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Abstract
A novel diversity combining scheme, in conjunction with the complex-valued decision-directed least mean square (CV-DD-LMS) algorithm, is evaluated, and a real-time experimental validation is presented. This proposed scheme employs the CV-DD-LMS algorithm to concurrently perform beam combination and carrier phase recovery (CPR), thereby effectively reducing the overall complexity of digital signal processing. Furthermore, in the numerical simulation, under a low signal-to-noise ratio (SNR), a scheme utilizing the CV-DD-LMS algorithm effectively avoids cycle slips (CS) and outperforms schemes employing independent CPR modules. We experimentally validate this novel scheme by implementing it on an FPGA in a real-time 2.5Gb/s QPSK diversity-receiving system with three inputs. The back-to-back sensitivity is assessed using static received optical power, while the dynamic performance is evaluated by employing variable optical attenuators (VOAs) to simulate a power fluctuation at a frequency of 100kHz. The result proves that the implementation of the CV-DD-LMS algorithm yields stable performance while effectively reducing computational complexity.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In recent years, free-space optical (FSO) communication has become an attractive candidate in the satellite communication system [1–3]. Compared to microwave communication technology, FSO can provide a higher data rate, better security, and ease the shortage of spectral resources [4,5]. In the FSO channel, one major challenge is the fluctuation of the power and the phase of the received signals caused by atmospheric turbulence [1,6].
To solve this problem, the diversity receiving technique in the FSO system has been proposed, including aperture diversity and mode diversity [3,7–9]. Coherent optical communication was also introduced in FSO due to its better receiver sensitivity than the intensity modulation/direct detection scheme [10]. To implement diversity receiving in a coherent optical communication system, one practicable method is using the beam combination (BC) algorithms in digital signal processing (DSP). The purpose of the BC algorithm is to optimize the combining weights for the diversity of branches. However, when it comes to implementing the BC algorithm on a real-time DSP chip, its complexity becomes a crucial concern. At present, many studies and demonstrations about the diversity receiving system have been reported. In [4], the performance of maximum ratio combining (MRC) and selection combining (SC) was discussed in a 3-aperture diversity system by numerical simulation, and the result showed that the MRC outperforms the single-input, single-output scheme and the SC scheme. However, as the MRC uses the estimated SNR to determine the value of weights, it brings calculation latency and does not consider carrier phase differences between different received signals. In contrast, the decision-directed least mean square (DD-LMS) algorithm turned out to be capable in the coherent system. In [11], a BC scheme based on real-valued DD-LMS (RV-DD-LMS) with parallel structure is evaluated offline, where three separate coherent receiving DSP systems were used for clock recovery, channel equalization, and carrier recovery. In [6], researchers investigated the performance of the complex-valued decision-directed least mean square (CV-DD-LMS) algorithm in a 3-mode diversity receiving system with offline DSP. It was proved that this scheme can lower the required transmitted power by 4 dB. In this scheme, the BC and MIMO equalization is performed using a 2 × 6 FIR filter controlled by the CV-DD-LMS algorithm. Except for the studies mentioned above, there are many other investigations about the DSP-based diversity receiving algorithm including SC, equal gain combining (EGC), MRC, constant modulo algorithm (CMA), and DD-LMS [2,12–14], and all of them are based on numerical simulation or offline DSP.
Up to now, all the investigations about BC algorithm for FSO are evaluated using offline DSP and there has not been any report about verification on the real-time DSP platform such as FPGA or ASIC. Besides, in the FSO link, the failure of blind CPR caused by power fluctuation has not been deeply studied.
