## 2 Nonlinear Characteristics of Different LEDs in the VLC System  
In the ideal condition, we consider an ideal LED as a distortion-free diode. We also define the input port signal as the driving current and the output port signal power as the emitted optical power. LEDs also exhibit nonlinearities, introducing distortions on the emitted signal [10]. Since the physical models, which include the dynamic rate equation model [10], failed to approximate practical LEDs, we model the static transfer function of the LED output power, \(P_{out}(t)\), with polynomials shown in Equation (1).  
\[P_{out}(t)=\sum_{n=0}^{\infty}b_{n}[I_{in}(t)-I_{DC}]^{n} \tag{1}\]  
where \(I_{in}(t)\) is the driving current, \(I_{DC}\) is the bias current, and \(b_{n}\) is the \(n\)-th order power coefficient of the transfer function. Although polynomial orders are required to be \(n=5\) to realistically model transfer functions, a second-order polynomial is proven to be a fair description [11]. The polynomial function in question is  
\[P_{out}(t)= b_{0}+b_{1}(I_{in}(t)-I_{DC})+b_{2}(I_{in}(t)-I_{DC})^{2} \tag{2}\]  
In this paper, we set the normalized current \(I_{DC}=0.5\), and \(b_{0}\), \(b_{1}\), and \(b_{2}\) are the Direct Current (DC) constant, the linear coefficient, and the second-order nonlinearity coefficient. Moreover, it is also known that an LED has constant behavior. As a result, the derivation of \(P_{out}(t)\) with respect to \(I_{in}(t)\) must be \(0\leq P_{out}(t)<1\).  
To describe the degree of nonlinearity, we define the nonlinearity parameter as \(\zeta\) in the source transfer function. It is the normalized output power corresponding to the input current. For example, we assume that the LED has \(\zeta=0.5\), derived as the linear line in the transfer function shown in Figure 2. If the transfer function is concave, \(\zeta>0.5\). If it is convex, \(\zeta<0.5\). LED is the prime example that has the concave characteristic [12]. The coefficients of Equation (2) can be expressed in \(\zeta\) as follows [11].  
\[b_{0}=\zeta,\hskip 14.226378 ptb_{1}=1,\hskip 14.226378 ptb_{2}=-4\zeta+2 \tag{3}\]  
Figure 2 shows the concave and convex curve examples. In this figure, the concave curve is based on the red LED coefficient, but white and infrared LEDs also have concave  
Figure 1: A general example scenario of Visible Light Communication applications for mobility services in indoor environments. The bicycle and scooters are driving or paused in the building or tunnel in this scene. Note that the line of sight between LED and mobility devices is available with multiple reflections. (Scooter icon in Vehicle and Travel Pack designed by Wishforge Games, www.wishforge.games), accessed on 3 January 2022.  
curves. The experimental values of \(\zeta\) according to LED type 85 are 0.732 in red LEDs, 0.541 in infrared LEDs, and 0.582 in white LEDs [11]. However, the examples of fixed coefficient values given LED types in the assumption are not consistent in the natural environment. Depending on types and actual production, the coefficient value can be off. For that matter, it would be reasonable to solve LED distortion issues under two assumptions: (1) The LED nonlinear parameter value is a fixed one and consistent. (2) The LED nonlinear parameter value can vary for each LED and is inconsistent. Note that our term inconsistency does mean the slight change, not significant disruption, that the LED light-emitting type may be changed. Given these two assumptions, we need to investigate the solutions for the VLC system.