index \(k\). This ensures spectral containment while allowing overlap between subsymbols and subcarriers, which enhances bandwidth efficiency and time-frequency flexibility. \[x[n]=\sum_{k=0}^{K-1}\sum_{m=0}^{M-1}d_{k,m}\cdot g_{k,m}[n]. \tag{5}\]  
The total transmitted signal \(x[n]\) is formed by summing all shaped and modulated symbols over subcarriers and subsymbols, as defined by Equation (5). This results in a complex baseband signal of length \(N\), which constitutes one GFDM block. By allowing overlapping in both the time and frequency domains, GFDM achieves improved spectral efficiency compared to OFDM, although this may introduce inter-symbol interference (ISI). Alternatively, the transmit signal can be expressed in a more compact and computationally efficient matrix form as \(\mathbf{x}=\mathbf{A}\cdot\mathbf{d}\), where \(\mathbf{A}\) is the modulation matrix composed of column vectors representing the filters \(g_{k,m}[n]\), and \(\mathbf{d}\) is the vector of data symbols. This matrix formulation not only simplifies the implementation, but also facilitates the analytical design of demodulators and equalizers.