## 2 Theoretical Formula Derivation
### Derivation of BER with Changes in Transmission Signal Duty Cycle  
This paper proposes a scheme to mitigate the impact of trailing effects by changing the transmission signal duty cycle. The duty cycle can be calculated using the following formula:  
\[D=\frac{PW}{T}\times 100\%, \tag{20}\]  
Figure 4: The current variation curve during LED emission tailing.  
where \(PW\) represents the duration of the high level of a signal, \(T\) denotes the total time of one complete period, and \(D\) represents the duty cycle of the signal.  
The specific implementation of the proposed scheme is to appropriately reduce the duty cycle of LED emission, i.e., to decrease the duration of high-level emission of the ultraviolet LED chip. The untreated duty cycle of LED emission is generally 50%. Building upon OOK modulation, the duty cycle of symbol \(1\) can be shortened to \(\zeta\). At this stage, the formula for the LED's luminous energy per symbol unit is expressed as follows:  
\[IL_{one}(\xi)\cdot T=(\eta V/R)\cdot T-\eta(\int\limits_{t=0}^{t=l_{\xi}}I_{ on}(t)Vdt)+\eta(\int\limits_{t=l_{\xi}+T}^{t=2T}I_{off}(t)Vdt), \tag{21}\]  
\[IL_{zero}(\xi)\cdot T=\eta(\int\limits_{t=0}^{t=l_{\xi}}I_{on}(t)Vdt)+\eta( \int\limits_{t=l_{\xi}+T}^{t=2T}I_{off}(t)Vdt), \tag{22}\]  
The ratio of this noise intensity to the signal light intensity is expressed as follows:  
\[\begin{array}{ll}\sigma(\xi)&=\dfrac{IL_{zero}(\xi)\cdot T}{IL_{one}(\xi) \cdot T}=\dfrac{\eta(\int\limits_{t=0}^{t=l_{\xi}}I_{on}(t)Vdt)+\eta(\int \limits_{t=l_{\xi}+T}^{t=2T}I_{off}(t)Vdt)}{(\eta V/R)\cdot T-\eta(\int \limits_{t=0}^{t=l_{\xi}}I_{on}(t)Vdt)+\eta(\int\limits_{t=l_{\xi}+T}^{t=2T}I_ {off}(t)Vdt)}\\ &\approx\dfrac{\eta(\int\limits_{t=0}^{t=l_{\xi}}I_{on}(t)Vdt)+\eta(\int \limits_{t=l_{\xi}+T}^{t=2T}I_{off}(t)Vdt)}{(\eta V/R)\cdot T},\end{array} \tag{23}\]  
Utilizing the BER formula associated with the Poisson distribution probability formula, the variation curve of the BER when the duty cycle changes can be calculated.  
By adjusting the duty cycle, it is possible to reduce the luminous intensity of symbol \(0\), thereby reducing the probability of misjudgment, as shown in Figure 5. However, at the same time, the luminous intensity of symbol \(1\) will also decrease as the duty cycle decreases. Therefore, for specific rates, there exists an optimal duty cycle selection position. Additionally, since the RC charging and discharging time is fixed, the influence of trailing effects on the BER will diminish for lower rates. In such cases, the choice of duty cycle tends to approach 50%. In other words, changing the duty cycle has a smaller impact on lower rates.  
Figure 5: The conduction current variation curve of LED when the duty cycle changes.  
The theoretical derivation in this section reveals that, under a specific rate, improving the duty cycle of the transmitted signal can ameliorate the degradation in BER caused by the LED's trailing effect in UV communication, thereby enhancing communication stability and quality. In the subsequent chapters, Sections 3 and 4, we will verify and analyze this scheme through computer simulation and outdoor field hardware experiments, respectively.