# 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 Reduction Using Adaptive Differential Equalization  
In general, when an arbitrary signal (\(x\)) is transmitted through any transmission medium, noise (\(d\)) generated during the transmission is added, and then the output signal (\(y\)) is produced as described in Equation (9). \[y=x+d \tag{9}\]  
where \(y\) is an output signal, \(x\) is an input signal, \(d\) represents the noise generated in the process of passing through the transmission medium. Using Equations (6)-(8), Equation (9) can be summarized as Equation (10):  
\[ \begin{array}{l}I_{r}(t)=\frac{\mathcal{R}}{2}A(t)+\frac{\mathcal{R}}{2} \sqrt{2\bigg{[}a_{0}+\sum\limits_{i=1}^{n1}\begin{array}{c}a_{i}\cos(2\pi(10 0i-50)t)\\ +b_{i}\cos(2\pi(100i)t)\end{array}\bigg{]}}+\\ \\ \frac{\mathcal{R}}{2}\sqrt{2\bigg{[}c_{0}+\sum\limits_{i=1}^{n2}\begin{array}{c }c_{i}\cos(2\pi(100i-50)t)\\ +e_{i}\cos(2\pi(100i)t)\end{array}\bigg{]}}+\frac{\mathcal{R}^{2}\sum\limits _{\infty}^{\infty}E_{2}^{2}(\tau)\hat{\mu}_{I}^{2}(t-\tau)d\tau}{8\mathcal{B}} +\frac{\mathcal{R}^{2}\int_{-\infty}^{\infty}E_{flux}^{2}(\tau)\hat{\mu}_{I}^{ 2}(t-\tau)d\tau}{8\mathcal{B}}+\\ \\ \frac{\mathcal{R}^{2}\int_{-\infty}^{\infty}E_{2}^{2}(\tau)\hat{\mu}_{I}^{2}(t -\tau)d\tau}{8\mathcal{B}}+\frac{\mathcal{R}}{2}I_{dc-shot}\end{array}  \tag{10}\]  
As shown in Equation (10), an NRZ-OOK signal with a digital waveform is designated as an input signal (\(x\)), and the interference noise and shot noise (\(d\)) generated during the optical wireless transmission are expressed as harmonic distortion and random white noise with Poisson function, respectively. In this way, various signal processing techniques based on total variation (TV) techniques have been proposed to reduce the corresponding noise when the waveform of the input signal is different from that of the noise [30; 31; 32]. If a conventional TV technique is used to recover the NRZ-OOK signal, Equation (10) is changed into Equation (11):  
\[\begin{array}{l}A(t)=\underset{P(t)}{\text{argmin}}\|\frac{\mathcal{R}}{2} A(t)-I_{r}(t)\|_{2}^{2}+\lambda\|A(t)\|_{TV}\\ \\ =\underset{P(t)}{\text{argmin}}\|\frac{\mathcal{R}}{2}\sqrt{2\bigg{[}a_{0}+ \sum\limits_{i=1}^{n1}\begin{array}  
the regularization parameter and adapts to the waveform of the signal, and a non-local weighting function that is robust to noise with a random analog waveform. In addition, the Poisson distribution function was added to the weighting function to reduce the non-stationary and stationary shot noise effectively.  
The ADE technique is mathematically formulated as follows: Since it is implemented based on digital signal processing (DSP), the NRZ-OOK signal, interference noise, and shot noise are digitally sampled. Therefore, when converting from continuous-time signal to discrete-time signal, the restored NRZ-OOK signal, \(A(t)\), is changed to \(A(i)\) in case that \(\Psi(i)\) is the whole set of sampling points, \(i\). Here, the polynomial function, \(\Gamma_{p}(i)\) mathematically implemented to recover the NRZ-OOK signal using the ADE technique can be expressed by Equation (12).  
\[\Gamma_{p}(i)=\operatorname*{argmin}_{f\in\mathcal{X}_{L}}\sum_{j\in\Psi(i)} \left|f_{r}(i)-A(j)\right|^{2}\text{T}(j,i)+\lambda\left\|f\right\|_{TV} \tag{12}\]  
where \(\mathcal{X}_{L}\) is the set of all polynomial functions of degree L or less. \(\text{T}(j,i)\) is a weight function to which a Poisson distribution function is applied, as shown in Equation (13). \[\text{T}(j,i)=exp\left(-\frac{\sum_{v\in K}\left\|I_{r}(j+v)-I_{r}(i+v)\right\| ^{2}S_{\sigma}(\left\|v\right\|)}{h^{2}}\right) \tag{13}\]  
where \(S_{\sigma}\) is the Poisson distribution function with a standard deviation of \(\sigma\). \(K\) is a small set of sampling points located around sampling points of \(i\) and \(j\), and \(h\) is a filtering parameter. As a result, \(\Gamma_{p}(i)\) obtained by using Equation (12) becomes a recovered NRZ-OOK signal.