## 2 Materials and Methods
### OFDM System Working Principle  
IFFT/FFT technology is proposed by [PERSON] and [PERSON] 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 \(f_{k}=\
icefrac{{k}}{{\tau}}(k=0,1,\ldots,N-1)\) and assumes that \(t=\
icefrac{{nT}}{{N}}(n=0,1,\ldots,N-1)\). The discrete OFDM modulation signal is:  
\[S_{n}=\sum_{k=0}^{N-1}\sum_{m}s_{k}(m)e^{j2\pi in/N} \tag{1}\]  
Figure 1: QD multi-spot position detection system based on OFDM. Figure 2: The schematic diagram of the OFDM communication system. where \(T\) is the duration of the OFDM symbol. \(s_{k}(m)\) represents the \(m\)-th symbol on the \(k\)-th subcarrier. Equation (2) is obtained after converting OFDM signals from serial to parallel:  
\[\frac{1}{N}S_{n}=\frac{1}{N}\sum_{i=0}^{N-1}s_{i}(t)e^{j2\pi in/N}=IDFT(s_{i}) \tag{2}\]  
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. \[S_{n}=\sum_{i=0}^{N-1}s_{i}(t)e^{-j2\pi in/N}=DFT(s_{i}) \tag{3}\]  
Assume that a signal \(x(t)=\cos(\Omega t+\varphi)\) is used at the transmitter to intensity modulate the incident light with light field intensity \(a\), beam frequency \(w\), and phase \(\varphi\), then the modulated light field intensity \(P_{s}(t)\) is [11]:  
\[ \begin{array}{ll}P_{s}(t)&=\frac{a^{2}}{2}\left[1+k_{p}\cdot x(t)\right]\cos ^{2}(wt)\\ &=\frac{a^{2}}{2}\left[1+k_{p}\cdot\cos(\Omega t+\varphi)\right]\cos^{2}(wt) \end{array}  \tag{4}\]  
where \(k_{p}\) is the proportion coefficient. Then, the photocurrent \(I\) outputted by the detector corresponding to the incident light is:  
\[I=c\cdot\frac{a^{2}}{2}\left[1+k_{p}\cdot\cos(\Omega t+\varphi)\right] \tag{5}\]  
where \(c=e\gamma/h\upsilon\) is the photoelectric conversion coefficient, \(\gamma\) is the quantum efficiency, \(\upsilon\) is the incident light intensity, \(h=6.626\times 10^{-34}J\cdot s\) represents [PERSON]'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 \(x(t)\). 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:  
\[V(t)=s(t)+n(t)=KA\cos(\Omega t+\varphi)+n(t) \tag{6}\]  
where \(K\) represents the total gain of the signal channel between the QD output and ADC; \(A=ca^{2}/2\) represents the signal amplitude; \(s(t)\) is the signal of the detector to the incident light; and \(n(t)\) is Gaussian white noise with a mean of 0 and a variance of \(\sigma^{2}\). \(s(t)\) is rewritten by [PERSON]'s formula and transformed by FFT [5]:  
\[S(W)=\frac{KA}{2}\left[e^{j\varphi}\cdot\delta(W+\Omega)+e^{-j\varphi}\cdot \delta(W-\Omega)\right] \tag{7}\]  
Here, \(W\) is the intensity variable. The energy of the output photocurrent signal corresponding to the detector can be concentrated at the intensity point \(\Omega\) 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.