## 2 Principle and Methods  
This section analyzes the mechanism of spread-spectrum technology when employed in QD spot position detection and communication. A block diagram of the QD system utilized for beacon spread-spectrum modulation is shown in Figure 1. Communication data are spread-spectrum-modulated by PRNs. This involves performing an exclusive OR (XOR) operation between the communication data and a pseudo-random code. Subsequently, the spread-spectrum data are modulated onto the laser, thereby enabling the transmission of laser signals in the space link. The output optical power signal after modulation is represented by the following:  
\[S_{t}(t)=P_{s}\cdot d(t)\ XOR\ c(t) \tag{1}\]  
where \(S_{t}(t)\) represents the output optical power signal of the laser, \(P_{s}\) represents the average optical power output of the laser, \(d(t)\) represents the communication data used for the beacon, and \(c(t)\) represents the PRN data. At the same time, the names and descriptions of the variables in Section 2 are listed in Table 1.  
Figure 1: Block diagram of QD system for beacon spread-spectrum modulation. ADC represents the Analog-to-Digital Converter. TIA represents the transimpedance amplifier, and LPF represents the low-pass filter. Early, Punctual, and Late represent the Early code, the Punctual code, and the Late code. \(R_{I}\) represents the correlation values between the Punctual code and the received signal. The PRN is a deterministic sequence that exhibits statistical properties similar to random noise. It possesses good autocorrelation characteristics and relatively weak cross-correlation characteristics. This article refers to the PRN design of Global Positioning System (GPS) during the design of pseudo-random codes and selects Gold sequences as the spreading codes [24; 25; 26]. The Gold sequence has small cross-correlation values, and both its autocorrelation and cross-correlation functions are bounded. Its autocorrelation function is expressed as follows [27; 28; 29]:  
\[R(\tau)=\frac{1}{T_{c}}{\int_{0}^{T_{c}}{c_{i}(t)c_{i}(t+\tau)dt}},\tau\in(-T_ {s}/2,T_{s}/2) \tag{2}\]  
where \(R(\tau)\) represents the autocorrelation value of the PRN, \(T_{c}\) represents the period of communication data bits, \(T_{s}\) represents the bit period of the PRN, and \(\tau\) represents the clock delay between PRNs. According to the definition of correlation, when \(\tau=0\) (that is, when the phases between the PRN sequences are aligned), we can obtain the following:  
\[R(0)=\frac{1}{T_{c}}{\int_{0}^{T_{c}}{c(t)c(t)dt}}=\frac{1}{T_{c}}{\int_{0}^{ T_{c}}{c^{2}(t)dt}}=1 \tag{3}\]  
when \(\tau\
eq 0\), if \(\tau=iT_{c}\) and \(i\) is a non-zero integer, \(\tau\) is the integer multiple of the chip length, \(n\) is the bit width of the PRN register, and \(N\) represents the depth of the PRN periodic sequence, Thus, according to Gold theory, the results of the cross-correlation triples are as follows:  
\[R_{i}(iT_{c})=\left\{\frac{-1}{N},\frac{-1-2^{\lfloor(n+2)/2\rfloor}}{N}, \frac{2^{\lfloor(n+2)/2\rfloor}-1}{N}\right\} \tag{4}\]  
when \(iT_{c}<\tau<(i+1)T_{c}\) (that is, when \(\tau\) is not an integer multiple of the chip length), the cross-correlation result is given by \(\Delta\tau=\tau-iT_{c}\):  
\[R(\tau)=R(iT_{c})\left(\frac{\Delta\tau}{T_{c}}\right)+R((i+1)T_{c})\left( \frac{T_{c}-\Delta\tau}{T_{c}}\right) \tag{5}\]