# Ambient LED Light Noise Reduction Using Adaptive Differential Equalization in Li-Fi Wireless Link
## 2 Ambient Light Noise Model and Adaptive Differential Equalization
### Interference and Shot Noise Model of Fluorescent Lamp and Dimmable LED  
Figure 1 shows the process of reducing interference noise generated from ambient light sources (fluorescent lamp and dimmable LED) using the proposed adaptive differential equalization technique in the Li-Fi wireless link. The white LED light exhibits similar characteristics to an uncorrelated broadband optical source, such as amplified spontaneous emission (ASE). Therefore, the white LED light, \(E_{s}(t)\) directly modulated by the NRZ-OOK signal and two ambient lights (fluorescent lamp, \(E_{flu}(t)\) and dimmable LED, \(E_{dim}(t)\)) with interference, noises can be expressed as Equations (1)-(3), respectively [26]. \[E_{s}(t)=A(t)\sum_{k=-M_{0}}^{M_{0}}exp[j\{2\pi(f_{0}+k\delta f_{0})t+\phi_{k}\}] \tag{1}\]  
\[E_{flu}(t)=N_{1}(t)\sum_{k=-M_{1}}^{M_{1}}exp[j\{2\pi(f_{1}+k\delta f_{1})t+ \theta_{k}\}] \tag{2}\]  
\[E_{dim}(t)=N_{2}(t)\sum_{k=-M_{2}}^{M_{2}}exp[j\{2\pi(f_{2}+k\delta f_{2})t+ \psi_{k}\}] \tag{3}\]  
where \(A(t)\) is an NRZ-OOK signal, \(N_{1}(t)\) and \(N_{2}(t)\) are interference noises, which are generated from fluorescent lamps with electrical circuits and dimmable LEDs, including LED driver circuits with a dimming controller. \(f_{0}\), \(f_{1}\), and \(f_{2}\) are the center optical frequencies in the spectral region of three different lights. \(\delta f_{0}\), \(\delta f_{1}\), and \(\delta f_{2}\) are elemental frequency intervals of each light. \(\phi_{k}\), \(\theta_{k}\), and \(\psi_{k}\) are random optical phases uncorrelated for each \(k\). \(B_{0}\), \(B_{1}\), and \(B_{2}\) are the spectral regions of each light. \(M_{0}\) is \(B_{0}/2\delta f_{0}\). \(M_{1}\) is \(B_{1}/2\delta f_{1}\), and \(M_{2}\) is \(B_{2}/2\delta f_{2}\). Accordingly, the received optical signal, \(E_{r}(t)\) in the receiver (Rx) can be described as Equation (4). \[E_{r}(t)=E_{s}(t)+E_{flu}(t)+E_{dim}(t) \tag{4}\]  
During the optical to electrical conversion in the photodetector of Rx, shot noise is generated in addition to the NRZ-OOK signal and interference noise. Here, two different  
Figure 1: Ambient light noise reduction using adaptive differential equalization. types of shot noise are generated in Rx. In more detail, they are stationary shot noises generated in the white LED as well as two ambient lights biased by DC current and non-stationary shot noises from the time-varying photocurrent generated by the NRZ-OOK signal and interference signal. The time-varying and non-stationary shot noise current can be realized by using Equation (5) by Personic analysis [PERSON] (1980):  
\[I_{shot}(t)=\frac{\mathcal{R}\int_{-\infty}^{\infty}p(\tau)h_{I}^{2}(t-\tau)d \tau}{2B} \tag{5}\]  
where \(\mathcal{R}\) is the responsivity of the photodetector, \(q\) is the electronic charge, \(p(t)\) is the received optical power and \(h_{I}(t)\) is the impulse response of the optical receiver, including bias circuit, transimpedance amplifier and equalizer. The output photocurrent, including interference noise and two kinds of shot noise, can be expressed as Equation (6) [28]:  
\[=\frac{\mathcal{R}}{2}\left\{ \begin{array}{c}A(t)+N_{1}(t)+N_{2}(t)+\frac{ \mathcal{R}\int_{-\infty}^{\infty}E_{2}^{2}(\tau)h_{I}^{2}(t-\tau)d\tau}{4B}+ \\ \frac{\mathcal{R}\int_{-\infty}^{\infty}E_{2T}^{2}(\tau)h_{I}^{2}(t-\tau)d \tau}{4B}+\frac{\mathcal{R}\int_{-\infty}^{\infty}E_{2T}^{2}(\tau)h_{I}^{2}(t -\tau)d\tau}{4B}+I_{dc-shot}\end{array} \right\} \tag{6}\]  
where \(I_{dc-shot}\) is a non-varying stationary shot noise caused by a photodetector, \(B\) is 3 dB bandwidth of photodetector. The terms from the fourth to the sixth on the right side of Equation (6) refer to the non-stationary shot noise generated by the photocurrent that changes due to the NRZ-OOK signal and interference noise, respectively. For reference, it was reported that a non-stationary shot noise generated by a time-varying photocurrent from a modulated light has a white spectrum with a Poisson probability distribution and has a correlation with a modulated signal [PERSON] and [PERSON] (2018).