###### Abstract  
For high-speed optical communication systems, laser phase noise (LPN) stands as a pivotal factor influencing the quality of the received signal. Therefore, the employment of a highly accurate carrier phase recovery (CPR) algorithm at the receiving end is indispensable to ensure the reliability of transmission. While a CPR algorithm called principal component-based phase estimation (PCPE) has been proven to be capable of achieving low-complexity and high-performance phase recovery for even-bit quadrature amplitude modulation (QAM) (i.e., square QAM) signals, it is not compatible with traditional cross-shaped odd-bit QAM signals. To circumvent this problem, a signal constellation design scheme based on geometric shaping (GS) is proposed. The pair-wise optimization (PO) algorithm is used to optimize the constellation structure of 32 QAM and 128 QAM signals in order to obtain results that are compatible with the PCPE algorithm. Monte Carlo simulation results reveal that for odd-bit QAM signals utilizing PCPE for phase recovery, the proposed GS constellations enhance the mutual information (MI) performance across the entire measured signal-to-noise (SNR) range. Moreover, compared with regular 32 QAM and 128 QAM const constellations using the well-known blind phase search (BPS) algorithm, the proposed GS and PCPE scheme can achieve SNR gains of 1.10 dB and 2.59 dB, respectively, when considering the 20% soft-decision forward error correction (SD-FEC) overhead. Verification through commercial simulation software corroborates these findings, demonstrating that the proposed GS constellations are particularly suitable for the PCPE algorithm, especially under conditions of high optical signal-to-noise ratio (OSNR). To the best of our knowledge, this is the first time that the incompatibility between the PCPE algorithm and odd-bit QAM signals has been investigated, and the proposed GS scheme has broadened the application scope of the low-complexity CPR algorithm. laser phase noise; carrier phase recovery; geometric shaping; principal component-based phase estimation; mutual information +
Footnote †: journal: _Photonics_  
low-carbon, energy-saving, and emission reduction while coping with the rapid surge of computing power traffic in the artificial intelligence (AI) era [11]. On the one hand, the most direct and effective solution to increase the capacity of the DCI system is to use a high-order modulation format, which can improve the single-wavelength rate without changing the baud rate and channel interval settings by improving the spectral efficiency [12]. On the other hand, the higher-order modulation format has a lower tolerance for damage such as phase noise and non-linearity in the transmission system, and often needs to use complex digital signal processing (DSP) algorithms to achieve equalization and compensation, which will pose new challenges to the power consumption of the system [13; 14; 15; 16]. Therefore, how to design and use low-complexity digital signal processing algorithms to reduce the resource occupation required for hardware implementation as much as possible and reduce system power consumption on the premise of ensuring reliable system communication is a very noteworthy problem in DCI networks. As an emerging low-complexity carrier phase recovery (CPR) algorithm, principal component-based phase estimation (PCPE) is very promising in the field of high-speed optical communications, due to its low cycle-slip rate (CSR), and better mutual information (MI) performance under low signal-to-noise ratio (SNR) compared with the wildly used blind phase search (BPS) algorithm [17]. However, the existing research is only limited to the application of PCPE in the square-quadrature amplitude modulation (QAM) constellations [17; 18; 19; 20]; few studies in the literature deal with the compatibility of PCPE with odd-bit QAM constellations.