# A Noise-Tolerant Carrier Phase Recovery Method for Inter-Satellite Coherent Optical Communications
###### Abstract  
Coherent free-space optical communication offers significant advantages in terms of communication capacity, making it particularly suitable for high-speed inter-satellite transmission within satellite communication networks. Nonetheless, the presence of Doppler frequency offset (FO) and phase noise (PN) associated with lasers adversely affects the bit error rate (BER) performance of these communication systems. Conventional methods for FO and phase estimation are usually hindered by high computational demands and phase cycle slips, especially in environments characterized by elevated channel noise. To address these challenges, a noise-tolerant method is proposed to facilitate accurate carrier phase recovery (CPR) with reduced complexity. This method merges a second-order feedback loop and a feedforward stage to achieve accurate estimation. The simulation results indicate that the proposed method surpasses traditional methods in terms of noise tolerance and resource efficiency. Particularly, the BER of the proposed method can be decreased to \(6.7\times 10^{-3}\) at a signal-to-noise ratio (SNR) of 4.5 dB, in contrast to a BER of 0.25 for the traditional method. Additionally, the resource consumption of the proposed method can be decreased by 64% under equivalent conditions. Furthermore, the experimental results reveal that the phase estimation error and BER for the proposed method are \(2.1\times 10^{-4}\) and \(7.5\times 10^{-4}\), respectively, when the received power is \(-41\) dBm. These values are significantly lower than those achieved with traditional methods, which obtain errors of \(1.85\times 10^{-3}\) and a BER  
This causes a Doppler frequency offset (FO), where the transmitted signal frequency shifts relative to the local oscillator (LO). This dynamic FO in satellite communications can degrade the signal integrity, necessitating accurate compensation for effective data retrieval. Furthermore, phase noise (PN), caused by fluctuations in the phase of the received signal, poses an additional challenge. In coherent optical communication, PN can significantly reduce th signal quality and increase errors in phase-sensitive demodulation [7; 8; 9]. Consequently, estimating and compensating for both Doppler FO and PN are crucial for reliable system performance. FO estimation techniques can be categorized into frequency domain and time-domain methods. Frequency domain methods, specifically fast Fourier transform-based FO estimation (FFT-FOE) [10; 11; 12], exhibit excessively high computational complexity, making them unsuitable for resource-constrained satellite payloads. Conversely, time-domain methods, known as differential FO estimation (Diff-FOE), utilize phase difference information between adjacent symbols. This approach employs block estimation to smooth data, thereby mitigating the effects of noise. [PERSON] et al. [13] proposed an FO estimation method based on the differentiation of adjacent symbols. [PERSON] et al. [14] introduced a two-stage estimation method that employed differentiation across multiple symbol intervals. [PERSON] et al. [15] simplified the fourth-power (4 th) operation through the use of absolute value operations. Although Diff-FOE methods demand lower computational resources, the discrepancies in estimation between blocks can adversely affect the performance of subsequent phase recovery. Phase estimation methods can be broadly sorted into blind and data-aided estimation techniques. Data-aided estimation involves the periodic insertion of known pilot information to assist in estimating the carrier phase, using specific pilot estimation techniques at the receiver. However, rapid fluctuations in laser phase noise limit the effectiveness of pilot estimation to short time intervals, necessitating increased pilot insertion frequency and, consequently, higher pilot overhead. Conversely, blind estimation does not rely on pilot information, offering advantages in spectral efficiency and resource utilization while enabling the continuous estimation of the carrier phase.