## 3 Detection Precision and Sensitivity Analysis
### Precision Analysis of Quadrant Detector under Beacon Spread-Spectrum Modulation
At the same time, since all four \(p\)-n junction photodiodes of the QD need to receive the beacon light simultaneously for the correct spot output position, the centroid position \(x_{0}\) of the spot must be less than the radius \(\omega\) of the optical plate when the model is valid. When the centroid of the light spot is positioned at the origin of the detector, the spot radius is 0.7 mm, and \(R_{\text{SNRE}}\) = 20 dB, according to Equations (15) and (17). The spreading gain \(G_{P}\) of the system and the precision of the position detection performed by the QD at the center position are shown in Table 2. PRN depths of 1023, 511, 255, 127, 63, 31, and 1, respectively, represent pseudo-random sequence periods of 1023, 511, 255, 127, 63, and 31 for the modulation of communication data, in addition to direct detection without spread-spectrum technology. \(\sigma\) represents the theoretical precision of the position detection performed by the QD according to Equation (17). \(\sigma_{\mathrm{C_{p}=1}}/\sigma\) represents the theoretical ratio of the position-detection accuracy achieved by the QD when modulating the PRN at different depths to that of the unspread spectrum, which is used to characterize the optimization of the position-detection precision. From Table 2, it can be inferred that the precision of position detection achieved by the detector without spread-spectrum technology is 100\(\upmu\)m. When the modulation PRN period is 1023, the precision of position detection achieved by the QD at the center position is 3.12 \(\upmu\)m. In this case, the precision of the position detection under spread-spectrum technology is 15.05 dB higher than the precision of the position detection achieved without spread-spectrum technology. Therefore, adopting beacon spread-spectrum modulation significantly improves the precision of position detection. When the \(R_{\mathrm{SNRE}}\) is within the range of 10 dB to 30 dB, as the depth of modulation of different spreading sequences increases from 31 to 1023, the simulated performance of the QD position detection is as follows:  
It is evident from Figure 2 that when the spreading gain is kept constant, increasing the SNR effectively improves the precision of position detection achieved by the detector. At the same SNR, as the spreading gain increases, i.e., the depth of the modulating PRN becomes greater, the precision of the position detection achieved by the QD improves. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Pseudocode Sequence Depth** & **1** & **31** & **63** & **127** & **255** & **511** & **1023** \\ \hline \(G_{P}\) (dB) & 0 & 14.91 & 17.9 & 21.03 & 24.06 & 27.08 & 30.09 \\ \hline \(\sigma\) (\(\upmu\)m) & 100 & 17.96 & 12.59 & 8.87 & 6.26 & 4.42 & 3.12 \\ \hline \(\sigma_{\mathrm{C_{p}=1}}/\sigma\) (dB) & 0 & 7.45 & 8.99 & 10.52 & 12.03 & 13.54 & 15.05 \\ \hline \end{tabular}
\end{table} 
Table 2: When \(R_{\mathrm{SNRE}}\) = 20 dB, the precision of the position detection achieved by the QD at the center position under different spreading gains. Figure 2: The various levels of position-detection accuracy achieved by the detector under different spreading gains and SNRs.