## 2 Principle and Methods
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variable Name** & **Description** & **Variable Name** & **Description** \\ \hline \(P_{s}\) & Average optical power output of the laser & \(I_{j}(t)\) & \(j\)-th quadrant \\ \hline \(d(t)\) & Communication data & \(j\) & Takes values of I, II, III, and IV \\ \hline \(c(t)\) & PRN data & \(\eta\) & The responsivity of the QD \\ \hline \(S_{t}(t)\) & Output optical power signal of the laser & \(P_{j}\) & \(j\)-th quadrant \\ \hline \(R(\tau)\) & Autocorrelation value & \(n_{j}(t)\) & \(j\)-th quadrant \\ \hline \(T_{c}\) & Period of communication data bits & \(K\) & Transimpedance amplifier gain of the circuit \\ \hline \(T_{s}\) & Bit period of the PRN & \(V_{j}(t)\) & \(j\)-th quadrant photoelectric signal collected by the ADC \\ \hline \(\tau\) & Clock delay between pseudo-random codes & \(R_{j}\) & \(j\)-th quadrant \\ \hline \(i\) & A non-zero integer & \(R_{sj}(\tau)\) & Correlation value between the j-th quadrant \\  & & beacon signal and the PRN \\ \hline \(N\) & The depth of the PRN periodic sequence & \(R_{nj}(\tau)\) & \(j\)-th quadrant and the PRN \\ \hline \(n\) & Bit width of the PRN register & \((x_{r},y_{r})\) & The relative position coordinates of the \\  & & & optical spot center \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Names and descriptions of the variables in Section 2. Therefore, when \(R(\tau)\) takes a non-integer value, the value of the correlation result changes linearly between \(R(iT_{c})\) and \(R((i+1)T_{c})\). The receiver uses a QD to detect the beacon light, which consists of four identical \(p\)-n junction photodiodes. When the QD receives the beacon light, each quadrant independently outputs a photocurrent that is proportional to the received optical power. The position of the optical spot is estimated by the ratio of the amplitudes of the photocurrent signals from the four quadrants. The photocurrent outputs from each quadrant are represented as follows:  
\[I_{j}(t)=\eta\cdot P_{j}\cdot d(t)\cdot c(t) \tag{6}\]  
where \(j\) takes values of I, II, III, and IV, representing the four quadrants of the quadrant detector; \(I_{j}(t)\) represents the photocurrent signal collected by the j-th quadrant; \(\eta\) represents the responsivity of the QD; and \(P_{j}\) represents the optical signal power received by the \(j\)-th quadrant. The weak photocurrent signals are amplified into voltage signals using the transimpedance amplifier (TIA), and the amplified signals are then collected using an ADC. The voltage signals collected from each quadrant are as follows:  
\[V_{j}(t)=K\cdot\eta\cdot p_{j}\cdot d(t)\cdot c(t)+n_{j}(t) \tag{7}\]  
where \(V_{j}(t)\) represents the \(j\)-th quadrant photoelectric signal collected by the ADC, \(K\) is the transimpedance amplifier gain of the circuit, and \(n_{j}(t)\) represents the background light noise and detector noise received by the \(j\)-th quadrant, both of which are Gaussian white noise [30]. The digital signal processing unit is used to receive the signal, despread it, and output the position of the detected optical spot on the QD and the beacon communication data. Initially, the receiver constructs digital delay-locked loops for each quadrant's collected voltage signals in order to generate three sets of local PRNs: the Early code, the Punctual code, and the Late code; these are identical to the modulated PRN at the transmitter and differ in phase by half a chip. Then, each set is subjected to cross-correlation operations that are performed with the received voltage signal using a correlator. The process of cross-correlation between the local pseudo-random sequence and the received signal can be expressed as follows:  
\[ \begin{array}{ll}R_{j}(\tau)&=E\big{[}V_{j}(t)\cdot c(t+\tau)\big{]}=\frac{1 }{T_{c}}\int_{0}^{T_{c}}K\cdot\eta\cdot p_{i}\cdot d(t)c(t)\cdot c(t+\tau)+n( t)\cdot c(t+\tau)dt\\ &=R_{sj}(\tau)+R_{nj}(\tau)\\ &\end{array}