In the past, there have been numerous applications and studies in the field of free-space optical communication^[1–4]. Recently, numerous megaconstellation satellite systems have been both planned and deployed by various nations, and the demand for intersatellite optical wireless communication (IsOWC) is continuously increasing. Consequently, technologies related to IsOWC have garnered significant attention^[5,6]. Due to the significant link losses associated with the long-distance transmission^[7] and the alignment errors induced by the acquisition, tracking, and pointing (ATP) system^[8], the received optical power (ROP) in IsOWC systems is generally quite low. This implies that in IsOWC systems, the ratio of bit energy to noise power spectral density (Eb/N0) is typically low. Thus, the coherent optical communication system plays a critical role in IsOWC, due to the ROP sensitivity advantages. For phase-estimation algorithms, link distance, wavelength deviation, and limited computational resources in IsOWC in the same orbit are problems^[6,7,9].

The Viterbi and Viterbi phase-recovery algorithm (V&V algorithm) is widely used in many systems that are suitable for various modulation formats^[10–12]. It is also frequently incorporated as a key component within multistage phase-estimation frameworks^[13,14]. The V&V algorithm initially employs an Mth power operation to remove the modulation phase. Subsequently, it utilizes multisymbol averaging to mitigate the effects of noise, and finally, it estimates the phase error through angular computation^[15]. However, a notable drawback of the Mth power operations is their significant computational complexity. Additionally, in IsOWC systems, computational resources are limited. Many efforts have been made to simplify the Mth power operation to save multipliers^[16–18]. References [16,17] introduced a simplification of the Mth power operation by employing the absolute value method. This approach demonstrated marginal improvements in performance over the V&V algorithm in both quadrature phase shift keying (QPSK) and quadrature amplitude modulation (QAM) systems. Reference [18] applied the concept of exponential expansion to approximate the Mth power calculation, achieving phase recovery for QPSK with little performance degradation compared to the V&V algorithm. Reference [19] employed only logic operations and one real-number addition. This method matches the performance of the V&V algorithm while avoiding complex multiplications and angle computations. However, it requires the constant-modulus algorithm (CMA) to ensure accurate phase tracking.

In our previous work, we have proposed a multiplication-free phase-estimation algorithm^[20]. While it shared some similarities with the approach discussed in Ref. [19], our method introduced a step-size adjustment mechanism for phase tracking. In Ref. [20], we have validated the feasibility of our algorithm, demonstrating the significant advantages in terms of algorithmic complexity and computational delay. In this paper, we further explore the impact of various parameters of our proposed algorithm through a combination of theoretical derivation, simulation, and experiments. The main contributions of this work are shown as follows:

1. 1)QPSK phase-recovery algorithm with high performance and low complexity: Different from the traditional V&V algorithm, the proposed algorithm is based on constellation diagram partitioning; thus, no multiplication is needed in phase estimation. Additionally, due to the step-feedback tracking method employed by our algorithm, phase tracking exhibits minimal jitter.
2. 2)Guidance for optimizing phase-ecovery performance: Through a combination of theoretical analysis and simulation, this work provides a detailed explanation of how to determine the key parameters for implementing the proposed phase-recovery algorithm. This work offers guidance and references for achieving real-time phase recovery, optimizing the algorithm’s performance for enhanced accuracy and efficiency.
3. 3)Superior performance and robustness: Our proposed method exhibits better sensitivity, with an improvement of 1 dB at the bit error rate (BER) of the hard-decision forward-error-correction (HD-FEC) limit in the experiments. What is more, simulation results demonstrate that our algorithm maintains phase-tracking stability under low Eb/N0 conditions (4 dB @ QPSK). This robustness ensures reliable communication, which is critical for maintaining stable connections in IsOWC systems.

The proposed algorithm in this paper is based on the phase trajectories, illustrated in Figs. 1 and 2. The proposed algorithm processes the symbols obtained from the quadrature demodulation to achieve phase recovery. Notably, the proposed algorithm does not employ any multiplication operations during this phase estimation process.

