Research on Quadrant Detector Multi-Spot Position Detection Based on Orthogonal Frequency Division Multiplexing

Abstract

Free space optical communication is developing towards laser communication networking. A novel method of quadrant detector (QD) multi-spot position detection based on orthogonal frequency division multiplexing (OFDM) is proposed for multi-spot laser communication systems. The mathematical model is constructed, and the Monte Carlo method is used to validate it. The position calculation of three beams incident on QD is simulated. The influence of key parameters on the accuracy of position detection is analyzed when the spots are at the same position and a different position. The results show that when the SNR of the system is 31.74 dB, the radius of the Gauss spot is 2 mm, the number of FFT (fast Fourier transform) points is 1024, and the center of the multi-spot is at the same position of the detector target; the accuracy of signal position detection calculated by the equation is 1.433 μm, and the simulation results are 1.351 μm, 1.354 μm, and 1.389 μm, respectively. When the center position of the multi-spot is at different positions of the detector target, the detection accuracy calculated by the formula is 1.438 μm, 1.433 μm, and 1.434 μm, respectively, and the simulation results are 1.419 μm, 1.387 μm, and 1.346 μm, respectively. The experimental results verify the effectiveness of the proposed multi-target simultaneous detection method. This article proposes a new multi-spot position detection method which can not only achieve one-to-multiple node laser communication but also improve the accuracy of point position detection.

1. Introduction

Space laser communication has many advantages, such as a high communication rate, strong anti-interference ability, and high confidentiality. It has been widely used in the information transmission of various laser communication links [1,2,3,4]. In the process of building a laser communication link, it has to be ensured that the beam of the transmitter and receiver are pointed (PAT), captured, and tracked, which is a prerequisite for stable communication. And the accuracy of spot position detection determines the performance of the communication system. Considering that the quadrant detector (QD) has the advantages of a fast response speed, high measurement accuracy, and simple data processing, it has developed into a core device in the space laser communication system [5,6,7].

In the practical system, the relationship between the output signals of QD and the real position of the spot on the detector target is complicated. Therefore, in recent years, some researchers have conducted a lot of work on how to improve the spot position detection accuracy of QD. In [8], the relationship between the spot position and the output voltage is analyzed when the spot shape and the spot energy distribution are different, and the linear measurement range of QD is extended. In [9], the relationship between the position of the spot position and the output signal of QD is obtained by combining the opposite error characteristics of the infinite integral method and Boltzmann method, which reduces the position detection error of a different spot radius. In [10], an error compensation factor and the gap size of the detector are introduced into the traditional spot position detection model. The high-precision spot position information of the QD in a complex environment is obtained by using the improved spot position detection model. In [11], the effect of the signal-to-noise ratio (SNR) on the accuracy of QD position detection is analyzed, and the cyclic cross-correlation method is used to denoise the modulated light, which improves the accuracy of spot position detection. Reference [12] used the Kalman filter to estimate the real output of four electrical signals of QD, which decrease the root-mean-square errors of detection using the improved detection method compared with the direct detection method.

However, more research focuses on the traditional space laser communication system, which can no longer meet the transmission needs of space information by point-to-point mode. Therefore, the laser communication system should have the ability to communicate with multiple nodes simultaneously to form a high-speed communication network [13]. However, traditional modulation technology cannot realize one-to-many laser communication. Therefore, researchers have been exploring the application of multiplexing technology in space laser communication systems to achieve one-to-many laser communication.

Reference [14] proposes the use of frequency division multiplexing technology combined with phase detection technology to achieve the simultaneous detection of multi-spot on a single position sensing detector (PSD), but the frequency of the modulation signals is low. Reference [15] proposes the use of the integral approximation method to fit the offset of distributed light spot, which is used as the initial position. The Boltzmann function is used to compensate for the calculation error to simplify the calculation of the multi-spot offset of QD, but its application has limitations. Reference [16] proposes applying a square wave signal to modulate the beacon light at the transmitter. At the receiver, the same-frequency square wave signal is used to perform a correlation operation with the PSD output photocurrent signal to calculate the spot position. However, the method needs to ensure synchronization between the signals of the transmitter and the receiver. Therefore, orthogonal frequency division multiplexing (OFDM) is a special multi-carrier modulation technology. The frequency of the modulation signals is orthogonal. It has high spectrum utilization and can use a fast discrete Fourier inverse/forward transform (IFFT/FFT) algorithm to realize signal modulation and demodulation. It has been widely used in space laser communication systems [17,18,19,20].

