# A Noise-Tolerant Carrier Phase Recovery Method for Inter-Satellite Coherent Optical Communications
###### Abstract
Blind estimation encompasses methods such as blind phase search (BPS) [16] and Viterbi-Viterbi phase estimation (VV-PE) [17]. The process of BPS entails the computation of numerous phase sets to identify the phase that minimizes the error function, resulting in increased computational demands. VV-PE, on the other hand, estimates the carrier phase by eliminating the modulation phase, avoiding the requirement to process the phase set, resulting in significantly lower computational demands [18; 19; 20]. Furthermore, by selecting specific points on the mapped constellation [21] or a particular segmentation of the constellation [22], VV-PE can be adapted for carrier phase recovery (CPR) in higher-order modulation scenarios. However, these methods are susceptible to severe burst errors induced by phase cycle slips, particularly in conditions of low signal-to-noise ratios (SNRs) [23]. To facilitate comparison, Table 1 provides a concise overview of the advantages and disadvantages of various methods for frequency offset and phase estimation. To address these issues, a noise-tolerant CPR method combining a feedback mechanism and a feedforward component is proposed. The feedback mechanism helps to achieve a stable coarse estimate through filtering correction, while the feedforward component enables the accurate estimation of rapidly fluctuating phase noise. This novel method demonstrates superior performance in phase estimation and bit error rate (BER). The BER, defined as the ratio of erroneous bits to total transmitted bits, is a commonly used metric for evaluating communication system performance--lower BER values indicate better performance. Particularly, in low-SNR conditions, the proposed method exhibits enhanced noise tolerance capability. Compared to the traditional differential cascaded fourth-power (Diff-4 th) methods [13; 14; 15], the proposed approach achieves improved noise robustness while significantly reducing the computational complexity. The contributions of this research are delineated as follows:  
* We propose the noise-tolerant method for CPR in inter-satellite coherent optical communication, and investigate how the natural frequency and estimation block length parameters impact its performance. * A comparative analysis is conducted between the proposed noise-tolerant method and the conventional Diff-4 th methods, with a focus on the logarithmic root-mean-square error (LRMSE) in phase estimation and BER. The findings indicate that the LRMSE of the proposed method approaches the modified Cramer-Rao lower bound (MCRLB). * An assessment of the computational complexity of the noise-tolerant method reveals that it requires only 36% of the multiplications and 0.8% of the additions needed by the traditional Diff-4 th methods. * A coherent optical communication hardware system has been developed, validating the efficacy of the proposed method for Doppler tracking under a maximum carrier frequency change rate of 10 MHz/s. The rest of this paper is organized as follows. Section 2 outlines the system model for inter-satellite coherent optical communication. In Section 3, the proposed noise-tolerant method for CPR is detailed. Section 4 provides the statistical simulation parameters and compares the results with those derived from the Diff-4 th methods, including an analysis of the computational complexity. Section 5 describes the design of the experimental platform and analyzes the results obtained. Finally, Section 6 concludes with a summary of the findings.