## 2 System Model  
The scenario under consideration comprises a SISO downlink VLC system. We consider a noncoherent LED transmitter, oriented horizontally downwards and installed on the ceiling of the ideal indoor environment at a vertical distance from the horizontal plane containing the receiver, made up of a PD pointing vertically upwards as shown in Figure 2. Furthermore, we assume that the LED emulates a uniformly distributed planar radiation source distributed throughout its area, such as \(A_{L}=w_{L}l_{L}\) with \(w_{L}\) and \(l_{L}\) being the sources distributed along the x-direction and the y-direction, respectively. By adhering to a widely accepted convention, each individual point on the transmitter's aperture exhibits a characteristic radiation pattern, conforming to the principles of Lambertian emission. This radiation pattern is characterized by a Lambertian order denoted as \(m\). This concept is of particular significance in the domain of optical and communication systems and contributes to the uniform distribution of light and signals emitted across space. The Lambertian radiation pattern captures the behavior of light emitted from a point source in a manner that adheres to [PERSON]'s cosine law. This law dictates that the radiant intensity of light varies with the cosine angle between the observer's line of sight and the direction of emission. By extension, the Lambertian radiation pattern encapsulates  
Figure 2: VLC scenario with optical propagation geometric scheme and IRS positioning. the way light disperses more broadly when emitted from a surface. The parameter \(m\) defines the degree of Lambertian behavior, ranging from fully Lambertian (\(m=0\)) to less Lambertian-like as \(m\) increases. Therefore, the radiance of a general point on the LED is defined as [23]:  
\[L_{(\theta_{L})}=\frac{P_{t}(m+1)}{2\pi A_{L}^{2}}\cos^{m-1}(\theta_{L}), \tag{1}\]  
where \(m=-1/\log_{2}[\cos(\phi_{1/2})]\), \(\phi_{1/2}\) is the half power beam-width, \(\theta_{L}\) is the radiance angle of the LED, and \(P_{t}\) is the power transmitted by the LED. On the other hand, we assume a vertically upward-oriented PD, which is based on hemispherical lenses that follow a Lambertian detection pattern with a respective FoV. The PD optical gain, denoted \(g(\theta_{P})\), is given as follows [24]:  
\[g(\theta_{P})= \begin{cases}\frac{\eta^{2}}{sin^{2}(\theta_{P})},&\text{if }0 \leq\theta_{P}\leq\Theta\\ 0,&\text{if }\theta_{P}\geq\Theta\end{cases} , \tag{2}\]  
where \(\eta\) is the reflective index of the PD, \(\theta_{P}\) is the incidence angle on the PD, and \(\Theta\) is the FoV. Once the LED emission and PD reception models have been defined, the LoS CIR of the VLC channel that governs the proposed scenario can be expressed as [25; 26]  
\[h(t)=\frac{A_{\text{eff}}(m+1)}{2\pi d^{2}}\cos^{m}(\theta_{L})\cos(\theta_{P} )\delta(t-\frac{d_{0}}{c}), \tag{3}\]  
where \(A_{\text{eff}}\) is the effective signal collection area for the LoS path, \(c\) is the speed of the light in free space, and \(\delta(t-\frac{d_{0}}{c})\) represents the signal propagation delay. In general, in real indoor scenarios, there are obstacles or obstructions that significantly affect the LoS component. In this context, a physical phenomenon that conditions these environments is shadowing. Due to its significant dependence on having a line of sight, the direct light component is very affected by the effect of this phenomenon. In general, people, immobile objects, and moving elements produce shadowing, so we consider it coherent to introduce an error probability coefficient in the mathematical expression of the LoS component, as follows [24; 27; 28]:  
\[h(t)_{sh}=h(t)P, \tag{4}\]  
where \(P\) is a weighting function that represents the probability that the LoS link will not be blocked [24; 27; 28]. This coefficient is a parameterized value between 0 and 1. That is, if there is a total blockage of the LoS link, its value will be 0. Otherwise, it will be 1 if there is no blockage.