## 2 Theoretical Formula Derivation
### Derivation of Ideal BER  
In this section, the formula for the system BER under ideal LED emission conditions is derived. The relationship between the emission power of an LED and its driving current can be described through the optoelectronic characteristics of the LED. These optoelectronic characteristics can be expressed by the following equation:  
\[P_{LED}=I_{LED}\times U_{LED}, \tag{7}\]  
Here, \(P_{LED}\) represents the emission power of the LED, \(I_{LED}\) denotes the driving current, and \(U_{LED}\) stands for the operating voltage.  
The emission characteristics of LED in the ideal state are illustrated in Figure 2.  
The theoretical emitted light energy for symbols 1 and 0 within a unit symbol time can be obtained with the following formula:  
\[IL_{zero}\cdot T=\eta\cdot\int\limits_{t=T}^{t=2T}I_{off}(t)Vdt, \tag{8}\]  
\[IL_{one}\cdot T=\eta\cdot\int\limits_{t=0}^{t=T}I_{on}(t)Vdt, \tag{9}\]In the above formulas, _IL_ is defined as the luminous power of the LED per unit time. _IL-T_ represents the optical energy radiated during a single symbol time. The duration of symbol emission \(T\) represents the duration of light emission for a single symbol in the system and is related to the system's communication rate. \(\eta\) represents the electro-optical conversion efficiency of the LED, \(I(t)\) denotes the current at a given time \(t\), and \(V\) stands for the voltage applied to the LED.  
For a UV communication system, the sources of photons for symbol 0 mainly include the following three aspects: 1 PMT dark count noise; 2 ambient light noise; 3 LED emission tail noise. Therefore, for an ideal UV communication system, the mean expressions for the number of received photons for symbols 1 and 0 at the receiver are as follows:  
\[\lambda_{s}=N_{dk}+N_{lt}+IL_{one}\cdot T\cdot\alpha/E_{photon}, \tag{10}\]  
\[\lambda_{b}=N_{dk}+N_{lt}+IL_{zero}\cdot T\cdot\alpha/E_{photon}, \tag{11}\]  
In the above expression, \(N_{dk}\) represents the number of photons generated by the dark current noise introduced into the communication system per symbol, and \(N_{lt}\) represents the number of photons generated by introduced light noise per symbol. \(E_{photon}\) denotes the energy of a single UV photon, \(\alpha\) represents the system attenuation coefficient.  
By obtaining the values of \(\lambda_{s}\) and \(\lambda_{b}\) from the above equation, and substituting them into the Poisson distribution expression, it is possible to describe the probability of symbols 1 and 0 generating a certain number of received photons \(n\) at the receiver.  
During the signal recovery process, it is necessary to determine the threshold for symbol 0/1 decision. The decision threshold is related to the Poisson distribution probability expression. The BER can then be expressed as follows:  
\[BER(x_{0})=\frac{1}{2}(\sum_{n=x_{0}+1}^{\infty}P_{zero}(n))+\frac{1}{2}(1- \sum_{n=x_{0}+1}^{\infty}P_{one}(n)), \tag{12}\]  
Here, \(x_{0}\) represents the 0/1 decision threshold, and \(BER(x_{0})\) denotes the bit error rate performance at the \(x_{0}\) threshold. By calculating the above equation, the BER of the UV communication system can be obtained in the ideal scenario where there is no LED emission tailing effect.  
Figure 2: Current variation curve of an LED in ideal operation mode.