This paper introduces the QD multi-spot position detection method based on OFDM. Initially, a mathematical model is formulated to assess the accuracy of QD position detection, drawing insights from the operational principles of OFDM and QD multi-beam position detection. Subsequently, an analysis of the SNR of the photocurrent signal in the OFDM system after FFT demodulation is presented. The Monte Carlo method is employed to simulate the practical scenario of light spot incidence on the QD in real-world systems. The proposed mathematical model is validated using LabView, demonstrating the congruence between simulation and formula-based results. This substantiates that the proposed mathematical model is capable of directly computing the accuracy of multi-spot position detection within specified parameters. Finally, the QD multi-spot position detection method based on OFDM is shown to facilitate efficient one-to-many laser communication. The primary contributions of this paper are delineated as follows:

(1)

The presented mathematical model establishes a connection between the position detection accuracy and key factors such as the Gaussian spot radius, spot center position, SNR, and the quantity of FFT points. This correlation facilitates the concurrent detection of multiple spot positions, ultimately enabling efficient one-to-many laser communication.

(2)

The incorporation of FFT operation results in an enhancement of the SNR in the photocurrent signal, thereby elevating the precision of spot position detection.

The remainder of this paper is organized as follows: The working principle of the OFDM system is introduced in Section 2. Section 3 shows the construction and analysis of the simulation system. The analysis of the simulation results is presented in Section 4. Finally, Section 5 provides the concluding remarks for this paper.

2. Materials and Methods

Figure 1 represents the block diagram of the QD multi-spot position detection system based on OFDM. The modulated signal light is simultaneously illuminated on the detector target surface. The photocurrent signal is amplified by a trans-impedance amplifier (TIA) to obtain a voltage signal, and the analog signal is converted into a digital signal through the ADC. The DC signal is converted into an AC signal through the digital DC-AC conversion module to achieve DC isolation processing of the signal. Finally, FFT is performed on these signals to obtain the amplitude value of the photocurrent signal in each quadrant. The spot center positions are calculated using the QD position calculation equation.

2.1. OFDM System Working Principle

IFFT/FFT technology is proposed by Weinstein and Eben to apply in the modulation and demodulation of OFDM systems, which reduces the complexity of multi-carrier systems and solves the problem of the strict orthogonality of subcarriers [21]. Therefore, IFFT/FFT is usually used to realize the modulation and demodulation of OFDM signals in practical applications. Figure 2 shows the schematic diagram of the OFDM communication system.

In OFDM systems, the k-th subcarrier frequency satisfies $f_k=\raisebox{1ex}{$k$}\!\left/ \!\raisebox{-1ex}{$T$}\right.\left(k=0,1,\dots ,N-1\right)$ and assumes that $t=\raisebox{1ex}{$nT$}\!\left/ \!\raisebox{-1ex}{$N$}\right.\left(n=0,1,\dots ,N-1\right)$. The discrete OFDM modulation signal is:

$$S_n={\displaystyle \sum _{k=0}^{N-1}{\displaystyle \sum {}_m}{s}_k\left(m\right)}{e}^{j2\pi in/N}$$

(1)

where $T$ is the duration of the OFDM symbol. $s_k\left(m\right)$ represents the $m$-th symbol on the $k$-th subcarrier. Equation (2) is obtained after converting OFDM signals from serial to parallel:

$$\frac{1}{N}{S}_n=\frac{1}{N}{\displaystyle \sum _{i=0}^{N-1}{s}_i\left(t\right)e^{j2\pi in/N}}=IDFT\left(s_i\right)$$

(2)

Therefore, the modulation process of OFDM can be completed using inverse discrete Fourier transform (IDFT), while at the receiver, discrete Fourier transform (DFT) is needed to separate the OFDM subcarrier signal.

$$S_n={\displaystyle \sum _{i=0}^{N-1}{\widehat{s}}_i\left(t\right)e^{\raisebox{1ex}{$-j2\pi in$}\!\left/ \!\raisebox{-1ex}{$N$}\right.}}=DFT\left(s_i\right)$$

(3)

