# A New COVID-19 Detection Method Based on CSK/QAM Visible Light Communication and Machine Learning
## 4 Methodology
### Model of a Channel Based on CSK/QAM Mapping
The explicit CIE xyY coordinates of the symbol \(\mathbf{s}_{k}\) are denoted as \(\mathbf{s}_{k}=[x_{k},y_{k}]^{\mathrm{T}}\). The symbol \(\mathbf{s}_{k}\) can be expressed as a radiant flux vector \(\Phi=[\Phi_{\mathrm{R}},\Phi_{\mathrm{G}},\Phi_{\mathrm{B}}]^{\mathrm{T}}\in \mathbb{R}_{\geq 0}^{3}\), where \(\Phi_{\mathrm{R}},\Phi_{\mathrm{G}},\Phi_{\mathrm{B}}\) represent the radiant fluxes to be emitted by the red, green and blue LEDs respectively. The radiant flux vector \(\Phi\) can be obtained by solving the system of Equation (8). Figure 4: 4-CSK constellation mapping into 4-QAM. Figure 3: General CSK/QAM assignment. \[x_{k} =\[PERSON]_{\rm R}x_{\rm R}+\[PERSON]_{\rm G}x_{\rm G}+\[PERSON]_{\rm B}x_{\rm B}\] \[y_{k} =\Phi_{\rm R}y_{\rm R}+\[PERSON]_{\rm G}y_{\rm G}+\[PERSON]_{\rm B}y_{\rm B} \tag{8}\] \[1 =\Phi_{\rm R}+\Phi_{\rm G}+\Phi_{\rm B}\]  
The centroid is determined by multiplying the four points \((x_{k},y_{k})(x_{\rm R},y_{\rm R})(x_{\rm G},y_{\rm G})\)\((x_{\rm B},y_{\rm B})\) by their components and \(\Phi\), which enables the transmission of white light between the points. For separator planes, such as using a support vector machine, it is simpler to map these four points in a square constellation. Using the cursors from the CSK, these points are mapped once more into a square regular M-QAM constellation. Data are processed by the M-QAM modulator, which then maps them into a plane with an in-phase and quadrature component. \[s=a+jb\ {\rm where}\ a,b\in\{\pm 1,\pm 3,\ldots,\pm(\lceil\sqrt{M}\rceil-1)\} \tag{9}\]  
Suppose we have \(k\) cursors according to Equation (A2), which indicates that all \(2^{2^{i}}\), \((i=3)\) M-QAM elements of constellation have been addressed. The fastest method would be to undergo transmission in a single cycle, but this is unnecessary because transmittion as a matrix \(2^{2^{i}}\times 2^{2^{i}}\), \((i=1)\) could occur, in which case the tables would be smaller according to \(\mathcal{G}\). Given a set of LEDs with \(2^{2^{i}}\) colours with the same characteristics, except for having different Semiconductor Photo Detectors (SPDs), given by \(\Phi_{1}(\lambda)\), \(\Phi_{2}(\lambda)\)... \(\Phi_{2^{2^{i}}}(\lambda)\) respectively, spatially grouped so that their positions in space can be approximated from a sufficiently large distance and DPs, each with a spectral response \(R(\lambda)\) and surface area \(A_{R}\) and filtered by an optical filter of one of the \(2^{2^{i}}\) colours with spectral gains of \(\mathcal{G}_{1}(\lambda)\), \(\mathcal{G}_{2}(\lambda)\)...\(\mathcal{G}_{2^{2^{i}}}(\lambda)\), respectively, and spatially grouped in such a way that spatial positions can be approximated from a sufficient distance, if the distance \(d\) between the LEDs and PDs is large enough that a single emitter and receiver position is a reasonable approximation, then the gain of the UM-VLC DC electro-optical channel from the \(i\)-th LED to the \(j\)-th PD with \(i,j=1,2,3,...2^{2^{i}}\) can be given by:  
\[H_{\rm el}[i,j]=H_{\rm LoS}^{\rm el}[i,j]+H_{\rm NLoS}^{\rm el}[i,j]\,, \tag{10}\]  
where \(H_{\rm LoS}^{\rm el}[i,j]\) is the DC gain of the LoS link, which can be expressed as:  
\[H_{\rm LoS}^{\rm el}[i,j] =\frac{1}{P_{T,i}}\int R(\lambda)\mathcal{A}_{0,j}(\lambda)\Phi_ {i}(\lambda)\,d\lambda \tag{11}\] \[\mathcal{G}=\Big{\{}a_{1}\mathbf{s}_{\mathrm{R}}+a_{2}\mathbf{s}_{\mathrm{C}} +a_{3}\mathbf{s}_{\mathrm{B}}\in\mathbb{R}^{2}\Big{\}};\quad a_{1}+a_{2}+a_{3 }=1\wedge\forall a_{1},a_{2},a_{3}\geq 0 \tag{7}\]0  
where \(P_{T,i}=\int\Phi_{i}(\lambda)\,d\lambda\).