## 2 Operation Principle of the Proposed Scheme  
Complex signal corruption entering a carrier recovery module is mainly the result of frequency offset between the transmitter laser and the local oscillation (LO) laser, as well as carrier phase errors caused by atmospheric turbulence, laser linewidth, and ASE noise. Other damage to the carrier signal was compensated for in the previous process module. The kth observed signal, \(s_{k}\), at the receiver can be modeled as [14]  
\[s_{k}=\exp[j(\theta_{k}+2\pi\Delta fkT+\varphi_{k})]+n_{k} \tag{1}\]  
where \(\theta_{k}\) is the modulation phase of the \(k\)th signal, \(\Delta f\) denotes the carrier FO, \(T\) is the symbol duration, \(\varphi_{k}\) denotes the carrier phase noise (which includes the phase noise introduced via the laser linewidth and atmospheric turbulence), and \(n_{k}\) denotes the ASE phase noise, which presents zero-mean Gaussian distribution. The block diagram of the proposed JCSCR is shown in Figure 1, where \(s_{k}\) represents the complex signal of the kth-input carrier recovery module. In the first stage of the JCSCR, the received signal will need to be preprocessed. To remove the signal modulation phase more efficiently, we used a four-level absolute operation instead of a fourth-power operation, adding and subtracting the absolute values of the real and imaginary parts [15]. The flow chart of the first-stage phase operation is shown in Figure 2. The real function () in the figure represents the operation of the complex real part, and the imag function \(0\) represents the operation of the imaginary part; \(s_{4,k}\) indicates that the kth output complex signal after the phase angle is expanded to four times that of the original. For convenience of discussion, the effects resulting from additive ASE noise, \(n_{k}\), were not considered during the derivation of the first-stage equations. Analysis of \(n_{k}\) will be explained later [16;17]. First, the received complex signal is operated via calculating the real part and the imaginary part:  
\[I_{1,k}=real(s_{k})=cos(\theta_{k}+2\pi\Delta fkT+\varphi_{k})=cos(\phi_{k})\]  
\[Q_{1,k}=imag(s_{k})=sin(\theta_{k}+2\pi\Delta fkT+\varphi_{k})=sin(\phi_{k}) \tag{2}\]  
where \(\phi_{k}=\theta_{k}+2\pi\Delta fkT+\varphi_{k}\). We considered expanding the phase angle of the trigonometric function by a factor of two using the twofold-angle formula. \[cos(2\phi_{k})=cos^{2}(\phi_{k})-sin^{2}(\phi_{k})\]  
\[sin(2\phi_{k})=2 cos(\phi_{k})\cdot sin(\phi_{k})\]  
\[=\frac{1}{2}\Big{\{}[cos(\phi_{k})+sin(\phi_{k})]^{2}-[cos(\phi_{k})-sin(\phi _{k})]^{2}\Big{\}} \tag{3}\]  
As shown in Figure 3, the square operation of trigonometric function and the absolute-value operation are very close to the waveform; they have the same period and approximate change law. To reduce the complexity of the calculation, we replaced the square operation of the trigonometric function with the absolute-value operation of the real numbers for approximation instead, realizing the twofold operation of the phase angle. Figure 1: Block diagram of the proposed scheme. Figure 2: Block diagram of the signal-preprocessing stage. Therefore, the trigonometric bisector cosine term in Equation (3) can be approximated as follows:  
\[cos(2\phi_{k})\approx\left|cos(\phi_{k})\right|-\left|sin(\phi_{k})\right| \tag{5}\]  
The sine term in Equation (3) can be approximated with the following equation:  
Figure 2.0  
Figure 2.1  
Figure 2.2  
The cos-term coefficient of Equation (5) is multiplied by \(\sqrt{2}\) to balance with the sin-term coefficient in Equation (6); then, Equation (3) can be rewritten as  
Figure 2.3  
Figure 2.4  
Similarly, to achieve the quadruple-phase angle, it is only necessary to perform the corresponding angle expansion operation using the above method on top of the double-phase angle. The final result was  
Figure 2.5  
Figure 2.6  
We reassembled the real and imaginary terms into a complex signal.