Assume that a signal $x\left(t\right)=\cos \left(\mathsf{\Omega }t+\phi \right)$ is used at the transmitter to intensity modulate the incident light with light field intensity $a$, beam frequency $w$, and phase $\phi$, then the modulated light field intensity $P_s\left(t\right)$ is [11]:

$$\begin{array}{cc}{P}_s\left(t\right)& =\frac{a^2}{2}\left[1+k_p\cdot x\left(t\right)\right]{\cos }^2\left(wt\right)\hfill \\ & =\frac{a^2}{2}\left[1+k_p\cdot \cos \left(\mathsf{\Omega }t+\phi \right)\right]{\cos }^2\left(wt\right)\hfill \end{array}$$

(4)

where $k_p$ is the proportion coefficient. Then, the photocurrent $I$ outputted by the detector corresponding to the incident light is:

$$I=c\cdot \frac{a^2}{2}\left[1+k_p\cdot \cos \left(\mathsf{\Omega }t+\phi \right)\right]$$

(5)

where $c=e\gamma /hv$ is the photoelectric conversion coefficient, $\gamma$ is the quantum efficiency, $v$ is the incident light intensity, $h=6.626\times {10}^{-34}J\cdot s$ represents Planck’s constant, and $e=1.602\times {10}^{-19}C$ represents the charge constant. According to Equation (5), the detector output photocurrent signal after being intensity modulated contains the information of the modulation signal $x\left(t\right)$. The photocurrent signal is also a single-frequency signal after being processed by the AC coupling circuit. This can be rewritten into a voltage expression as:

$$V\left(t\right)=s\left(t\right)+n\left(t\right)=KA\cos \left(\mathsf{\Omega }t+\phi \right)+n\left(t\right)$$

(6)

where $K$ represents the total gain of the signal channel between the QD output and ADC; $A=c{a}^2/2$ represents the signal amplitude; $s\left(t\right)$ is the signal of the detector to the incident light; and $n\left(t\right)$ is Gaussian white noise with a mean of 0 and a variance of ${\sigma }^2$. $s\left(t\right)$ is rewritten by Euler’s formula and transformed by FFT [5]:

$$S\left(W\right)=\frac{KA}{2}\left[e^{j\phi }\cdot \delta \left(W+\mathsf{\Omega }\right)+e^{-j\phi }\cdot \delta \left(W-\mathsf{\Omega }\right)\right]$$

(7)

Here, $W$ is the intensity variable. The energy of the output photocurrent signal corresponding to the detector can be concentrated at the intensity point $\mathsf{\Omega }$ after intensity modulation of the incident light. And the beam energy is also proportional to the amplitude value at this intensity point. Therefore, the FFT operation can be used at the receiver to demodulate the combined photocurrent signals output from the QD to obtain the amplitude values of the corresponding quadrant photocurrent signals. Comparing the amplitude can determine the offset of the spot center relative to the QD origin. And then the spot position information of the incident light can be calculated. The energy of Gaussian white noise will be distributed over a wider frequency band. The FFT operation can improve the SNR of the photocurrent signal at this intensity point. The system SNR of the OFDM system is analyzed after FFT demodulation as follows.

Define an output signal $y\left(t\right)$ containing cosine signals of different frequencies with Gaussian white noise as:

$$y\left(t\right)=A_1\cos \left(2\pi f_{1}t+{\theta }_1\right)+A_2\cos \left(2\pi f_{2}t+{\theta }_2\right)+\dots +A_M\cos \left(2\pi f_{M}t+{\theta }_2\right)+n\left(t\right)$$

(8)

where $A_1,A_2,\dots ,A_M$ represent the signal amplitude calculated by $A=c{a}^2/2$, and ${\sigma }_n^2$ is the variance of Gaussian white noise. Take $T_s$ as the sampling period and discretize $y\left(t\right)$ at point $N$. $y\left(k\right)$ is obtained, where $k=0,1,2,\dots ,N-1$.

$$y\left(k\right)=A_1\cos \left(2\pi f_{1}k{T}_s+{\theta }_1\right)+A_2\cos \left(2\pi f_{2}k{T}_s+{\theta }_2\right)+\dots +A_M\cos \left(2\pi f_{M}k{T}_s+{\theta }_2\right)+n\left(k\right)$$

(9)

Then, the SNR of signal $y\left(k\right)$ can be expressed as:

$$SNR=\frac{P_s}{P_n}=\frac{A_1^2+A_2^2+\dots +A_M^2}{2\cdot {\sigma }_n^2}$$

(10)

