## 3 Simulation System Construction  
This Section uses LabView to build the QD multi-spot simultaneous detection system. The Monte Carlo method is used to analyze the proposed mathematical model. The overall block diagram of the simulation system model is shown in Figure 3. The simulation system consists of three parts, divided into generating a Gaussian spot signal, calculating the spot positions, and calculating the standard deviation. Building a simulation system based on the Monte Carlo simulation method includes the following three steps:  
1. Generating Gaussian spot signal  
According to the analysis in Section 2, the cosine signal is generated in the simulation system by Equation (6), which is considered to be the modulated incident light signal. The signal value is used to characterize the number of Monte Carlo random points \(n\). According to Equation (14), the energy distribution of a Gaussian spot follows a two-dimensional normal distribution, and x-axis and y-axis can be represented by two independent one-dimensional normal distributions. Generating random points that obey a two-dimensional Gaussian distribution is considered to characterize the Gauss facula model. The x-axis and y-axis are controlled to produce random points that obey a normal distribution by adjusting the number of standard deviations \(\sigma\) and random points \(n\). which are used to simulate the distribution of photons in the Gaussian spot on the x-axis or y-axis. The spot position can be changed by adding x-offset and y-offset. And we can express the Gaussian spot radius as follows directly using Equation (22):  
\[\omega=2\sigma \tag{22}\]  
Figure 3: Overall block diagram of simulation system model. In order to verify the feasibility of the Monte Carlo simulation of Gaussian light spots, statistical analysis is conducted on the generation of photons at different positions. The distribution of Gaussian spot light field distribution simulated by Monte Carlo shown in Figure 4a,b is the projection in the x-axis and y-axis direction of Figure 4a. Therefore, it can be concluded that using the Monte Carlo method to simulate Gaussian spot is feasible. Compared to the method of directly generating optical signals, we applied the Monte Carlo method in the signal generation process to make the simulated Gaussian facula closer to the actual system. In actual communication systems, signal noise can be approximately expressed as Gaussian white noise. Therefore, Gaussian white noise is employed in the simulation process to simulate the noise accumulated during the channel transmission process. After FFT operation, the standard deviation of Gaussian white noise will decrease. Calculating the spot position  
The number of random points in each quadrant is counted according to the radius of QD, the initial coordinates of x-axis and y-axis, and the Gauss value. The number of random points represents the number of photons received in each quadrant. It can be seen that the noise introduced by the detector during operation can be approximately expressed as Gaussian white noise in the actual system according to the analysis in Section 2. In the simulation system, the Gaussian white noise with distribution \(\left(0,\sigma_{i}^{2}\right)\) is used to approximate the QD noise. The four-channel detector noise is superimposed on the detector output signal, respectively. The four-channel output signal containing noise is processed by FFT operation to obtain the amplitude value of the output signal. Finally, the relative positions \(\left(\hat{x}_{j},\hat{y}_{j}\right)\) and actual positions \(\left(x_{j},y_{j}\right)\) of different spot positions are calculated according to Equations (17) and (20). Figure 4: (**a**) Gaussian spot light field distribution simulated by Monte Carlo method; (**b**) the projection in the x-axis and y-axis.