## 4 Simulation and Analysis
### The Impact of System SNR on Position Detection Accuracy  
Assume that \(\omega\) is 2 mm and \(N\) is 1024. The channel noise parameters and the detector noise standard deviation remain unchanged, and the SNR will change when the input signal power is reduced gradually. Figure 8a below shows the relationship between the position detection accuracy and the SNR when the three spots are located at the center of the QD. Figure 8b is the relative error curve of Figure 8a. Figure 8c shows that the spots are at a different position, respectively, and its relative error curve is Figure 8d. It can  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Gaussian Spot** & \(\mathbf{e_{mean\_x1}}\) & \(\mathbf{e_{mean\_x2}}\) & \(\mathbf{e_{mean\_x3}}\) \\ \hline Same Position & 4.01\% & 3.61\% & 3.29\% \\ Different Positions & 3.91\% & 2.33\% & 4.51\% \\ \hline \hline \end{tabular}
\end{table}
Table 2: The average relative error between position detection accuracy value by simulation and value calculated by formula when changing the spot radius.  
Figure 7: (**a**) The influence of spot radius on position detection accuracy when the spots are at the same position. (**b**) The relative error curve when the spots are at the same position. (**c**) The influence of spot radius on position detection accuracy when the light spots are at different positions. (**d**) The relative error curve when the spots are at different positions.  
be seen from Figure 8a,c that the multi-spot position detection accuracy decreases as the SNR increases. In the actual system, the values of \(\omega\), \(\left(x_{j},y_{j}\right)\), and \(N\) will be limited. For example, when the SNR is 29.68 dB, the position detection accuracy of the spot position at the center of the detector is 1.847 \(\upmu\)m, 1.835 \(\upmu\)m, and 1.859 \(\upmu\)m, respectively. However, when the SNR is 31.74 dB, the detection accuracy of the spot position is 1.351 \(\upmu\)m, 1.354 \(\upmu\)m, and 1.389 \(\upmu\)m, respectively. This significantly improves the accuracy of position detection. It can be seen that if the SNR can be improved, then better spot position detection accuracy can be achieved.  
When the SNR is changed, the average relative error of the simulation value relative to the calculated value of the formula is shown in Table 3. As can be seen from Table 3, when the spots are at the same position, the AREs are 2.58%, 2.92%, and 2.43%. When the spots are at different positions, the AREs are 2.52%, 2.63%, and 2.75%, respectively. This indicates that the simulation value of the position detection accuracy is close to the mathematical model.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Gaussian Spot** & \(\mathbf{e_{meanR_{max}}}\)\_x1 & \(\mathbf{e_{meanR_{max}}}\)\_x2 & \(\mathbf{e_{meanR_{max}}}\)\_x3 \\ \hline Same Position & 2.58\% & 2.92\% & 2.43\% \\ Different Positions & 2.43\% & 2.63\% & 2.75\% \\ \hline \hline \end{tabular}
\end{table}
Table 3: The average relative error between position detection accuracy value by simulation and value calculated by formula when changing the SNR.  
Figure 8: (**a**) The influence of the SNR on position detection accuracy when the spots are at the same position. (**b**) The relative error curve when the spots are at the same position. (**c**) The influence of the SNR on position detection accuracy when the spots are at different positions. (**d**) The relative error curve when the spots are at different positions.