In the phase rotating detection module, the QPSK constellation is divided into eight regions shown in Fig. 2. Let ski′ represent the real part of the kth symbol after correction, and skj′ represent the imaginary part of the kth symbol after the correction calculation. The rotating direction for the demodulated QPSK signal is defined using the following formula: Dk=h(ski′,skj′)=−sign(ski′)·sign(skj′)·sign(ski′−skj′),(1)where the sign(X) function is defined by sign(X)={1,X>00,X=0−1X<0.(2)

Considering that the phase rotating direction estimation based on a single symbol is susceptible to system noise, a smooth filter is added to achieve better estimation performance. Let nsum represent the number of summation symbols, and Dk′ represent the statistical results of the rotation direction. Thus, the Dk′ can be calculated using the following formula: $$\begin{gathered} Dk′=∑k=m1m2Dk, \\ m1=k|nsum·nsum,m2=(k|nsum+1)·nsum. \end{gathered}$$(3)

For final compensation, it is necessary to map the directions to the phases. Since the proposed algorithm in this paper is a feedback system, its compensation is not instantaneous. The phase compensation value φk for a feedback system is given as $$\begin{gathered} φk={0k≤ndelayφm3+g(Dk−ndelay′)k>ndelay, \\ m3=(k−1)|nsum·nsum, \end{gathered}$$(4)where ndelay represents the number of delayed symbols for calculation and compensation, and the g(X) function is defined by g(Dk′)=α·sign(Dk′),(5)where α represents the tracking step for each iteration. The range of α will be discussed in Sec. 4. From the above discussion, it is evident that no multiplication operations are used during the phase estimation. Only simple logic operations and additions are used.

After obtaining the φk, the phase correction results can be calculated through the following equation: ski′+iskj′=(ski+iskj)·eiφk,(6)where ski represents the real part of the symbol before correction, and skj represents the imaginary part of the symbol before correction.

Taking into account the unavoidable residual frequency offset on both the transmitter (Tx) and receiver (Rx) sides for the IsOWC system, the compensated phase deviation can be given by Δθk={C360°·fc·Tsymbol+Δθk−1−φk0≤k≤ndelayk>ndelay,(7)where C is the initial phase deviation, Δfc is the residual frequency offset after prior signal processing, and Tsymbol represents the symbol duration. To ensure that the proposed phase-recovery algorithm can effectively track the phase shifts, the following condition must be satisfied: limk→∞Δθk=0.(8)

If nsum is set to 1, we can directly analyze the impact of the signal processing chain. By integrating Eqs. (4), (5), (7), and (8), we can derive the following expression: α>360°·fc·Tsymbol.(9)

When the smooth filter is applied, the update of the compensation value is no longer immediate. Instead, the compensation value is updated only after a group of symbols. Consequently, the condition derived from Eq. (9) needs to be adjusted to reflect the delayed update mechanism under these circumstances. Therefore, the expression becomes α>360°·fc·nsum·Tsymbol.(10)

We validate the performance of the proposed algorithm through simulation and experiment. Table 1 summarizes the key system parameters.

As shown in Fig. 3, the QPSK signal is generated offline and mapped. We used PRBS15 with the generator polynomial X15+X14+1 to generate the binary data stream. Subsequently, the binary data was grouped into pairs of two bits and mapped to a Gray-coded QPSK constellation: {00} was mapped to 1+1j, {01} to 1−1j, {10} to −1+1j, and {11} to −1−1j. The generated QPSK signal has a periodic frame structure, with each frame containing 1000 PRBS15 data bits.

For the simulation, since the wavelength deviation does not directly affect the phase-recovery module, it is first mitigated by the laser temperature control circuit and preceding symbol synchronization stages before reaching the phase-recovery module. Consequently, in practical systems, the residual frequency offset handled by the phase-recovery module is typically in the range of tens of kilohertz to slightly over a hundred kilohertz. Therefore, in our simulations, a fixed 200 kHz frequency offset was added to the generated QPSK signal (significantly larger than the frequency offset in actual systems), and additive white Gaussian noise (AWGN) was introduced to simulate intersatellite link loss to test the proposed algorithm. In the receiver digital signal processing (Rx DSP), the processed signal is then subjected to symbol synchronization, followed by phase recovery. In the phase-recovery procedure, we compare the proposed algorithm with the classic V&V algorithm through the BER after demapping.

