# Performance Improvement by FRFT-OFDM for Visible Light Communication and Positioning Systems
## 2 Principle
### System Model  
We have designed an indoor VLC and VLP system based on FRFT-OFDM, as shown in Figure 1. The input serial bit streams are processed using QAM to produce the output \(\mathbf{X}[u]\). To ensure that the transformed output signal is real, we apply the RPFRFT method described in Section 2.1 to separate the real and imaginary parts of \(\mathbf{X}[u]\). Subsequently, we perform serial-to-parallel (S/P) conversion on the real and imaginary parts independently. At this stage, the real part of \(\mathbf{X}[u]\) corresponds to subcarriers \(1,\cdots,N/2\), while the imaginary part corresponds to subcarriers \(N/2+1,\cdots,N\). This can be represented as \(\mathbf{X}_{c}[u]~{}=~{}[\text{Re}(\mathbf{X}[u]),\text{Im}(\mathbf{X}[u])]^{\mathsf{T}}\). The separated signals undergo the RPFRFT, followed by a parallel-to-serial (P/S) conversion to produce \(\mathbf{x}_{c}[n]\). The real-valued output \(\mathbf{x}_{c}[n]\) obtained at this stage can be expressed as  
\[ \begin{split}\mathbf{x}_{c}[n]~{}=~{}\mathbf{R}_{-p}\mathbf{X}_{c}[u]\end{split}  \tag{13}\]  
Due to the chirp periodicity of the signal, a chirp cyclic prefix (CP) is required at the transmitter instead of the conventional CP. By selecting the optimal FRFT order, the chirp CP effectively mitigates inter-symbol interference (ISI). However, the chirp CP has a significant drawback: it necessitates a feedback channel between the transmitter and receiver to relay the optimal FRFT order, which complicates the overall system. To simplify the system, we have adopted the approach from [38] and utilized zero padding (ZP) in place of the chirp CP. This adjustment results in an output \(\mathbf{x}_{zp}[n]\) that incorporates ZP while simultaneously reducing the transmit power required by the system. Due to the limited dynamic range of LEDs, it is necessary to scale the signal \(\mathbf{x}_{zp}[n]\) before adding a DC bias to ensure the signal falls within an appropriate range. We have  
\[\mathbf{x}_{in}[n]\ =\ \frac{\zeta I_{max}\mathbf{x}_{zp}[n]}{(\zeta+1)\cdot\max\{|\mathbf{x }_{zp}[n]|\}} \tag{14}\]  
where \(I_{max}\) is the maximum current desired after scaling and biasing, and \(\zeta\) is the modulation index, which is given as  
\[\zeta\ =\ \frac{I_{max}-I_{\text{DC}}}{I_{\text{DC}}} \tag{15}\]  
where \(I_{\text{DC}}\) represents the DC-bias current. Various studies have attempted to model LED nonlinearities, employing approaches such as the Volterra series [39] and the Wiener [40], Hammerstein [41], and Wiener-Hammerstein [42; 43] models. The authors of [41] surveyed nonlinear effects in VLC and discussed several classical models based on polynomial expansions. Among these, the Volterra series provides the most comprehensive description of nonlinearities with memory [44]. However, it requires a substantial amount of training data and incurs significant computational complexity. To mitigate these challenges, the Hammerstein and Wiener models have been simplified based on the Volterra series by assuming separable memory effects over nonlinearity [45; 46; 47]. In [48], the author demonstrated that the Hammerstein model is more effective than the Wiener model for describing LED nonlinearities. We have adopted this model, which consists of a memoryless nonlinear module followed by a linear time-invariant (LTI) system (see Figure 2). Figure 1: Block diagram of RPFRFT-based OFDM system. Figure 2: Hammerstein model. The LED nonlinear module is represented by a third-order polynomial [49], while the LTI module is modeled as a first-order low-pass filter [12]. The output \(x_{out}\) of the intensity-modulated LED is expressed as  
\[x_{out}(t)\ =\ f_{\rm NL}(x_{in}(t))\otimes h_{\rm LED}(t) \tag{16}\]  
where \(f_{\rm NL}\) and \(h_{\rm LED}\) are given by  
\[f_{\rm NL}(x)\ =\ \left\{ \begin{array}{ll}0.15,&x<0.1\\ 1.2897x^{3}-3.2738x^{2}+3.3919x-0.1577,&0.1\leq x<1\\ 1.25,&x\geq 1\end{array} \right. \tag{17}\]  
\[h_{\rm LED}(t)\ =\ e^{-2\pi B_{m}t} \tag{18}\]  
where \(B_{m}\) is the modulation bandwidth, and \(\otimes\) denotes the time-domain convolution.