## 2 Operation Principle of the Proposed Scheme
Figure 2.7  
Figure 2.8  
Figure 2.9 Comparison of the results of the absolute-value operation and the square operation of the (**a**) cosine functions and the (**b**) sine functions. Subsequently, after the preprocessing stage, the signal, \(s_{4,k}\), enters the frequency-deviation-estimation stage. In this stage, the phase error caused by frequency deviation between the transmitter laser and the LO laser can be easily calculated as  
\[I_{1,k}=real(s_{k})=cos(\theta_{k}+2\pi\Delta fkT+\varphi_{k})=cos(\phi_{k})\]0  
After elimination of the effects caused by FO, the third stage of the carrier-phase-recovery module estimates and compensates for phase errors in the signal. \[I_{1,k}=real(s_{k})=cos(\theta_{k}+2\pi\Delta fkT+\varphi_{k})=cos(\phi_{k})\]1  
The POE (\(\hat{\varphi}_{k}\)) detection module uses two adjacent phase offset estimations for comparison. If adjacent phase offset estimations jump, a phase period is compensated [18]:  
\[I_{1,k}=real(s_{k})=cos(\theta_{k}+2\pi\Delta fkT+\varphi_{k})=cos(\phi_{k})\]2  
\[I_{1,k}=real(s_{k})=cos(\theta_{k}+2\pi\Delta fkT+\varphi_{k})=cos(\phi_{k})\]3  
\[I_{1,k}=real(s_{k})=cos(\theta_{k}+2\pi\Delta fkT+\varphi_{k})=cos(\phi_{k})\]4  
where \(\Phi_{1}=0\). The actual final compensated POE in the proposed scheme is  
\[I_{1,k}=real(s_{k})=cos(\theta_{k}+2\pi\Delta fkT+\varphi_{k})=cos(\phi_{k})\]5  
Finally, the final phase offset estimation, \(\hat{\varphi_{k}}^{\prime}\), and the frequency offset estimation, \(\Delta\hat{f}\), derived from the previous frequency offset estimation module are jointly compensated into the received signal. It should be noted that random fluctuations in the amplitude and phase of the received signal, caused by refractive index changes due to atmospheric turbulence, usually have frequencies between 20 Hz and 1 kHz [19; 20]. In a CFSO transmission system with a communication rate on the order of GHz, the phase noise variation caused by the laser linewidth and atmospheric turbulence will vary relatively slowly, and its influence of two adjacent symbols can be considered the same [19; 21]. The fourth-power operation was nonlinear; after this fourth-power operation, Equation (1) generated many cross-multiplication terms containing the signal and ASE noise. In taking the power of two as an example--\(\left(s+n\right)^{2}=s^{2}+2s{\cdot}n+n^{2}\)--only the first term is a useful signal; the remaining terms can be regarded as amplified noise. This means that the fourth-power operation will expand system noise and the 4 th-PD-VV algorithm is less resistant to noise. On the contrary, the absolute-value operation of real numbers in the proposed scheme can directly offer sign judgment without generating cross-multiplication terms, so it has less impact on subsequent modules during the carrier recovery. Additionally, instead of viewing FOE and CPE as two separate processing flows in the proposed scheme, a preprocessing stage was chosen to be added prior to the conjugate multiplication of the CFOE. In this stage, four-level absolute-value operation is performed to remove the signal-modulation-phase operation. The preprocessed signal is fed to the subsequent carrier recovery compensation module. This approach avoids the repetitive four-level absolute-value operation of the CPE module in the third stage and minimizes the complexity of the whole scheme. Finally, the scheme uses joint compensation of the FOE and the CPE. This approach reduces the number of multipliers used, which further reduces the complexity of the scheme.