## 3 Detection Precision and Sensitivity Analysis
### Analysis of Detection Sensitivity for QD under Spread-Spectrum Technology  
During the process of detection using the QD system, the correlation operation performed during despreading suppresses noise factors, thereby providing a certain spreading gain that enhances the SNR. Consequently, a further analysis of the sensitivity of the QD with regard to the use of spread-spectrum technology is warranted. In laser communication using intensity modulation, the relationship between the bit error rate (BER) and the Receiver SNR of the detector output photocurrent is represented by the following [35; 36]:  
\[BER=\frac{1}{2}erfc\bigg{(}\frac{1}{2\sqrt{2}}\sqrt{SNR}\bigg{)} \tag{18}\]  
when the detector receives beacon light, it is affected by detector noise; this includes thermal noise, shot noise, etc. Therefore, the expression used to represent the Receiver SNR Enhancement (\(R_{\text{SNRE}}\)) of the detector receiver is as follows [37]:  
\[R_{\text{{SNRE}}}=\frac{\rho^{2}P_{in}^{2}M^{2}}{2e(\rho P_{s}+I_{D}N_{D})M^{2 }F(M)\Delta f+4N_{D}k_{B}T\Delta f/R_{L}} \tag{19}\]  
where \(P_{in}\) represents the received optical power, \(\rho\) represents the detector responsivity, \(M\) represents the avalanche multiplication factor, \(N_{D}\) represents the number of quadrants of the detector, \(e\) represents the elementary charge, \(\Delta f\) represents the bandwidth of the receiver, \(T\) represents the temperature of the quadrant detector, \(F(M)\) represents the excess noise factor of the quadrant detector, \(R_{L}\) represents the load resistance, \(I_{D}\) represents the dark current of the quadrant detector, and \(k_{B}\) represents the Boltzmann constant. When the system bit error rate is 1 \(\times\) 10\({}^{-6}\)[38], the received optical power of the detector can be obtained from Equations (20) and (21); as such, the receiver detection sensitivity and its expression are as follows:  
\[P_{\text{min}}=\sqrt{\frac{R_{SNE}[2e(\rho P_{in}+I_{D}N_{D})M^{2}F(M)\Delta f +4N_{D}k_{B}T\Delta f/R_{L}]}{\rho^{2}M^{2}}} \tag{20}\]  
when receiving low-power beacon light, the main sources of detector noise are thermal noise and dark current. Therefore, according to Formula (22), we know that \(P_{\text{min}}\propto\sqrt{R_{SNE}}\). According to Formula (15), when spread-spectrum technology is used, the output SNR after despreading is \(G_{P}\) times the SNR of the received optical signal. Therefore, the detection sensitivity of the quadrant detector will increase by a factor of \(\sqrt{G_{P}}\). According to Formula (22), the detection performance of the QA-4000 quadrant detector from the First Sensor company is simulated. The detector's parameters are as follows: the photosensitive area diameter is 4 mm, the responsivity \(\rho\) is 0.36 A/W, the avalanche multiplication factor \(M\) is 100, the bandwidth \(\Delta f\) is 50 MHz, the dark current in each quadrant \(I_{D}\) is 7 nA, and the load resistance \(R_{L}\) is 10 k\(\Omega\). Since the experiment was conducted at room temperature and thermal noise is influenced by temperature variations, we used 296 K in the simulation to simulate a room-temperature environment. The excess noise factor \(F(M)\) is 1. The change in the received optical power and bit error rate of the detector under different spreading gains is depicted in the following figure:  
It is evident from Figure 3 that, under the same spreading gain, the SNR increases and the bit error rate decreases as the received optical power increases. When considering different spreading gains, a larger spreading gain results in the detector having a lower required optical power at the same bit error rate. For example, if the system operates under a threshold condition (bit error rate of 1 \(\times\) 10\({}^{-6}\)), with spreading gains of 1023, 255, 127, 63, 31, 15, and 1, the sensitivity of the QD is \(-62.6\) dBm, \(-61.1\) dBm, \(-59.6\) dBm, \(-58\) dBm, \(-56.5\) dBm, \(-54.9\) dBm, and \(-47.1\) dBm, respectively.