# A Python-Based Indoor Channel Model with Multi-Wavelength Propagation for Color Shift Keying
## 1 Introduction
\begin{bmatrix}x_{d}&y_{d}\end{bmatrix} = \begin{bmatrix}P_{i}&P_{j}&P_{k}\end{bmatrix}  _{d}\cdot \begin{bmatrix}x_{i}&y_{i}\\ x_{j}&y_{j}\\ x_{k}&y_{k}\end{bmatrix}  \tag{1}\]  
where \((x_{i},y_{i})\) are the intrinsic color coordinates of band i, \((x_{j},y_{j})\) are the intrinsic color coordinates of band j and \((x_{k},y_{k})\) are the intrinsic color coordinates of band k. Based on this linear relation, the color-to-intensity encoder computes the optical power in each band to generate a specific color by including a constraint of constant total optical power, which can be expressed as  
\[P_{i}+P_{j}+P_{k}=1. \tag{2}\]  
CSK is a promising modulation scheme within the realm of VLC, primarily due to its ability to enhance spectral efficiency by utilizing wavelength division multiplexing. This enables the transmission of more data within a given bandwidth, resulting in higher data rates, making CSK suitable for applications requiring substantial data throughput [4]. Regarding lighting performance, one of its key strengths lies in its capacity to modulate information by varying the intensity of multiple colored LEDs, typically red, green, and blue, without altering the overall luminous flux. This ensures non-flickering light, providing a seamless user experience for illumination while facilitating efficient data transmission. A distinct feature of CSK is its ability to deliver tunable perceived color by modifying the constellation design, which allows for adjustment of the light's color temperature to meet various aesthetic or functional requirements in indoor environments. This flexibility in color tuning makes CSK particularly well suited for smart indoor lighting systems, where both communication and lighting quality are paramount. Furthermore, CSK schemes can achieve a color rendering index (CRI) greater than 80 [5], ensuring that the lighting accurately represents the colors of objects within the space. This is a critical factor in environments where high-quality lighting is essential, such as retail spaces, healthcare facilities, and art galleries. For indoor VLC applications, the power received by the photodetector depends on both line-of-sight (LoS) and non-line-of-sight (NLoS) components. The spectral power distribution (SPD) of the NLoS component is modified according to the spectral reflectance of the objects and walls inside the indoor space. Due to the wide spectral emission of LEDs and the wide spectral response photodetectors, the power radiated by an LED at a central wavelength is detected by the photodetectors at different wavelengths. This effect is known as inter-channel interference, or cross-talk [6]. According to [2], the inter-channel interference in CSK can be modeled as a gain matrix, and determines the communication performance of the communication system; on the other hand, VLC systems for indoor applications must fulfill lighting quality requirements such as illumination level, color temperature, and color quality. These parameters also depend on the SPD of the light received from LoS and NLoS. Therefore, an adequate analytical estimation of cross-talk and lighting parameters for indoor CSK systems must consider both LoS and NLoS components. Different channel models are widely used for indoor applications. More specialized channel models can compute the channel impulse response (CIR) function, which completely characterizes the channel performance. These models consider realistic geometries such as walls, objects, people, and furniture [7; 8; 9; 10]. Simpler models only calculate some characteristics for empty rooms with rectangular, cylindrical, or spherical shapes [11; 12; 13; 14; 15; 16]. Concerning the models used to study the CSK performance, several limitations of CSK  
Figure 1: Functional block diagram of the CSK modulation format. Figure 2: Color point mapping for 4-CSK, 8-CSK, and 16-CSK.