# TRC-Based High-Precision Spot Position Detection in Inter-Satellite Laser Communication
## 3 The Response of QD to Modulated Light Beam  
A typical single frequency signal is a cosine (or a sine) signal, so the modulation signal can be defined as:  
\[d(t)\ =\ \cos(\Omega t+\varphi_{1}) \tag{3}\]  
The incident light field of the beacon light is \(E_{s}\), the light intensity \(P_{s}\) can be written as:  
\[P_{s}(t)\ =\ E_{s}^{2}(t)\ =\ a^{2}\cdot\cos^{2}(wt+\varphi_{0}) \tag{4}\]  
Figure 1: The no independent beacon light acquisition, tracking, and aiming (ATP) system.  
where \(a\) is the light field intensity, \(w\) is the light frequency, and \(\varphi_{0}\) is the phase of the incident light field. Then, the intensity of the modulated light can be expressed as:  
\[\begin{array}{ll}P_{s}(t)&=\frac{a^{2}}{2}[1+k_{p}\cdot d(t)]\cos^{2}(wt+ \varphi_{0})\\ &=\frac{\Phi}{2}[1+k_{p}\cdot\cos(\Omega t+\varphi_{1})]\cos^{2}(wt+\varphi_{ 0})\end{array} \tag{5}\]  
where \(k_{p}\) is proportional coefficient. Then, the QD output current corresponding to the incident light signal is:  
\[I_{beam}\ =\ \eta\cdot\frac{a^{2}}{2}[1+k_{p}\cdot\cos(\Omega t+\varphi_{1})] \tag{6}\]  
where \(\eta\) is the photoelectric conversion coefficient. It can be seen from Equation (6) that the output photocurrent contains the information of the modulation signal \(d(t)\). After passing through the AC coupling circuit, the photocurrent signal become a single frequency cosine signal, which can be rewritten as a voltage form:  
\[V(t)\ =\ s(t)+n(t)\ =\ KA\cos(\Omega t+\varphi_{1})+n(t) \tag{7}\]  
where \(K\) is the total gain of the signal channel from the QD output to the analog-to-digital converter (ADC) and \(A\ =\ k_{p}\cdot\eta\cdot a^{2}/2\) is the signal amplitude. \(s(t)\) responses to the light signal received by QD; \(n(t)\) is the noise. According to [PERSON]'s formula, \(s(t)\) can be transformed as:  
\[\begin{array}{ll}s(t)&=\frac{KA}{2}(e^{i(\Omega+\phi_{1})}+e^{-j(\Omega+ \phi_{1})})\\ &=\frac{KA}{2}(e^{i\phi_{1}}\cdot e^{i\Omega}+e^{-j\phi_{1}}\cdot e^{-j\Omega} )\end{array} \tag{8}\]  
After Fourier transform, we can get:  
\[S(W)\ =\ \frac{KA}{2}[e^{i\phi_{1}}\cdot\delta(W+\Omega)+e^{-i\phi_{1}} \cdot\delta(W-\Omega)] \tag{9}\]  
From Equation (9), if the intensity of received light beam is modulated by a single frequency signal, the amplitude values of the certain spectrum lines output from the four APDs will also reflect the energy distribution ratio of the beacon light. Therefore, the line spectrum detection method can be used to extract the amplitudes of the photocurrents, and the amplitudes can be substituted into Equation (1) to calculate the spot position.  
After intensity modulating, the spectrum of the photocurrent signal output by QD is a line spectrum signal. In contrast, the energy of Gaussian noise is distributed over a wider frequency band. In addition, the amplitude value of the spectrum line at this frequency point is proportional to the energy of the beam incident on the QD target surface. Therefore, the amplitude of the spectrum line at the frequency point \(\Omega\) can be detected by fast Fourier transform (FFT) and substituted into Equation (1) to calculate the spot position.  
However, the modulation only concentrates the valid signal energy and improves the SNR at the local frequency point, but cannot suppress noise. Under the condition of extremely low SNR, the interference of strong noise will greatly reduce the detection accuracy of the direct FFT method. Therefore, this paper proposes to use the TRC algorithm with window function to suppress noise, and then perform the FFT operation, which can provide a higher SNR gain and achieve the purpose of accurately detecting the spectrum line amplitude of the weak signal.