## 1 Introduction
Optimizing pilot placement and density is therefore crucial for maintaining robust phase noise compensation in practical implementations. To address the ever-increasing demand for capacity, recent research has explored the use of probabilistic shaping to increase the Euclidean distance between constellation points and enhance digital signal processing (DSP) algorithms [26]. A promising approach combines probabilistic shaping with extended Kalman filtering [27]. Specifically, a rate-adaptive coded modulation scheme with probabilistic signaling has been integrated with the BPS algorithm [28] in order to reduce overhead while optimizing system performance. Previous studies have explored methods for correcting phase cycle slips by integrating coarse phase detection with interpolators, along with the use of a maximum likelihood (ML) algorithm to correct residual phase offsets [29]. Additionally, advancements in FEC techniques have enabled the reduction of required pre-FEC signal-to-noise ratio (SNR) values, allowing signals to be recovered at lower SNR levels. These developments require more robust phase recovery and improved performance in challenging communication environments. In this study, we present the implementation of pilot-aided carrier phase recovery (PA-CPR) in a burst-mode configuration, where pilot symbols are inserted into every \(L\) consecutive symbol. To optimize efficiency, the burst length was set at \(L=32\), limiting the overhead to approximately 3% of the total number of symbols. Initially, a linear regression interpolation (LRI) method was used for carrier phase recovery. This approach was chosen over frequency filter-based interpolation to avoid dependence on the entire signal for phase tracking. To further enhance performance, the LRI method was improvedby integrating a hard-decision filter. This filter uses a noise variance threshold to identify and eliminate outliers, significantly improving accuracy compared to the standalone LRI method. We then explore an advanced technique to refine and stabilize phase tracking. Although smooth fitting curves have been a staple in signal processing for decades [30,31], our novel approach incorporates a cosine filter to remove outliers, followed by a localized weighted smoothing technique to improve phase tracking accuracy. This approach achieves significant computational efficiency by using pilot symbols for phase tracking. Using only the information between the pilot symbols in a burst mode, we eliminate the need to process the entire signal for phase interpolation, resulting in substantial computational savings. The structure of this paper is organized as follows. Section 2 describes the system model, providing a detailed explanation of the PA-CPR technique, the LRI model, and the locally weighted regression (LWR) method. In addition, relevant pseudocodes are included to facilitate the replication of the proposed algorithms. Section 3 presents the simulation setup and results, offering a comparative analysis of the various methods studied. Section 4 analyzes the computational complexity of the cosine-robust locally weighted scatterplot smoothing algorithm (Cosine-RLOWESS) compared to the BPS method and the run-time comparison. In Section 5, the results of our research are further examined by comparing the performance of pilot-based techniques and evaluating their computational complexity. Finally, Section 6 summarizes the key findings and conclusions of this work.