## 2 System Model and Problem Formulation  
### System Model  
Consider a SISO VLC system with \(K\) tunable IRS units employing the IM/DD scheme, as depicted in Figure 1. The reflectance of the IRS units, denoted by \(\mathbf{r}=(r_{1},\ldots,r_{K})\), can be modulated coherently by the central control unit (CCU) with the transmit signal. The VLC signal, represented by \(X\), is emitted from the LED and travels to the photo-detector (PD) receiver through the LoS channel, the IRS reflected NLoS channels and the diffusely reflected NLoS channels. We ignore the diffusely reflected NLoS channels due to its negligible gain [1], so the received signal is formulated as  
\[Y=(h_{L}+\mathbf{r}^{T}\mathbf{h}_{NL})X+Z, \tag{1}\]where \(h_{L}\) is the direct channel (DC) gain of the LoS link, which can be calculated according to the geometric positions of the transceivers [8],  
\[h_{L}=\left\{\begin{array}{cc}\frac{A_{P}(m+1)}{2\pi a_{L}^{2}}\cos^{m}(\theta) T_{s}(\psi)g(\psi)cos(\psi),&0\leq\psi\leq\Psi_{c}\\ 0,&\text{Otherwise}\end{array}\right. \tag{2}\]  
where \(A_{P}\) is the area of the PD, and \(d_{L}\) is the distance between the LED and the PD. \(\theta\) is the angle of irradiance, and \(m=-1/\log_{2}(\cos(\Theta_{1/2}))\) is the Lambertian radiation index, where \(\Theta_{1/2}\) is the half radiation angle; \(\psi\) is the angle of incidence; \(\Psi_{c}\) is the semi-angle of the field of view (FoV); \(g(\psi)\) is the optical filter gain; and \(T_{s}(\psi)\) is the optical concentrator gain, which can be calculated by  
\[T_{s}(\psi)=\left\{\begin{array}{cc}\frac{n^{2}}{\sin^{2}\Psi_{c}}&0\leq\psi \leq\Psi_{c}\\ 0&\text{Otherwise}\end{array}\right., \tag{3}\]  
where \(n\) is the refractive index.  
\(\mathbf{h}_{NL}=(h_{1}^{NL},\ldots,h_{N}^{NL})\) represents all of the IRS-reflected NLoS path loss without reflection loss, where \(h_{k}^{NL}\) can be calculated by [8]  
\[h_{k}^{NL}=\left\{\begin{array}{cc}\frac{A_{P}(m+1)\cos^{m}(\theta_{R_{k}}^{ \odot})}{2\pi(\|R_{k}\|_{2}+\|R_{k}D\|_{2})^{2}}\frac{\cos(\theta_{R_{k}}^{ \odot})\cos(\theta_{R_{k}}^{\mathrm{MS}})}{\mathbf{N}_{k}^{T}\widehat{R_{k}D }}T_{s}(\cos(\theta_{R_{k}}^{D}))g(\cos(\theta_{R_{k}}^{D})),&0\leq\theta_{R_{ k}}^{D}\leq\Psi_{c}\\ 0,&\text{Otherwise}\end{array}\right. \tag{4}\]  
where, \(\|R_{k}S\|_{2}\) and \(\|R_{k}D\|_{2}\) represent the distances from the \(k\)-th IRS unit to the LED (S) and the PD (D), respectively. \(\theta_{R_{k}}^{S}\) is the angle of irradiance from the LED to the \(k\)-th IRS unit. \(\theta_{R_{k}D}^{\mathrm{MS}}\) is the angle of irradiance from the \(k\)-th IRS unit to the PD, where \(\hat{N}_{k}^{T}\) represents the unit normal vector of the \(k\)-th IRS unit. \(\theta_{R_{k}}^{D}\) is the angle of incidence from the \(k\)-th IRS unit to the PD. \(\widehat{R_{k}D}\) is the unit vector from the \(k\)-th IRS unit to the PD. Thus, \(\mathbf{r}^{\mathrm{T}}\mathbf{h}_{NL}\) is the IRS reflected channel gain. \(Z\sim N(0,\sigma^{2})\) is the additive white Gaussian noise.  
Due to LEDs' lighting property, real-valued signals are modulated on the intensity of the LED light, so a non-negative constraint and a peak intensity constraint are imposed on the amplitude as  
\[0\leq X\leq A, \tag{5}\]  
Figure 1: A tunable IRS-aided SISO VLC System.  
where \(A\) is the peak light intensity of the LED. The actual perceived illumination intensity of human eyes depends on the expectation of \(X\), which is supposed to be adjusted according to the user's dimming requirement:  
\[E[X]=\xi A, \tag{6}\]  
where \(0\leq\xi\leq 1\) is the dimming coefficient.