After performing $N$-point FFT operation on signal $y\left(k\right)$, the signal power $P_{ss}$ is

$$P_{ss}={\displaystyle \sum _{j=1}^M{\left(\frac{N}{2}{A}_j\right)}^2}$$

(11)

If the Gaussian white noise with standard deviation ${\sigma }_n$ has a time domain length of $N$, then after the $N$-point FFT operation, the standard deviation becomes $\sqrt{N}{\sigma }_n$, so the noise power can be written as:

$$P_{nn}={\left(\sqrt{N}{\sigma }_n\right)}^2=N{\sigma }_n^2$$

(12)

The $〈SNR〉$ is

$$〈SNR〉=\frac{P_{ss}}{P_{nn}}=\frac{N}{2}SNR$$

(13)

2.2. Working Principle of Multi-Beam Position Detection

QD is a photoelectric position detector which is composed of four identical photodiodes. When incident light strikes the photosensitive surface of the QD, the four quadrants will output corresponding photocurrents. The photocurrent is proportional to the energy of the beam. Suppose there are $M$ beams irradiating the $i$-th quadrant of the QD at the same time, the energy density of the $j$-th Gaussian beam incident on the detector target surface is $p_{ij}\left(x,y\right)$ [22], and the corresponding photocurrent is $I_{ij}$.

$$p_{ij}\left(x,y\right)=\frac{2p_0}{\pi {\omega }^2}\exp \left\{-\frac{2\left[{\left(x-x_j\right)}^2+{\left(y-y_j\right)}^2\right]}{{\omega }^2}\right\}$$

(14)

where $p_0$ is the total energy of the Gaussian beam, $\omega$ is the beam waist radius of the Gaussian spot, and $\left(x_j,y_j\right)$ is the coordinate of the spot center. Then, the total photocurrent $I_i$ received by the $i$-th quadrant of the detector can be recorded as [23].

$$I_i={\displaystyle \sum _{j=1}^M{p}_{ij}\left(x,y\right)dxdy}={\displaystyle \sum _{j=1}^M\eta {\displaystyle {\iint }_{{\displaystyle \sum i}}{p}_{ij}\left(x,y\right)}dxdy}={\displaystyle \sum _{j=1}^M{I}_{ij}}, i=A,B,C,D$$

(15)

Here, $\eta$ is the photoelectric conversion efficiency, and $\sum {}_i$ is the integration area of the $i$-th quadrant.

Equation (15) shows that the photocurrent output by QD in response to multiple light beams is linearly superposed. If single-frequency signals of different intensities are used to intensity modulate the incident beam separately, the photocurrent signal output by the detector is a superposition of single-frequency signals of different intensities.

Assume that the photocurrent signal values generated in each quadrant after the $j$-th laser light which illuminates the detector are $I_{A_j},I_{B_j},I_{C_j},I_{D_j}$, respectively. After FFT operation, the amplitude value are $A_{A_j},A_{B_j},A_{C_j},A_{D_j}$. ${\widehat{x}}_j$ and ${\widehat{y}}_j$ represent the offset of the center position of the target spot measured by the $j$-th laser light relative to the center of the QD on the x-axis and y-axis, which can be obtained according to the addition and subtraction algorithm theory [24]:

$$\left\{\begin{array}{l}{\widehat{x}}_j=\frac{\left(I_{A_j}+I_{D_j}\right)-\left(I_{B_j}+I_{C_j}\right)}{I_{A_j}+I_{B_j}+I_{C_j}+I_{D_j}}=\frac{\left(A_{A_j}+A_{D_j}\right)-\left(A_{B_j}+A_{C_j}\right)}{A_{A_j}+A_{B_j}+A_{C_j}+A_{D_j}}\\ {\widehat{y}}_j=\frac{\left(I_{A_j}+I_{B_j}\right)-\left(I_{D_j}+I_{C_j}\right)}{I_{A_j}+I_{B_j}+I_{C_j}+I_{D_j}}=\frac{\left(A_{A_j}+A_{B_j}\right)-\left(A_{D_j}+A_{C_j}\right)}{A_{A_j}+A_{B_j}+A_{C_j}+A_{D_j}}\end{array}\right.$$

(16)

${\widehat{x}}_j$ and ${\widehat{y}}_j$ are the relative position information of the spot center solved by Equation (16) under the ideal situation without introducing noise. In the position detection system, the x-axis and the y-axis are independent and symmetrical. So, this paper only derives the mathematical model of position detection in the x-axis direction. In the actual detection process, the noise at the QD output signals includes thermal noise, shot noise of the detector, and the noise introduced by light, which are all Gaussian white noise. And the power spectral density can be regarded as evenly distributed over the entire frequency band, which is independent of each other and not affected [25,26].

