# Experimental Study on Chromaticity Control in Visible Light Communication Systems
## 2 Theoretical Basis  
To analyze the color rendering of LEDs, the Gaussian mathematical model (Y-model) proposed by [PERSON] et al. agrees well with the actual spectral distribution of LEDs and was widely accepted. This model can be represented as follows [17]:  
\[S_{LED}(\lambda,\lambda_{0},\Delta\lambda_{0.5})=\gamma_{c}\frac{g(\lambda, \lambda_{0},\Delta\lambda_{0.5})+2g^{5}(\lambda,\lambda_{0},\Delta\lambda_{0.5 })}{3}, \tag{1}\]  
where \(g(\lambda,\lambda_{0},\Delta\lambda_{0.5})=\exp[-((\lambda-\lambda_{0})/\Delta \lambda_{0.5})^{2}]\); \(\lambda\) is the wavelength; \(Y_{c}\) is the spectral power at peak wavelength \(\lambda_{0}\); and \(\Delta\lambda_{0.5}\) is the half peak width.  
The tristimulus value of a light source refers to the basic optical characteristics of the light source and is usually used to describe and measure the color of the light source. According to the spectral mathematical model of LED, the tristimulus values of \(X\), \(Y\), and \(Z\) of the light source to be tested can be calculated, and the calculation formula is [18] as follows:\[\left\{\begin{array}{l}X=K_{m}\int_{380}^{780}S(\lambda)\overline{x}(\lambda)d \lambda\\ Y=K_{m}\int_{380}^{780}S(\lambda)\overline{y}(\lambda)d\lambda\\ Z=K_{m}\int_{380}^{780}S(\lambda)\overline{z}(\lambda)d\lambda\end{array} \right., \tag{2}\]  
where \(K_{m}=683 lm/W\) as the constant; \(S(\lambda)\) is the relative spectral power distribution; and \(\overline{x}(\lambda)\), \(\overline{y}(\lambda)\) and \(\overline{z}(\lambda)\) are the color matching function, which is the average data of the CIE 1931 standard observer, also known as the spectrum triple stimulation function [18].  
In the actual calculation, the CIE 1931 standard chromaticity tristimulus value of the light source to be tested is obtained by using the method of discrete summation and approximate integration calculation. The expression can be expressed as follows:  
\[\left\{\begin{array}{l}X=K_{m}\underset{380}{\overset{780}{\sum}}S(\lambda) \overline{x}(\lambda)\Delta\lambda\\ Y=K_{m}\underset{380}{\overset{780}{\sum}}S(\lambda)\overline{y}(\lambda) \Delta\lambda\\ Z=K_{m}\underset{380}{\overset{780}{\sum}}S(\lambda)\overline{z}(\lambda) \Delta\lambda\end{array}\right., \tag{3}\]  
where \(\overline{x}(\lambda)\), \(\overline{y}(\lambda)\), and \(\overline{z}(\lambda)\) can be determined by looking up tables, and the spectral sampling interval of \(\Delta\lambda\)= 1 is often taken.  
According to the obtained light source three stimulus values, the corresponding color coordinates can be found. After the normalization of \(X\), \(Y\), and \(Z\), the color coordinates \(x\), \(y\), and \(z\) of the light source to be measured can be obtained:  
\[\left\{\begin{array}{l}x=\frac{X}{X+Y+Z}\\ y=\frac{Y}{X+Y+Z}\\ z=\frac{Z}{X+Y+Z}\end{array}\right., \tag{4}\]  
In lighting systems, color temperature is a physical quantity used to define the chromaticity of a light source, which can measure the lighting quality of the light source. The relevant color temperature of the light source can be calculated from the chromaticity coordinates, and the formula for calculating the color temperature is as follows [19]:  
\[T=-437n^{3}+3601n^{2}-6861n+5514, \tag{5}\]  
where \(n=\frac{x-0.3320}{y-0.1858}\); and \(x\), \(y\) is the color coordinate.  
The color index is a parameter to evaluate the degree to which the light source restores the true color of an object. It is generally believed that the color presented by objects under natural light is the most authentic, with a color index of 100. The color index of the light source for a standard sample can be expressed as follows [20]:  
\[R_{i}=100-4.6\Delta E_{i}, \tag{6}\]  
where \(\Delta E_{i}\) is the color difference of the sample under the reference light source and the light source to be tested, which can be obtained from the chromaticity coordinates.  
The average color index of the light source for color samples 1 to 8 is called the general color index \(R_{a}\):  
\[R_{a}=\frac{1}{8}\underset{i=1}{\overset{8}{\sum}}R_{i}, \tag{7}\]