# 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  
Due to its low cost and simplicity, intensity modulation with direct detection (IM/DD) is used in the majority of VLC systems. The transmitters in this kind of system are LEDs, and the instantaneous optical power \(\Phi_{c}(t)\) is modulated in proportion to the driving electrical current \(i_{t}(t)\), which is modulated in accordance with the data to be broadcast. The optical power signal travels down the channel and eventually reaches the receiver's surface, which is often a photodiode, also known as a photodetector (PD). In the photodiode, the received optical power causes a proportionate photocurrent, \(i_{r}(t)\). In this study, the Bit Error Rate (BER) was calculated using the Minimum Mean Square Error (MMSE) estimator for a variety of MIMO arrays [73]. In order to send the DNA samples that are in a sewer, an uplink process must be created. This assumes that there are \(N_{t}\) light sources capable of transmitting the signal and a receiving device with \(N_{r}\) Photo detector (PDs):  
\[\mathbf{y}= \begin{pmatrix}h_{1,1}&\ldots&h_{1,N_{t}}\\ &\ldots&\\ h_{n_{r},1}&\ldots&h_{n_{r},N_{t}}\end{pmatrix} \mathbf{x}+\mathbf{n}\, \tag{6}\]  
where \(\mathbf{y}\) is the \(N_{r}\times 1\) vector representing the signal received at each PD, \(\mathbf{x}\) is the transmitted signal vector of size \(N_{t}\times 1\), \(h_{ij}(i=1,\ldots,N_{r}\quad and\quad j=1,\ldots,N_{t})\) are the channel gain of the link between the \(j\)-th transmitter and the \(i\)-th PD, and \(\mathbf{n}\) is the \(N_{r}\times 1\) vector representing the noise at each PD, including all possible noise, which can be expressed as Equation (5). Furthermore, the mapping of an M-CSK constellation to an M-QAM constellation is proposed. Figure 2 represents an xyY diagram of the Commission Internationale de l'Eclairage (CIE) colour space from 1931. This was the first colour space based on experimental results of human colour perception. A colour space is a multidimensional collection of all colours that a certain colour model can generate. Historically, the IEEE 802.15.7 standard defines some guidelines for designing M-CSK constellations and directly applying them for modulation [74]. Figure 2: The xyY diagram of the Commission Internationale de l’Éclairage. Figure 3 illustrates the fundamental principle of mapping for the chromosomal length \(N^{DNA}\). Depending on the length of the DNA, \(N^{DNA}=2^{2^{i}}\) points in the M-CSK constellation are selected and mapped to the M-QAM constellation, which is depicted on the right side of Figure 3 as clouds of points, with a total of \(2^{2^{i}}\) points \(i=0,1,2,3\ldots\) shown. Figure 4 represents 4-CSK constellation mapping into 4-QAM. Although points in the CIE 1931 xyY space can be assigned in any arbitrary order, the four 4-CSK stars correspond to the four 4-QAM stars. Due to their proximity, it is more difficult to pass separator hyperplanes to detect the four points of the 4-CSK constellation. On the right is a 4-QAM constellation with more evenly spaced stars, which makes it simpler to deliver separator planes to them. The centre of the colour bands can be expressed in CIE 1931 xyY space coordinates as \(\mathbf{s}_{\mathrm{R}},\mathbf{s}_{\mathrm{G}},\mathbf{s}_{\mathrm{B}}\in \mathbb{R}^{2}\), which is known as the centre of band symbols. All colours that can be reproduced by LEDs via additive mixing form a triangle in CIE xyY space with the vertices \(\mathbf{s}_{\mathrm{R}},\mathbf{s}_{\mathrm{G}}\) and \(\mathbf{s}_{\mathrm{B}}\). The gamut of the system is the set of all colours that may be reproduced by the three LEDs and is mathematically defined as the convex combination of the centre of band symbols in the CIE xyY space, as shown in  
\[\mathcal{G}=\Big{\{}a_{1}\mathbf{s}_{\mathrm{R}}+a_{2}\mathbf{s}_{\mathrm{C}} +a_{3}\mathbf{s}_{\mathrm{B}}\in\mathbb{R}^{2}\Big{\}};\quad a_{1}+a_{2}+a_{3 }=1\wedge\forall a_{1},a_{2},a_{3}\geq 0 \tag{7}\]  
where \(\mathcal{G}\) denotes the system's gamut.