Taking the noise into account, the position solution is solved by Equation (17):

$$\begin{array}{cc}{\widehat{x}}_j& =\frac{\left(I_{A_j}+I_{nA}+I_{D_j}+I_{nD}\right)-\left(I_{B_j}+I_{nB}+I_{C_j}+I_{nC}\right)}{I_{A_j}+I_{B_j}+I_{C_j}+I_{D_j}+I_{nA}+I_{nB}+I_{nC}+I_{nD}}\hfill \\ \hfill & =\left(\frac{I_{A{D}_j}-I_{B{C}_j}}{I_{A{D}_j}+I_{B{C}_j}}+\frac{I_{nAD}-I_{nBC}}{I_{nAD}+I_{nBC}}\right)\cdot \left(\frac{1}{1+\frac{I_{nAD}+I_{nBC}}{I_{A{D}_j}+I_{B{C}_j}}}\right)\\ & \approx \frac{I_{A{D}_j}-I_{B{C}_j}}{I_{A{D}_j}+I_{B{C}_j}}+2\cdot \frac{I_{B{C}_j}{I}_{nAD}-I_{A{D}_j}{I}_{nBC}}{{\left(I_{A{D}_j}+I_{B{C}_j}\right)}^2}\hfill \end{array}$$

(17)

where $I_{A{D}_j}=I_{A_j}+I_{D_j}$ represents the sum of useful photocurrents in the first and fourth quadrants, and $I_{B{C}_j}=I_{B_j}+I_{C_j}$ represents the sum of useful photocurrents in the second and the third quadrants. $I_{nAD}=I_{nA}+I_{nD}$ represents the sum of noise photocurrent in the first and fourth quadrants, and $I_{nBC}=I_{nB}+I_{nC}$ represents the sum of noise photocurrent in the second and third quadrants. $I_{nA},I_{nB},I_{nC},I_{nD}$ represent the noise currents in the four quadrants. The variance ${\sigma }_{{}_{j_{xn}}}^2$ of the spot relative to position ${\widehat{x}}_j$ is used to characterize the effect on the position detection accuracy:

$$\begin{array}{cc}{\sigma }_{{}_{j_{xn}}}^2& =2\cdot \frac{{\left(I_{A{D}_j}\right)}^2+{\left(I_{B{C}_j}\right)}^2}{I^4}\cdot \left(I_{nAD}^2+I_{nBC}^2\right)\hfill \\ & =k\cdot \frac{I_n^2}{I^2}\hfill \end{array}$$

(18)

where $I^2$ represents the total power of useful photocurrent, $I_n^2=I_{nAD}^2+I_{nBC}^2$ represents the total current noise power, and $R_{SN}=I^2/I_n^2$ represents the system SNR. And the proportional coefficient $k$ can be written as:

$$k=2\cdot \frac{{\left(I_{A{D}_j}\right)}^2+{\left(I_{B{C}_j}\right)}^2}{I^2}$$

(19)

The ${\widehat{x}}_j$ solved by Equation (17) is the relative position of the spot. The actual position $\left(x_j,y_j\right)$ of the spot needs to be solved according to the distribution model of the spot. Assume that the energy distribution model of the laser spot is a Gaussian spot and $\omega$ is the spot radius of the Gaussian spot; when the spot position is near the center of the detector and $x_j,y_j\ll \omega$, the relative position value is substituted into the error function and the first-order approximation is made, then the actual position can be expressed as:

$$x_j\approx \frac{\sqrt{\pi }\omega }{2\sqrt{2}}{\widehat{x}}_j$$

(20)

The standard deviation ${\sigma }_{j_{xn}}$ and ${\sigma }_{j_{yn}}$ of the actual value of the position detection is considered to be the standard to judge the accuracy of position detection in the process of position detection. The standard deviation of the position detection in the x-axis direction can be obtained by Equations (13), (18) and (20):

$${\sigma }_{j_{xn}}\approx \frac{\sqrt{\pi }\omega }{2\sqrt{2}}\cdot \frac{1}{\sqrt{\frac{N{R}_{SN}}{2}}}\cdot \sqrt{1+er{f}^2\left(\frac{\sqrt{2}{x}_j}{\omega }\right)}$$

(21)

A mathematical model of the position detection accuracy of the multi spot of the QD can be established. From Equation (21), it can be seen that the position detection accuracy is related to the number of FFT points $N$, the Gaussian spot radius $\omega$, the actual position value of the spot centroid position $\left(x_j,y_j\right)$, and the system SNR $R_{SN}$.

