## 1 Introduction  
Free-space optical (FSO) communication represents a cutting-edge technology that leverages free space--such as vacuum, atmospheric, and oceanic environments--as the transmission medium, with lasers serving as the carriers of information [1]. This technology has garnered significant attention in the field of optical communication due to its substantial capacity, high bandwidth, and robust security features [2]. Notably, coherent detection methods exhibit considerable promise for advancing FSO communication systems [3], particularly when paired with high-order modulation techniques, which have spurred considerable research interest [4; 5; 6]. However, the performance of FSO communication systems is affected by various limiting factors such as weather changes and device manufacturing processes. Transmission through atmospheric channels can further degrade system performance, negatively impacting the received signals [7; 8; 9; 10]. To address these challenges, digital signal processing (DSP) is employed at the receiver end to reduce the overall bit error rate (BER) [11]. The DSP workflow is illustrated in Figure 1. Figure 1 illustrates the signal processing sequence as follows: The received signal first undergoes I/Q balancing, utilizing the algorithm from [12] to achieve amplitude and phase equalization and compensation between the I and Q paths. Subsequently, the timing recovery module employs the algorithm from [13] to track and compensate for frequency offset and phase jitter between the transmitter clock and the receiver ADC sampling clock, thereby preventing degradation in the system's BER performance. Next, polarization demultiplexing is primarily employed for channel equalization. The proposed Time-Tagged QPSK partitioning (TTQP) frequency offset estimation algorithm is then applied to estimate and correct the frequency offset, enhancingsystem performance and reducing the bit error rate (BER). Following this, carrier phase recovery is performed to estimate and compensate for phase distortion. The decision-directed least-mean-square (DD-LMS) algorithm is then utilized to improve signal quality. Finally, the system's BER is calculated to assess the error performance after DSP processing. Depending on the application scenario, DSP in a digital coherent receiver can be classified into two types: offline and online. Offline processing involves applying the DSP algorithm to stored signals using simulation software, which can only compensate for pre-stored data, but provides valuable simulation validation for online processing. In contrast, online processing entails real-time signal processing, considering both resource utilization and timing constraints [14; 15; 16]. While offline processing is limited to compensating stored data, it offers an effective means for the simulation verification of online processing [17]. Based on this, we present an offline design of a parallel DSP algorithm in this study. Currently, FOE algorithms can be broadly categorized into two types: frequency-domain carrier FOE and time-domain carrier FOE [18]. For example, the frequency-domain maximum spectrum line search algorithm based on the 4 th power fast Fourier transform (4 th-FFT) [19] offers high estimation accuracy, but is characterized by high complexity. Hardware implementations of this algorithm must balance complexity and performance, which can limit its practical applicability [20]. Although the algorithm is suitable for various modulation formats and provides high estimation precision [18; 21], its hardware implementation consumes significant resources and has certain limitations. Time-domain FOE does not involve frequency domain transformations and is generally less complex than frequency-domain FOE [22]. For instance, the training sequence (TS)-based FOE algorithm [23] utilizes a training sequence to eliminate the modulation phase of the received signal. This approach avoids the computationally intensive 4 th power operation, thereby reducing the complexity of the frequency offset estimation (FOE).