In this work, for the first time, a prototype of a 3-input coherent diversity receiver was realized based on a single FPGA chip (model: Stratix V 5SGSMD8K2F40C3). For a lower complexity and better stability, we also made a deep investigation into the CPR in the coherent diversity receiver. In the numerical simulation, we found that the CV-DD-LMS can mitigate the effect of cycle slip (CS) under low SNR, which is a common problem in traditional blind CPR algorithms like Viterbi-Viterbi phase estimation (VVPE) and can degrade the performance of combination [15,16]. Finally, we also carried out a real-time back-to-back transmission of 2.5Gbps QPSK with 3-lane diversity receiving using three variable optical attenuators (VOAs) to simulate the coupling efficiency variation. The results show that the FPGA prototype can achieve a bit error ratio (BER) of 7 × 10-3 under an ROP of −57dBm and achieve a BER lower than 1 × 10-3 under an ROP fluctuation from −43dBm to −53dBm with a frequency of 100KHz.
2. Principle of the diversity receiving system and beam combining algorithm
2.1 FSO diversity receiving system
The FSO communication system with mode diversity or aperture diversity receiving is presented in Fig. 1 [6,17]. In the following part of this paper, only the mode diversity receiving system will be considered. As the diagram shows, to suppress the degradation of signal quality brought by the atmospheric channel, a mode DEMUX is placed at the output of the optical antenna. The three modes (LP01, LP11a, and LP11b) of the optical field are distributed to the three output ports of the mode DEMUX and coupled into three single-mode-fibers (SMF). Here, three copies of optical signals carry the same information, while they are of different optical power since the atmospheric turbulence can randomly change the sum and distribution of their power. Then, these signals are sent into three individual coherent optical receivers and the following digital signal processing modules.
Fig. 1. Diagram of FSO diversity receiving system. (a) Diagram of mode diversity reception. (b) Diagram of aperture diversity reception. CTx, coherent optical transmitter; FMF, few-mode-fiber; SMF, single-mode-fiber; CRx, coherent optical receiver; DSP, digital signal processing.
2.2 DSP scheme of 3-lane diversity receiving system
As Fig. 1 shows, unlike the typical DSP scheme in the traditional coherent receiving system, when it comes to the diversity system, three coherent DSP modules are used to demodulate these input signals. Figure 2 illustrates the detailed view of a typical 3-lane diversity receiving DSP flow, where three input signals are firstly processed by the clock recovery and carrier frequency recovery (CFR) modules [11]. Then the Viterbi-Viterbi phase estimation (VVPE), which is one of the most commonly used blind phase recovery algorithms, is used to recover the carrier phase noise of signal branches [18]. The VVPE modules are optional depending on the scheme of combining we used. After that, the signals are sent to the BC algorithm with an optional real function. In the BC module, the output y equals the sum of N weighted inputs as (1). Based on the DDLMS algorithm, the loss function J is defined as (2), where e denotes the instantaneous error vector between the combining output y and the corresponding decision value yd. For the RV-DD-LMS and CV-DD-LMS, the weight update formulas are given by (3) and (4) respectively, where n denotes the n-th symbol. If only the real component of the product is considered, the BC algorithm will be switched to RV-DD-LMS; otherwise, it will be switched to CV-DD-LMS.
Fig. 2. (a) DSP flow of 3-lane diversity receiving system. CFR: carrier frequency recovery. VVPE: Viterbi-Viterbi phase estimation. (b) The implementation methods of CPR and BC in different schemes.
Considering the practical implementation of the DSP, three duplicates of VVPE can heavily increase the complexity. Besides, under strong power turbulence, the SNR can be quite low, leading to CS during the process of VVPE [15]. As is previously revealed in [19], except for performing BC, the complex-valued DD-LMS (CV-DD-LMS) is also capable of conducting the carrier phase estimation. Thus, to lower the overall complexity, and improve the stability, in the next section, we will prove the feasibility of scheme III, i.e., integrating CPR and BC into a CV-DD-LMS BC module via numerical simulation.
3. Numerical simulation of DD-LMS-based beam combining algorithm
In this section, the performance of three different schemes listed in Fig. 2(b) and selective combination (SC) will be assessed. The simulation setup is shown in Fig. 3.
As is shown in Fig. 3, the signals are subjected to carrier frequency offset (CFO) and carrier phase noise by employing two distinct lasers, one at the transmitting end and the other at the receiving end. At the receiver side, three input signals are digitalized by three ADCs and fed into the 3-lane diversity receiving DSP module which is previously illustrated in Fig. 2.
