## 3 Results and Discussion
### Bit Error Ratio  
In this section, we define a new CSK reference model whose CIR, for the channel between the \(i\)th and \(j\)th, is denoted by \(h_{\text{ref}}[i,j](t)\) and is defined as  
\[h_{\text{ref}}[i,j](t)=R_{\text{ref,}j}\cdot\mathcal{G}_{\text{ref,}j}\cdot h _{\text{LED}}(t)*h_{\text{ref}}(t)\,, \tag{103}\]  
Figure 9: RMS delay: (**a**) Between red LED and red PD; (**b**) Between green LED and green PD; (**c**) Between blue LED and blue PD; (**d**) Reference model. (**e**) Empirical CDF for RMS delay for red, green, and blue channels in the proposed model and for RMS delay in the reference model. where \(h_{\text{ref}}(t)\) is the reference channel defined in Equation (64). The reference channel defined by Equation (103) could be considered a monochromatic approximation of the CSK channel model proposed in this paper, where all the wavelength-dependency is approximated linearly, and the LEDs are considered to be monochromatic light sources of wavelengths \(\lambda_{1}\), \(\lambda_{2}\), and \(\lambda_{3}\) for the red, green, and blue LEDs, respectively. In order to make it a fair comparison, we selected  
\[ \begin{bmatrix}R_{\text{ref},1}\\ R_{\text{ref},2}\\ R_{\text{ref},3}\end{bmatrix} = \begin{bmatrix}R(\lambda_{1})\\ R(\lambda_{2})\\ R(\lambda_{3})\end{bmatrix} \,, \tag{104}\]  
and  
\[ \begin{bmatrix}\mathcal{G}_{\text{ref},11}&\mathcal{G}_{\text{ref},12}& \mathcal{G}_{\text{ref},13}\\ \mathcal{G}_{\text{ref},21}&\mathcal{G}_{\text{ref},22}&\mathcal{G}_{\text{ ref},23}\\ \mathcal{G}_{\text{ref},31}&\mathcal{G}_{\text{ref},32}&\mathcal{G}_{\text{ ref},33}\end{bmatrix} = \begin{bmatrix}\mathcal{G}_{1}(\lambda_{1})&\mathcal{G}_{2}( \lambda_{1})&\mathcal{G}_{3}(\lambda_{1})\\ \mathcal{G}_{1}(\lambda_{2})&\mathcal{G}_{2}(\lambda_{2})&\mathcal{G}_{3}( \lambda_{2})\\ \mathcal{G}_{1}(\lambda_{3})&\mathcal{G}_{2}(\lambda_{3})&\mathcal{G}_{3}( \lambda_{3})\end{bmatrix} \,, \tag{105}\]  
where \(R(\lambda)\) is the spectral responsivity used in the proposed model; \(G_{1}(\lambda)\), \(G_{2}(\lambda)\), and \(G_{3}(\lambda)\) are the transmittances of the optical filters used in the proposed model; and \(\lambda_{1}\), \(\lambda_{2}\), and \(\lambda_{3}\) are the center-of-band wavelengths of the red, green, and blue LEDs used in the proposed model. The parameters used in the BER simulation are the same as those defined in Table 2, with the addition of parameters from Table 3. In the BER simulation, \(\sigma_{j}\) was not calculated with Equation (22), but rather Equation (25) was used, such that  
\[\sigma_{j}=\sqrt{\frac{(I_{\text{avg}}[j,j])^{2}}{\text{SNR}}}\,, \tag{106}\]  
where SNR is an independent variable. Given an SNR value, the BER was calculated by generating random binary sequences, modulating them with CSK and applying the proposed and reference channel models (Equations (15) and (103), respectively) to the CSK symbols, adding the noise with the corresponding standard deviation value \(\sigma_{j}\) (using Equation (106)), demodulating them, and applying the color calibration technique described in the IEEE 802.15.7 standard [11], in which a sequence of 24 symbols containing codes known by the receiver is sent before data transmission. A maximum of 524,232 bits of payload is to be sent per frame [11], and each frame contains one sequence for color calibration. For each SNR value, 20 frames were transmitted, meaning that about \(10^{7}\) bits were used to calculate each BER data point. Channel coding was not used. The BER curves were calculated for two positions in the simulation scenario: \(\mathbf{R}_{1}=(2,1,1.8)\) m and \(\mathbf{R}_{2}=(4,1,1.8)\) m. These two positions were selected since they are geometrically symmetric with respect to TX, differing only in the shadowing component: \(R_{1}\) is heavily shadowed, while \(\mathbf{R}_{2}\) is not. Figure 10 presents the resulting BER curves, Figure 10a for 4-CSK, Figure 10b for 8-CSK, and Figure 10c for 16-CSK.