In the experiments, the generated electrical signal is loaded onto a 24-GHz@8bit arbitrary waveform generation (AWG, Tektronix, AWG7122C) operating at 10 GS/s. The output in-phase quadrature (IQ) signal from the AWG is then modulated through an IQ modulator controlled by a modulator bias controller (MBC) with a 1% feedback. Following this, a variable optical attenuator (VOA) is employed to adjust the ROP for evaluating the sensitivity. A 50–50 optical splitter is used to measure the ROP. At the receiver, the weak signal is implemented to a low-noise erbium-doped fiber amplifier (EDFA) for preamplification and filtered by an optical filter to eleminate the amplifier spontaneous emission (ASE) noise out of band. Then, the signal enters an integrated coherent receiver (ICR, Accelink, ICR-C-100-1) with a 20 GHz bandwidth. We use a heterodyne structure with an intermediate frequency of 2.5 GHz. The platform is shown in Fig. 4. The OE-converted signal from ICR is sampled by a 10 GS/s 8-bit analog-to-digital converter (ADC) through an oscilloscope (Tektronix, DSA 71254C) and processed offline. To achieve frequency synchronization, we use the oscilloscope to observe the intermediate frequency offset, then manually adjust the laser temperature to fine-tune the central wavelength. In digital signal processing, we also use fast Fourier transform (FFT), utilizing 8192 symbols for the frequency synchronization module, with an accuracy of 76 kHz and an estimation range of ±300MHz. Other processing steps remain consistent with the simulated receiver processing.

As demonstrated in Eq. (9), the tracking step α determines the angle change in each tracking iteration. For QPSK, the range for tracking step size is (0, 45°). The success of the tracking process is directly determined by the chosen step size. To evaluate the impact of the tracking step, the simulated BER performance as a function of tracking steps is shown in Fig. 5, where the number of symbols for smoothing is temporarily fixed at 18.

Shown in Fig. 5, a tracking step of 0.6 to 1.4 deg achieves optimal performance.

If the tracking step is below 0.6 deg, phrase tracing failure will occur. The minimum theoretical tracking step from Eq. (10) is ≈0.6deg, based on the system parameters shown in Table 1. This is consistent with the simulation results.

Moreover, when the step value is large, the step itself introduces phase adjustment noise into the signal. Additionally, the impact of misjudgments caused by noise becomes more severe, so the maximum value of the tracking step also has limitations.

Using a fixed step size for phase tracking, the proposed algorithm achieves greater stability. From Eqs. (4) and (5), it can be inferred that even if there is a misjudgment in the rotation direction, the impact of this error is constrained by the tracking step. The tracking step of the proposed algorithm is set to 1.4 deg, and the number of symbols for the smoothing filter is set to 18. We observe the phase estimation values for every 18 symbols to evaluate the accuracy of V&V and the proposed algorithm’s phase estimation through simulation.

As shown in Figs. 6 and 7, under the extreme low Eb/N0 conditions (4 dB), the V&V algorithm experiences significant jitter on the tracking angle. In contrast, even under such challenging conditions, the proposed method maintains minimal jitter in the tracking angle, ensuring accurate and stable phase estimation.

Increasing the number of symbols used for smoothing can reduce the impact of noise, but it also leads to a decrease in tracking rate in the proposed algorithm. The range for the numbers of smoothing symbols is [2,+∞). However, excessively large values are typically avoided, as they can complicate the implementation of the algorithm. To evaluate the impact of the number of symbols in the smoothing filter on performance, we conducted simulations. The tracking step is temporarily fixed at 1.4 deg, and Eb/N0 is set to 8 dB.

By substituting the simulation conditions into Eq. (10), we find that the number of symbols used for smoothing should be less than 49, which is consistent with the simulation results. Further, from Fig. 8, it is clear that the optimum smoothing symbol number is between 12 and 48. Employing an insufficient number of symbols for smoothing increases the probability of misjudgment, thereby compromising tracking performance. Conversely, utilizing an excessive number of symbols diminishes the frequency of feedback updates, which can result in performance degradation or even tracking failures.