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):

$$\omega =2\sigma$$

(22)

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.

(2)

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 $\left({\widehat{x}}_j,{\widehat{y}}_j\right)$ and actual positions $\left(x_j,y_j\right)$ of different spot positions are calculated according to Equations (17) and (20).

(3)

Calculating the standard deviation

The root-mean-square error (RMSE) ${\sigma }_{x_j}$ and ${\sigma }_{y_j}$ are considered to evaluate the average detection error of QD in the effective detection range, that is, the position detection accuracy.

$$\left\{\begin{array}{l}{\sigma }_{x_j}=\frac{{\displaystyle \sum _{m=1}^{N_s}{x}_{j_m}-{\overline{x}}_{j_m}}}{\sqrt{N_s}}\\ {\sigma }_{y_j}=\frac{{\displaystyle \sum _{m=1}^{N_s}{y}_{j_m}-{\overline{y}}_{j_m}}}{\sqrt{N_s}}\end{array}\right.$$

(23)

where $N_s$ represents the number of simulations, $x_{j_m}$ and $y_{j_m}$, respectively, represent the actual values of the x-axis and y-axis obtained from the $m$-th simulation of the $j$-th beam, which are values estimated using the Monte Carlo method. ${\overline{x}}_{j_m}$ and ${\overline{y}}_{j_m}$, respectively, represent the average values of the $j$-th beam, which are expectant values.

The comparison between the true spot position values and the actual spot position values of the x-axis is shown in Figure 5. The comparison between the real values and the actual values of signal 1, signal 2, and signal 3 is shown in Figure 5a when the three spots are located at the center of the QD. And the comparison between the real values and the actual values of signal 1, signal 2, and signal 3 is shown in Figure 5b when the three spots are located at the (0.1, 0), (0, 0), and (−0.05, 0) of QD. It can be seen that the positions of the different light spots can be calculated from Figure 5.

The assessment of the multi-spot position detection accuracy of the QD based on OFDM is principally categorized into the average relative error (ARE) and final relative error (FRE). The ARE represents the mean Euclidean distance between the true spot position values and the actual spot position values, while the FRE corresponds to the Euclidean distance between the true spot position values and the actual spot position values at the final temporal instance. The FRE $e_{x_j}$ and ARE $e_{mean{x}_j}$ can be written as Equations (24) and (25), where $p$ is the total number and $e_{x_{jf}}$ is $f$-th of the FRE.

$$e_{x_j}=\frac{\left|{\sigma }_{x_j}-{\sigma }_{j_{xn}}\right|}{{\sigma }_{x_j}}\times 100\%$$

(24)

$$e_{mean{x}_j}=\frac{{\displaystyle {\sum }_{t=1}^p{e}_{x_{jf}}}}{p}$$

(25)

4. Simulation and Analysis

From the analysis in Section 2, it can be seen that the multi-spot position detection accuracy of the QD based on OFDM is related to four key parameters: $\omega$, $\left(x_j,y_j\right)$, $R_{SN}$, and $N$. Section 4 analyzes the impact of the four key parameters on the accuracy of multi-spot position detection based on the Monte Carlo simulation system. It is assumed that the radius of the QD is 4 mm, and the light beams were excited by 1 MHz, 2 MHz, and 3 MHz Cosine signals, which are sampled at a frequency of 32 MHz. This paper will discuss conditions when the three spots are located at the center of the QD and at the (0.1, 0), (0, 0), and (−0.05, 0).

4.1. The Impact of FFT Points on Position Detection Accuracy

It is assumed that the SNR is 31.74 dB and $\omega$ is 2 mm. Figure 6 shows the relationship between the multi-spot position detection accuracy and $N$, where $N=2^m,m=5,6,7,\dots ,12$. Figure 6a is the relationship when the three spots are located at the same positions. Figure 6b is the relative error curve of Figure 6a. Figure 6c is the relationship when the three spots are located at different positions. Figure 6d is the relative error curve of Figure 6c. As can be seen from Figure 6a,c that the position detection accuracy decreases with the increase in $N$. The larger N is, the greater the signal-to-noise ratio gain, which improves the accuracy of position detection. But the impact on the position detection accuracy is smaller when $N$ increases to 512. The reason for this is that the signal length in this article is 1024. At the same time, the number of FFT points $N$ also determines the resolution and computational complexity of the spectral analysis. Therefore, the appropriate $N$ can be selected according to the actual system requirement on the premise of ensuring detection accuracy, such as signal resolution requirements, computational efficiency, and signal length.

