## 2 System Model and Analysis
### The Total Power Consumption  
In the VLC system via mirror array-based IRS, the total power consumption is composed of the transmit power, the hardware static power consumed in the transmitter and receiver, and IRS. The mirrors do not consume any transmit power since they are passive elements. The total power consumption model is shown in Figure 3.  
In the system, the purpose of the mirror is to get a suitable position by rotation and reflect the emitted light to PD. Therefore, the hardware static power of the mirror array-based IRS is mainly used to control the rotation angle of mirrors. According to Formula (4), the rotation angle of the mirror is related to the normal vector. The normal vector depends on the distance between each mirror and the transmitter or receiver. Because the interval of mirrors is much smaller than the distance between mirrors and transmitter or receiver, the difference of mirrors rotation angle in IRS is relatively small. Therefore, the power consumption for each mirror rotation can be regarded as the same. Based on the above considerations, the total power consumption of IRS-assisted VLC system can be expressed as  
\[P_{total}=P_{x}+P_{hsp}+NP_{m}=\varepsilon+I_{DC}^{2}+P_{hsp}+NP_{m}, \tag{24}\]  
Figure 3: The total power consumption model in the VLC system via mirror array-based IRS. Then, for (31) to hold, it is sufficient that  
\[\frac{\sum_{n=1}^{N}H_{n}}{1+\delta\Big{(}\sum_{n=1}^{N}H_{n}\Big{)}^{2}}\geq\frac {\sum_{n=1}^{N+1}H_{n}}{1+\delta\Big{(}\sum_{n=1}^{N+1}H_{n}\Big{)}^{2}}. \tag{33}\]  
If we set \(t=\sum_{n=1}^{N}H_{n}\), the inequality (33) can be written as an equivalent function  
\[f(t)=\frac{t}{1+\delta t^{2}}. \tag{34}\]  
If _f(t)_ is a monotonically decreasing function, then  
\[\frac{d(f(t))}{dt}=\frac{1}{1+\delta t^{2}}-\frac{2\delta t^{2}}{\left(1+ \delta t^{2}\right)^{2}}=\frac{1-\delta t^{2}}{\left(1+\delta t^{2}\right)^{2 }}\leq 0. \tag{35}\]  
So, when \(\delta\Big{(}\sum_{n=1}^{N}H_{n}\Big{)}^{2}\geq 1\), the inequality (33) can hold and \(R(N+1)-R(N)\geq R(N+2)-R(N+1)\). Hence the proof follows. \(\square\)  
Proposition 2: \(EE(N)\) _in (27) is a unimodal function._  
Proof of Proposition 2: Under the previously considered constraints of \(N\), \(R(N)\) is an increasing function that grows more and more slowly. For the denominator in the Formula (27), \(P_{m}\ll P_{else}\). \(NP_{m}+P_{else}\) increases with increasing of \(N\), and the growth rate becomes slower and slower. For \(EE(N)\), its changes are divided into two cases:  
1. It keeps increasing with the increasing of \(N\). The peak value of EE will not appear within the range of \(N\);
2. There exists an \(N^{\prime}\), \(EE(N)\) decreases monotonically when \(N\geq N^{\prime}\). At this time, \(EE(N^{\prime})\geq EE(N^{\prime}+1)\geq EE(N^{\prime}+2)\). When \(EE(N^{\prime})\geq EE(N^{\prime}+1)\), it holds \[\frac{R(N^{\prime})}{N^{\prime}P_{m}+P_{else}}\geq\frac{R(N^{\prime}+1)}{(N^{ \prime}+1)P_{m}+P_{else}},\] (36) and \[N^{\prime}\leq-\frac{P_{else}}{P_{m}}+\frac{R(N^{\prime})}{R(N^{\prime}+1)-R( N^{\prime})}.\] (37) Since \(R(N^{\prime})=-(R(N^{\prime}+1)-R(N^{\prime}))+R(N^{\prime}+1)\), (37) can be rewritten as \[N^{\prime}\leq-\frac{P_{else}}{P_{m}}-1+\frac{R(N^{\prime}+1)}{R(N^{\prime}+1)- R(N^{\prime})}.\] (38)  
In Proposition 1, we proof that when \(\delta\Big{(}\sum_{n=1}^{N}H_{n}\Big{)}^{2}\geq 1\), \(R(N+1)-R(N)\geq R(N+2)-R(N+1)\). So, the in Equation (38) implies  
\[N^{\prime}\leq-\frac{P_{else}}{P_{m}}-1+\frac{R(N^{\prime}+1)}{R(N^{\prime}+2 )-R(N^{\prime}+1)}. \tag{39}\]  
We have  
\[P_{total}=P_{x}+P_{hsp}+NP_{m}=\varepsilon+I_{DC}^{2}+P_{hsp}+NP_{m}, \tag{24}\]0  
It means that \(EE(N^{\prime}+1)\geq EE(N^{\prime}+2)\). So, if \(EE(N^{\prime})\geq EE(N^{\prime}+1)\), it can be proved that \(EE(N^{\prime})\geq EE(N^{\prime}+1)\geq EE(N^{\prime}+2)\geq EE(N^{\prime}+3) \geq EE(N^{\prime}+\cdots)\). To sum up, \(EE(N)\) is either monotonically increasing, or there exists an \(N^{\prime}\), with \(EE(N)\) monotonically decreasing when \(N\geq N^{\prime}\).