## 2 Principle and Methods
 \tag{8}\]  
where \(R_{j}\) represents the correlation value between the received signal and the PRN in the \(j\)-th quadrant, \(R_{sj}(\tau)\) represents the correlation value between the \(j\)-th quadrant beacon signal and the PRN, and \(R_{nj}(\tau)\) represents the correlation value between the noise in the \(j\)-th quadrant and the PRN. After the correlation operation, the main purpose of the correlation calculator when used for the Punctual code is data recovery and position calculation. The difference between the results of the correlation calculator regarding the Early code and the Late code is used to control the error signal in the loop. By using a low-pass filter (LFP) to control the phase delay of the input signal in the code generator update, the local pseudo-random sequence can track the phase of the input signal. This ensures that when the phase of the local PRN matches the phase of the input PRN signal (that is, \(\tau=0\)), the correlation result between the Punctual code in the local pseudo-random sequence and the received photoelectric signal can be obtained according to Equation (3):  
\[R_{sj}(\tau)=\frac{\eta\cdot p_{j}\cdot d(t)}{T_{c}}\int_{0}^{T_{c}}c(t)c(t+ \tau)dt=\eta\cdot p_{j}\cdot d(t) \tag{9}\]Therefore, in the \(j\)-th quadrant, the cross-correlation result between the received signal and the local pseudo-random sequence can be expressed as follows:  
\[R_{j}(\tau)=\eta\cdot p_{j}\cdot d(t)+\frac{1}{T_{c}}\int_{0}^{T_{c}}n(t)c(t+ \tau)dt \tag{10}\]  
Due to the lack of correlation between noise and the local PRN, according to Equation (10), the correlation operation between the noise and the Punctual code is significantly smaller than the correlation between the photoelectric signal and the Punctual phase of the local pseudo-random sequence. Therefore, the cross-correlation operation of the despreading algorithm effectively suppresses the noise and improves the SNR of the output signal. Considering the absolute value of the result after correlation eliminates the influence of the polarity of the communication data encoding \(d(t)\) Via substitution into the formula used to calculate the center of the optical spot [31], the relative position of the optical spot under the beacon spread-spectrum modulation detected by QD is as follows:  
\[ \begin{array}{l}x_{r}=\frac{(P_{\text{I}}+P_{\text{IV}})-(P_{\text{II}}+P_{ \text{III}})}{P_{\text{I}}+P_{\text{II}}+P_{\text{III}}+P_{\text{IV}}}=\frac{(| R_{\text{I}}|+|R_{\text{IV}}|)-(|R_{\text{II}}|+|R_{\text{III}}|)}{|R_{\text{I}}|+|R_{ \text{II}}|+|R_{\text{IV}}|+|R_{\text{IV}}|}\\ y_{r}=\frac{(P_{\text{II}}+P_{\text{II}})-(P_{\text{III}}+P_{\text{IV}})}{P_{ \text{I}}+P_{\text{II}}+P_{\text{III}}+P_{\text{IV}}}=\frac{(|R_{\text{I}}|+| R_{\text{II}}|)-(|R_{\text{II}}|+|R_{\text{IV}}|)}{|R_{\text{I}}|+|R_{\text{II}}|+|R_{ \text{IV}}|+|R_{\text{IV}}|}\end{array}  \tag{11}\]  
where \((x_{r},y_{r})\) represents the relative position coordinates of the optical spot center; \(P_{\text{I}}\), \(P_{\text{II}}\), \(P_{\text{III}}\) and \(P_{\text{IV}}\), respectively, denote the received optical power in each quadrant of the QD; and \(R_{\text{I}}\), \(R_{\text{II}}\), \(R_{\text{III}}\) and \(R_{\text{IV}}\), respectively, represent the correlation values between the Punctual code and the received signal. Therefore, it can be inferred from Equations (8)-(11) that applying spreading technology to QD beacon light detection effectively suppresses background light and detector noise during despreading operations. This enhances the SNR of the detection position signal, thereby improving the precision of optical spot position detection and reducing the bit error rate of communication data.