## 2 Operation Principle
### Frequency Offset Estimation  
For a spatial coherent laser communication system with a 16 QAM modulation format, and excluding the flicker effect of the atmospheric channel on the noise intensity of the transmission signal, the signal expression input to the frequency offset estimation and compensation module is as follows:  
\[S_{n}=\exp\left[j\big{(}\varphi_{n}+2\pi nf_{n}T+\varphi_{p}+\varphi_{ASE}\big{)}\right] \tag{2}\]  
where \(\varphi_{n}\) denotes the modulation phase of the nth signal in the input signal sequence, \(f_{n}\) represents the frequency offset value, T is the symbol duration, \(\varphi_{p}\) denotes the carrier  
Figure 4: The QPSK-TMS signal obtained after the time-tagging QPSK partitioning. Figure 3: Ideal constellation diagram and actual received signal sequence description for 16 QAM. (**a**) Ideal constellation diagram; (**b**) Actual received signal Sequence description. phase noise (including phase noise introduced by the laser linewidth and atmospheric turbulence), and \(\varphi_{ASE}\) represents the amplified spontaneous emission (ASE) phase noise, which follows a zero-mean Gaussian distribution [24]. In the second phase described in Figure 2, the input signal \(S_{n}\) is first multiplied by the conjugate of the previous signal \(S_{n-1}\) to obtain the signal \(C_{n}\). Next, the real and imaginary parts of \(C_{n}\) are extracted, and a low-complexity 4 th power operation is performed, ultimately achieving a fourfold increase in the phase angle of \(C_{n}\), resulting in \(C_{n}^{4}\). The following section outlines the principles behind the simplified 4 th power computation. Figure 5a compares the differences between the squared values of the sine function and their absolute values. Figure 5b compares the differences between the squared values of the cosine function and their absolute values. The label \"radians\" on the horizontal axis indicates that the angle \(\beta\) is measured in radians. The figures show that these values are very close, indicating that the squaring operations for the sine and cosine functions can be effectively replaced by absolute value operations. The core of the low-complexity 4 th power algorithm is based on this substitution, utilizing absolute value operations to achieve a fourfold increase in the phase angle, thereby avoiding the high-complexity 4 th power computations of traditional algorithms [29]. The design of the low-complexity 4 th power algorithm aims to simplify the computational complexity associated with traditional 4 th power operations. The following section provides the derivation of this algorithm. Step 1: The trigonometric functions corresponding to the real and imaginary parts are doubled. First, the real and imaginary parts of the signal can be expressed using the following equation:  
\[I_{n1}=real(S_{n})=\cos(\varphi_{n}+2\pi nf_{n}T+\varphi_{p}+\varphi_{ASE})= \cos(\beta_{n})\]  
\[Q_{n1}=imag(S_{n})=sin(\varphi_{n}+2\pi nf_{n}T+\varphi_{p}+\varphi_{ASE})=sin( \beta_{n}) \tag{3}\]  
Figure 5 demonstrates the similarity between the results obtained from squaring the trigonometric functions and performing absolute value operations, with the waveforms showing comparable patterns of variation. This confirms their interchangeability [24]. Therefore, the following equation can be obtained:  
\[I_{n1}{}^{2}=cos^{2}(\beta_{n})\approx|cos(\beta_{n})|\]  
Figure 5: The comparison between the results of squaring the sine and cosine functions and those obtained through absolute value operations demonstrates their similarity. (**a**) Differences between the squared values of the sine function and their absolute values; (**b**) Differences between the squared values of the cosine function and their absolute values. \[Q_{n1}{}^{2}=sin^{2}(\beta_{n})\approx|sin(\beta_{n})| \tag{4}\]  
Next, to achieve a twofold increase in the phase angle, we first use the double-angle formulas to expand \(cos(2\beta_{n})\) and \(sin(2\beta_{n})\), and then apply Equation (4) to complete the derivation of the double-angle simplification.