# Design of Indoor Visible Light Communication PAM4 System
## 4 Modulation Characteristic Analysis  
The noise sources of an intensity modulation/direct detection (IM/DD) optical communication system mainly include background optical noise, receiver noise, and light-intensity flicker noise caused by turbulence [22]. The noise after the background light passes through the receiver can be equivalent to Gaussian noise. The dark current and thermal noise of the receiver can be regarded as a Gaussian random process, obeying the Gaussian distribution [23]. The background optical noise and receiver noise can be regarded as additive noise. The light-intensity flicker noise caused by turbulence is much lower than the signal frequency and belongs to low-frequency multiplicative noise. Therefore, the mathematical model of the equivalent noise of the communication channel and the receiver can be expressed as [24]:  
\[y(t)=h(t)x(t)+n(t) \tag{14}\]  
where \(x(t)\) is the transmitted signal, \(n(t)\) is the additive white Gaussian noise, \(y(t)\) is the receiver output signal, the multiplicative noise \(h(t)\) is the channel state information, and \(h(t)\) obeys the lognormal distribution in the case of weak turbulence, and its PDF is [25]:  
\[f_{I}(h_{t})=\frac{1}{\sqrt{2\pi\sigma_{x}^{2}}}\frac{1}{2h_{t}}\exp\left[- \frac{\left(\ln h_{t}-2\mu_{x}\right)^{2}}{8\sigma_{x}^{2}}\right] \tag{15}\]  
where \(h_{t}\) is the channel fading coefficient, the state information of the reaction channel, and \(\mu_{x}\), \(\sigma_{x}\) is its mean and variance. According to the literature [26], \(\sigma_{x}^{2}\) can be expressed as [26]:  
\[\sigma_{x}^{2}=\frac{1.23C_{m}^{2}k^{2/6}l^{11/6}}{4} \tag{16}\]  
where \(C_{n}^{2}\) is the atmospheric refractive index structure constant, \(k=2\pi/\lambda\) represents the wavenumber, \(\lambda\) is the wavelength of light, and \(l\) is the transmission distance. For a random noise process \(N\) with a \(K\)-component Gaussian mixed distribution, the PDF is defined as [27]:  
\[f_{N-GM}(n)=\sum_{j=1}^{K}\varepsilon_{j}f_{j}=\sum_{j=1}^{K}\varepsilon_{j}N( \mu_{nj},\sigma_{nj})=\sum_{j=1}^{K}\frac{\varepsilon_{j}}{\sqrt{2\pi\sigma_{ nj}^{2}}}\exp\left[-\left\{(n-\mu_{nj})/\sqrt{2}\sigma_{nj}\right\}^{2}\right] \tag{17}\]  
where \(\varepsilon_{j}\) is the weight coefficient of the \(j\)-th component, and \(f_{j}=N(\mu_{nj},\sigma_{nj})\) is the PDF of the Gaussian distribution. It can be seen that the Gaussian mixture model (GMM) is actually the weighted sum of \(K\) Gaussian noise, so the GMM is still an additive noise. Since the average value of the detection output signal of the squared law of the optical field must be greater than 0, the Gaussian white noise with a mean value of 0 cannot accurately characterize the additive noise characteristics of the optical channel. [PERSON] proposed a noise model fitted with a double Gaussian function, which we call [PERSON]'s model [28], whose PDF is defined as:  
\[f(x)=\varepsilon_{1}\times\frac{1}{\sqrt{2\pi}\sigma_{1}}\times e^{\frac{(- \frac{(x-p_{1})^{2}}{2\sigma_{1}^{2}})}{2\sigma_{1}^{2}}}+\varepsilon_{2} \times\frac{1}{\sqrt{2\pi}\sigma_{2}}\times e^{(-\frac{(x-p_{2})^{2}}{2\sigma_ {2}^{2}})} \tag{18}\]  
where \(\varepsilon_{1}\), \(\varepsilon_{2}\) are the weight coefficients, ranging from 0  
seen from the setting of the parameters of the noise data Set 1 and the noise data Set 2. With the increase in parameters \(\sigma_{\text{GMM}}\), the BER performance deteriorated.