Firstly, we evaluate the performance of different schemes under different laser linewidth and SNRs. The SNRs of three received signals are set to the same value. To ensure the simulation aligns with reality, we have taken into account the impact of imperfect performance in the CFR module. The CFO computed by the CFR module exhibits a discrepancy with the actual value, which will impact the subsequent modules. This discrepancy is referred to as the residual CFO. The residual CFO after carrier frequency recovery is fixed at 100KHz. The block length of VVPE is set to the optimal value depending on the laser linewidth. Every result of BER shown in Fig. 4 is obtained by calculating the overall BER of 2.6 million received bits.
Fig. 4. BER of different BC schemes under different SNRs with the laser linewidth of (a) 25KHz; (b) 50KHz; (c) 75KHz; (d) 100KHz. Scheme I: VVPE followed by the RV-DD-LMS. Scheme II: VVPE followed by the CV-DD-LMS. Scheme III: CV-DD-LMS performing CPR and BC simultaneously.
As is presented in Fig. 4, when the SNR exceeds 7 dB, scheme III exhibits comparable performance to other schemes. Under low SNR, scheme III, where CV-DD-LMS is used for CPR and BC, outperforms other schemes due to the significant degradation caused by CS.
To further investigate the performance of CV-DD-LMS in carrier recovery and its resistance against CS, we calculate the residual carrier phase error by subtracting the carrier phase estimated by different schemes from the simulated laser’s phase noise. According to Fig. 4, considering a laser linewidth of 100 kHz and residual CFO of 100 kHz, two typical values of 3 dB and 5 dB are selected. The results under these two typical SNRs are exhibited in Fig. 5. Inset(a) and inset(b) show that, under low SNR, VVPE can frequently lead to CS, which makes the carrier phase error jump by an integral multiple of π/2. Moreover, lower SNR increases the likelihood of carrier sense (CS) occurrence. In that circumstance, the RV-DD-LMS and SC algorithms cannot deal with the CS as they are unable to correct carrier phase offsets between the signal copies. If the VVPE is followed by the CV-DD-LMS (Scheme II), sudden cycle slips brought by VVPE can be automatically corrected by adjusting the phase of weights as is revealed in Fig. 5(c) (in comparison with Fig. 5(a)) and Fig. 5(d) (in comparison with Fig. 5(b)): the little impulses indicate that, when CS occurs, the carrier phase error suddenly increases by an integral multiple of π/2 and is soon mitigated. However, a comparison between Fig. 5(c) and Fig. 5(d) reveals that this auto-correction mechanism can easily fail if the CS occurs too frequently under a lower SNR (see the circled part in Fig. 5(c)). Compared with scheme II, our proposed scheme III exhibits superior performance under these two typical SNRs. By employing our proposed scheme III, the VVPE is bypassed and both CPR and BC are simultaneously executed using the CV-DD-LMS BC algorithm. Consequently, it can be observed from Fig. 5(e) and Fig. 5(f) that CS does not occur, and the residual carrier phase error remains consistently close to zero.
Fig. 5. (a) (b)Residual carrier phase error of signals processed by VVPE with SNR of 3 dB and 5 dB respectively, measured at point A in Fig. 2; (c) (d) Residual carrier phase error of signals processed by VVPE and CV-DD-LMS (Scheme II) with SNR of 3 dB and 5 dB respectively, measured at point B in Fig. 2; (e) (f) Residual carrier phase error of signals processed by CV-DD-LMS without VVPE (Scheme III) with SNR of 3 dB and 5 dB respectively, measured at point B in Fig. 2.
To summarize, scheme III can effectively avoid the deterioration caused by the randomly occurring CS and achieve the lowest BER under low SNR. Moreover, when the SNR is sufficiently high to ensure negligible impact from CS on the quality of BC, scheme III exhibits comparable performance to alternative schemes.
