## 1 Introduction
Specifically, it reports OSNR gains of approximately 3.1 dB, 1.3 dB, and 0.6 dB (pre-FEC) for a 56 Gbit/s QPSK signal using a Viterbi-Viterbi (VV) algorithm cascaded with a pilot-aided phase-unwrapping (PAPU) technique. These gains were achieved with filter lengths of 8, 16, and 20, respectively, at a pre-FEC BER of \(1.8\times 10^{-2}\). Furthermore, the study claims post-FEC OSNR gains of 3 dB, 1 dB, and 0.5 dB under similar conditions. The ability to dynamically adapt modulation formats to network traffic conditions is crucial for optimizing energy efficiency, enabling significant energy savings in optical networks. The modulation format switch has shown the potential to reduce energy consumption by as much as 18% [17]. Green optical networks are essential for meeting the growing demands of artificial intelligence (AI), which requires substantial computational resources and data transfer. Using advanced optical technologies to minimize power consumption, these networks support AI scalability while maintaining sustainability, a vital balance as AI and environmental concerns converge. Likewise, coherent communications are being considered for short-range applications to meet the growing demand for bandwidth [18]. In addition to HCFs, free-space optical (FSO) links in low-earth orbit (LEO) satellites represent another critical technology driving 6G wireless communications. However, this technology imposes much stricter pointing requirements and is highly susceptible to atmospheric turbulence and the demanding precision needed for optical beam alignment. Coherent communications maximize system capacity by leveraging the advantages of coherent modulation in both amplitude and phase, directly connecting optical systems without electronic conversion, reducing processing requirements and leading to energy savings [19; 20]. Designing low-complexity carrier recovery algorithms for coherent FSO links remains a significant challenge. The Viterbi-Viterbi (VV) algorithm [21; 22], designed specifically for phase shift keying (PSK) modulation, exploits the constant amplitude and uniform phase transitions of PSK signals. It employs a 4 th-power operation to remove modulation and recover the carrier's frequency and phase. However, this power operation significantly amplifies the background noise in the received signal. To mitigate this, a low-complexity joint compensation scheme for carrier recovery (JCSCR) has been proposed [23], which relies on the addition and subtraction of real absolute values based on modulated phase information, effectively avoiding the noise-enhancing 4 th-power operation used in the VV algorithm for coherent FSO communications. In addition, other implementations utilize pilot-based channel estimation, bridging the gap between coherent fiber optic communications and free-space optical systems [24]. As discussed previously, these methods are inherently modulation dependent. For higher-order modulation formats, pilot-aided carrier phase recovery (PA-CPR) provides a modulation-format-independent approach to phase noise compensation. Moreover, PA-CPR exhibits significantly lower computational complexity compared to maximum likelihood (ML)-based detection and estimation algorithms, making it a more practical choice for high-speed optical communication systems. A key advantage of PA-CPR is its ability to continuously track laser phase noise over the full phase range of 0 to 2\(\pi\), thereby enabling the use of absolute phase information for data transmission. This capability eliminates the optical signal-to-noise ratio (OSNR) penalties typically associated with differential coding, leading to improved system performance [5]. However, the effectiveness of PA-CPR strongly depends on the design of the pilot symbol, particularly the pilot spacing. Rapid fluctuations in laser phase noise impose a fundamental limit on the temporal effectiveness of pilot-based estimation, restricting its accuracy to short time intervals, as noted in [25].