When the number of FFT points is changed, the average relative error of the value by simulation relative to the calculated value by formula is shown in

Table 1. The average relative error between position detection accuracy value by simulation and value calculated by formula when changing the N value.

Gaussian Spot Center Position	${\mathit{e}}_{\mathit{m}\mathit{e}\mathit{a}\mathit{n}\mathit{N}\_\mathit{x}1}$	${\mathit{e}}_{\mathit{m}\mathit{e}\mathit{a}\mathit{n}\mathit{N}\_\mathit{x}2}$	${\mathit{e}}_{\mathit{m}\mathit{e}\mathit{a}\mathit{n}\mathit{N}\_\mathit{x}3}$
Same Position	5.31%	5.03%	4.49%
Different Positions	4.48%	4.30%	5.79%

. As can be seen from Table 1, when the spots are at the same position, the AREs are 5.31%, 5.03%, and 4.49%, respectively. When the spots are at different positions, the AREs are 4.48%, 4.30%, and 5.79%, respectively. This indicates that the simulation value of the position detection accuracy is close to the mathematical model.

4.2. The Impact of Spot Radius on Position Detection Accuracy

Assume that the SNR is 31.74 dB and $N$ is 1024. The spot radius $\omega$ is controlled to increase from 1 mm to 3.6 mm in steps of 0.6 mm. Figure 7a shows the relationship between the position detection accuracy and the spot radius when three spots are located at the same positions. Figure 7b is the relative error curve of Figure 7a. Figure 7c is the relationship between the position detection accuracy and the spot radius when the three spots are located at different positions. Its relative error curve is Figure 7d. It can be seen from Figure 7a,c that the spot position detection can receive better accuracy with the decrease in the spot radius. The reason for this is that increasing the radius of the Gaussian spot will cause its energy to not be able to be less concentrated, which leads to a decrease in the position detection accuracy. Therefore, in practical systems, the position detection accuracy can be improved by reducing the spot radius within the detection range [27].

When the spot radius is changed, the average relative error of the value by simulation relative to the value calculated by formula is shown in

Table 2. The average relative error between position detection accuracy value by simulation and value calculated by formula when changing the spot radius.

Gaussian Spot Center Position	${\mathit{e}}_{\mathit{m}\mathit{e}\mathit{a}\mathit{n}\mathit{\omega }\_\mathit{x}1}$	${\mathit{e}}_{\mathit{m}\mathit{e}\mathit{a}\mathit{n}\mathit{\omega }\_\mathit{x}2}$	${\mathit{e}}_{\mathit{m}\mathit{e}\mathit{a}\mathit{n}\mathit{\omega }\_\mathit{x}3}$
Same Position	4.01%	3.61%	3.29%
Different Positions	3.91%	2.33%	4.51%

. As can be seen from Table 2, when the spots are at the same position, the AREs are 4.01%, 3.61%, and 3.29%, respectively. When the spots are at different positions, the AREs are 3.91%, 2.33%, and 4.51%. This indicates that the simulation value of the position detection accuracy is close to the mathematical model.

4.3. The Impact of System SNR on Position Detection Accuracy

Assume that $\omega$ is 2 mm and $N$ is 1024. The channel noise parameters and the detector noise standard deviation remain unchanged, and the SNR will change when the input signal power is reduced gradually. Figure 8a below shows the relationship between the position detection accuracy and the SNR when the three spots are located at the center of the QD. Figure 8b is the relative error curve of Figure 8a. Figure 8c shows that the spots are at a different position, respectively, and its relative error curve is Figure 8d. It can be seen from Figure 8a,c that the multi-spot position detection accuracy decreases as the SNR increases. In the actual system, the values of $\omega$, $\left(x_j,y_j\right)$, and $N$ will be limited. For example, when the SNR is 29.68 dB, the position detection accuracy of the spot position at the center of the detector is 1.847 μm, 1.835 μm, and 1.859 μm, respectively. However, when the SNR is 31.74 dB, the detection accuracy of the spot position is 1.351 μm, 1.354 μm, and 1.389 μm, respectively. This significantly improves the accuracy of position detection. It can be seen that if the SNR can be improved, then better spot position detection accuracy can be achieved.

