## 2 Pseudo-Flicker Weight Functionalized Derivative Equalization in Li-Fi Transmission
One is a non-varying stationary shot noise generated by the photodetector itself, and the other is a time-varying and non-stationary shot noise generated due to the direct modulation of the NRZ-OOK signal. Therefore, the output photocurrent after the photodetector can be expressed as Equation (4). \[ \begin{array}{ll}I_{r}(i)&=\mathcal{R}\sigma_{L}\left\{\frac{A_{s}(i)}{\sqrt {2}}\right\}^{2}\\ &=\frac{\mathcal{R}}{2}\sigma_{L}\sum\
olimits_{k=1}^{N}B(k,i)g\Big{(}\frac{( t(i)-kT)}{T}\Big{)}+\frac{\mathcal{R}}{2}\sigma_{L}\mathcal{C}(i)exp(j2\pi\tau f _{0}t(i)+\phi(i))\\ &+\frac{\mathcal{R}^{2}\sigma_{L}}{M\rightarrow\infty}\sum\
olimits_{n=1}^{M} A_{s}^{2}(n)h_{1}^{2}(t(i)-\tau(n))}{8B}+\frac{\mathcal{R}}{2}\sigma_{L}I_{ shot}\end{array}  \tag{4}\]  
where \(\mathcal{R}\) is the responsivity of the photodetector. \(\sigma_{L}\) is the optical loss during the optical wireless transmission. \(h_{1}\) is the impulse response of the photodetector with transimpedance amplifier. \(B\) is 3-dB bandwidth of photodetector. \(I_{shot}\) is a non-varying stationary shot noise, which is produced by photodetector. The first term on the right side of Equation (4) is an NRZ-OOK signal having a digital waveform, and the second term is a flicker noise signal having an analog waveform. The third term on the right side of Equation (4) means the time-varying and non-stationary shot noise current, which is generated by the direct modulation of NRZ-OOK signals. This is implemented using Personick analysis [22]. It is well known that this non-stationary shot noise has a white spectral characteristic that correlates with the modulated signal and shows a Poisson probability distribution [23]. Here, since the frequency of the imperceptible flicker noise, \(f_{0}\) (around 120 Hz), is usually within the NRZ-OOK signal spectrum, it is very difficult to reduce the flicker noise using a software-based finite impulse response (FIR) filter or a hardware-based matched RF filter. Therefore, we propose a pseudo-flicker weight functionalized derivative equalization technique to reduce the imperceptible flicker noise. The proposed technique is implemented based on the conventional total variation (TV) technique. However, since the conventional TV technique focuses on reducing random analog noise having a Gaussian distribution function as a variance, it may not be able to reduce the flicker noise with a sinusoidal waveform effectively. Accordingly, we implemented a non-local weight function with a random Gaussian phase exhibiting characteristics similar to the flicker noise and then adapted it to the conventional TV technique in order to maximize the flicker noise reduction. In this way, the proposed technique focuses more on reducing flicker noise than on random analog noise such as shot noise. The proposed technique can be expressed in Equation (5) as follows. \[z_{p}(i)=\underset{f_{p}\in\mathrm{H}_{f_{p}}}{\mathrm{argmin}}\sum\
olimits _{\tau\in\mathbb{R}(i)}\Big{|}f_{p}(i)-\widetilde{S(\tau)}\Big{|}^{2}\Theta( \tau,i)+\lambda\|\ f_{p}\ \|_{TV} \tag{5}\]  
where \(z_{p}(i)\) is a polynomial function of degree \(p\) or less. \(\widetilde{S(\tau)}\) is a received NRZ-OOK signal adding the flicker noise. \(\mathrm{H}_{f_{p}}\) is the full set of all polynomial functions of degree \(p\) or less. \(f_{p}\) is one of the polynomial functions in the full set, \(\mathrm{H}_{f_{p}}\). \(\lambda\) is a regularization parameter that controls the waveform of the recovered NRZ-OOK signal. \(\parallel\ \parallel_{TV}\) is TV norm, \(\parallel f_{p}\parallel_{TV}=\int_{\mathbb{R}}\left|\
abla f_{p}\right|\) with the gradient operator, \(\
abla\). \(\Theta(\tau,i)\) is a weight function with a random Gaussian phase that exhibits very similar properties to imperceptible flicker noise as shown in Equation (6).