When considering both tracking step size and the number of smooth symbols together, there is a range of optimal combinations, forming an optimal region. The optimal values for them are interdependent. Using a simulation-based approach, we evaluated the performance of different combinations of these two parameters by assessing the BER. The results are presented in Fig. 9, where the Eb/N0 is set to 8 dB.

As shown in Fig. 9, there is an optimal region for the tracking step and the number of symbols used for smoothing. The tracking step ranges from 0.2 to 1.5 deg, while the number of smooth symbols ranges from 1 to 45. This provides flexibility for practical implementation.

A low number of smooth symbols implies a high processing feedback rate, which might be unachievable in practice, as real-time processing on devices like the field-programmable gate array (FPGA) cannot work at clock speeds as high as the symbol rate. Therefore, by considering the actual number of symbols processed in parallel, an appropriate number of smooth symbols can be chosen, and an optimal point within this optimal region can be selected for practical use. Thus, our algorithm offers flexibility in FPGA implementation.

Loop delay can cause phase estimation lag, leading to inaccuracies in the estimated values. In real-time demodulation, the calculation delay will be generated by the calculation and phase correction in the FPGA or other devices. The unavoidable delays caused by computation and compensation can lead to performance degradation. In order to assess the impact of compensation delay, we performed numerous simulations, as shown in Fig. 10.

From Fig. 10, it can be observed that to ensure good algorithm performance, the impact caused by the delay should be minimized. A delay of 90 symbols is acceptable, at which the BER performance only degrades by 4% (Eb/N0=6dB).

To evaluate the performance of our algorithm, we further perform offline processing of data collected from the optical platform stated in Sec. 3. We have simulated intersatellite wireless communication over distances from 5000 to 13,300 km, where the ROP is significantly attenuated, ranging from −44 to −52dBm^[21,22].

The BER performance of both the V&V and proposed methods is compared in Fig. 11. The sensitivity of our proposed method reaches −49dBm with 5 Gbps QPSK (the HD-FEC limit). Our proposed method gains a better sensitivity of 1 dB compared with the V&V method. This results in a link distance improvement of 1100 km, providing additional redundancy for long-duration satellite communication. Furthermore, as the ROP continues to decrease, the difference in ROP required to maintain the same BER may expand to over 1 dB. This implies that if higher-performance codes [like the low-density parity-check (LDPC) code] are employed, our phase-recovery algorithm could deliver enhanced performance gains.

When the number of smooth symbols is N, the V&V algorithm for QPSK processing requires 8N real multipliers and 6N−1 real adders. The algorithm in Ref. [16] only requires 13N−1 real adders, while the algorithm in Ref. [19] requires 2N−1 real adders. Additionally, the algorithm in Ref. [19] requires the use of the CMA algorithm in the preceding stage to ensure its performance. By contrast, our proposed algorithm requires N real adders for comparing the modulus values of I and Q. For smoothing, it only involves N−1 simple additions of rotation correction directions valued at ±1 and 0. The computational load of each algorithm is shown in Table 2.

As observed in terms of computational load, the proposed algorithm does not require real multipliers and fewer real adders. This provides an advantage for implementing the algorithm on hardware platforms.

In this paper, we develop what we believe is a novel phase correction algorithm in QPSK-based IsOWC in the same orbit. From a theoretical perspective, we explain the process of the multiplier-free algorithm. Through simulations, we analyze several key parameters and provide guidance on their optimal selection. Additionally, we have demonstrated that our algorithm maintains stable phase-tracking capabilities, even at extremely Eb/N0 conditions (4 dB). This robustness ensures accurate and stable phase estimation, which is critical for maintaining the stable connections in IsOWC systems. In an ob2b transmission test, the sensitivity of our proposed method reaches −49dBm with 5 Gbps QPSK (the HD-FEC limit). It gains better sensitivity of 1 dB compared with the V&V algorithm. This results in a link distance improvement of 1100 km, providing additional redundancy for long-duration satellite communication. Our multiplier-free and robust algorithm meets the requirements of IsOWC systems and shows promise for future applications.