When the SNR is changed, the average relative error of the simulation value relative to the calculated value of the formula is shown in

Table 3. The average relative error between position detection accuracy value by simulation and value calculated by formula when changing the SNR.

Gaussian Spot Center Position	${\mathit{e}}_{\mathit{m}\mathit{e}\mathit{a}\mathit{n}{\mathit{R}}_{\mathit{s}\mathit{n}}\_\mathit{x}1}$	${\mathit{e}}_{\mathit{m}\mathit{e}\mathit{a}\mathit{n}{\mathit{R}}_{\mathit{s}\mathit{n}}\_\mathit{x}2}$	${\mathit{e}}_{\mathit{m}\mathit{e}\mathit{a}\mathit{n}{\mathit{R}}_{\mathit{s}\mathit{n}}\_\mathit{x}3}$
Same Position	2.58%	2.92%	2.43%
Different Positions	2.43%	2.63%	2.75%

. As can be seen from Table 3, when the spots are at the same position, the AREs are 2.58%, 2.92%, and 2.43%. When the spots are at different positions, the AREs are 2.52%, 2.63%, and 2.75%, respectively. This indicates that the simulation value of the position detection accuracy is close to the mathematical model.

4.4. The Impact of Spot Position on Position Detection Accuracy

From the analysis in Section 2, when $\omega$, $R_{SN}$, and $N$ are constant, the position detection accuracy is also related to the position of the Gaussian spot incident on the QD. Assume that $\omega$ is 2 mm and $N$ is 1024. The spot positions are controlled to change in the range of −0.22 mm~+0.22 mm with a step size of 0.01 mm. Figure 9a,c show the relationship between the position detection accuracy and the spot center position when the SNR is 29.69 dB and the SNR is 31.74 dB, respectively. These relative error curves are Figure 9b and Figure 9d, respectively. From Figure 9a,c, it can be seen that the position of the spot has little effect on the position detection accuracy. Under the same condition, the position detection accuracy can be improved obviously by increasing the SNR of the system.

When the spot position is changed, the average relative error of the simulation value relative to the calculated value of the formula is shown in

Table 4. The average relative error between position detection accuracy value by simulation and value calculated by formula when hanging the spot position.

SNR	${\mathit{e}}_{\mathit{m}\mathit{e}\mathit{a}\mathit{n}\mathit{p}\_\mathit{x}1}$	${\mathit{e}}_{\mathit{m}\mathit{e}\mathit{a}\mathit{n}\mathit{p}\_\mathit{x}2}$	${\mathit{e}}_{\mathit{m}\mathit{e}\mathit{a}\mathit{n}\mathit{p}\_\mathit{x}3}$
SNR = 29.68 dB	2.50%	2.26%	2.46%
SNR = 31.74 dB	2.22%	1.99%	1.90%

. As can be seen from Table 4, when the SNR is 29.68 dB, the AREs of signal 1, signal 2, and signal 3 are 2.56%, 2.26%, and 2.46%, respectively. When the SNR is 31.74 dB, the AREs of signal 1, signal 2, and signal 3 are 2.52%, 3.07%, and 2.75%, respectively. This indicates that the simulation value of the position detection accuracy is close to the mathematical model.

5. Conclusions

This paper analyzes the principle of QD multi-spot position detection based on OFDM and builds a mathematical model based on its principle to obtain the main influencing factors that affect the accuracy of multi-spot position detection: Gaussian spot radius, the actual position of the spot center of detector, the SNR, and FFT points. Finally, the Monte Carlo simulation method is used to verify the mathematical model using LabView. The results mean that the position detection accuracy value calculated by the proposed mathematical model is consistent with the simulation value. The results show that when the SNR is 31.74 dB, the $\omega$ is 2 mm, $N$ is 1024, and the multi-spot is at the center of QD; the accuracy of signal position detection calculated by the equation is 1.433 μm, and the simulation results are 1.351 μm, 1.354 μm, and 1.389 μm, respectively. Therefore, the method of QD multi-spot position detection based on OFDM proposed in this paper can directly calculate the position detection accuracy. Next, an experimental system will be established to further validate the effectiveness of the method. This method can provide theoretical guidance for engineering applications.