Secondly, we conduct a subsequent simulation to assess the long-term combining performance considering dynamic coupling efficiency. The power spectral inversion method is employed to generate a phase screen, enabling the simulation of atmospheric distribution in free space optical communication. The simulation is based on Kolmogorov power spectral density, which shows the spatial characteristic of turbulence [20]. Besides, the temporal characteristic is described by Taylor’s hypothesis of frozen turbulence [21]. Taylor's hypothesis believes that within a certain time scale, the spatial characteristics of turbulence are almost in a “frozen” state, that is, the spatial characteristics of turbulence at a certain point in space remain unchanged for a short time, while the temporal characteristics change with the lateral wind speed. In this simulation, the strength of the turbulence is defined as D/r0, where D is the far field spot diameter and r0 is the coherence length. The value of D is set to 0.1446 m and the value of r0 is set to 0.009 m, leading to D/r0 of ∼16. Besides, the link length is set to 5 km, the wind speed is 9.38 m/s, and the Greenwood frequency is 383 Hz, correspondingly. The Simulation steps are as follows: first, a large-scale phase screen of 10.5m × 10.5 m is generated, and then a series of continuous 1.5m × 1.5 m small-scale phase screens are obtained by randomly rotation and selection from the large screen, whose speed is determined by the wind speed.
After passing through atmospheric turbulence simulated using the phase screen, the light field is focused and coupled to the fiber core by the lens at the receiving end, and the end face of the fiber core is located on the back focal plane of the lens. Usually, optical fiber coupling efficiency ${\eta _f}$ is used to measure the spatial optical coupling effect, which is defined as the ratio of the optical power ${P_f}$ coupled to the optical fiber to the incident optical power ${P_i}$ of the receiving aperture plane [22]:
The coupling efficiency and the variation of three combining weights are shown in Fig. 6, while the power of AWGN for each signal remains constant, resulting in a fluctuating SNR with an average of 3 dB. The laser linewidth is set to 40KHz. The observation in Fig. 6(b) reveals that the variation of three weights, calculated by the CV-DD-LMS algorithm, exhibits a similar trend to the fluctuation of coupling efficiency, as indicated by corresponding changes in weight magnitudes.
Fig. 6. (a)The coupling efficiency and (b) corresponding variations of three combining weights calculated using the proposed Scheme III.
Figure 7 shows the residual phase error following CPR for different schemes along with the corresponding BER distribution. The residual phase error for scheme I is measured at point A shown in Fig. 2, while for schemes II and III, it is measured at point B shown in Fig. 2. Figure 7(a-c) show that, influenced by the turbulence, the signals of three modes suffer from CS after VVPE. Figure 7(d) shows that, in scheme II, the CS can be mitigated by the following CV-DD-LMS. However, when all three input modes are polluted by CS (after 6 ms), the performance of the followed CV-DD-LMS algorithm deteriorates and fails to correct the occurred CS in VVPE. In contrast, the proposed scheme III, as Fig. 7(e) shows, effectively maintains the residual phase error around zero, thereby achieving complete recovery of carrier phase noise and successful mitigation of CS during this period. Moreover, Fig. 7(f) reveals that, except for scheme III, the remaining two schemes exhibit a BER exceeding 0.1 in over 40% of the measured values due to the presence of CS. In contrast, the measured BER of scheme III demonstrates an improvement, with 98% of the measurements below 0.1, leading to an average BER of 2.45E-2. On the other hand, scheme II only achieves an average BER of 2.02E-1.
Fig. 7. Simulation results with time-varied coupling efficiency. The residual phase error of (a) LP01, (b) LP11a, and (c) LP11b processed by VVPE. The residual phase error of the combined signals processed by (d) scheme II and (e) scheme III. (f) The distribution of BER (measured for every 0.1 ms).
We also compare the computational complexity (in terms of real-value multiplier) of these algorithms with parallel structure, considering that the symbol rate typically exceeds the clock frequency of real-time DSP chips several times. The result is listed in Eq. (2–4), where m is the number of signal copies, NP is the number of parallel lanes for each branch and l is the number of symbols used for updating the weights. In Eq. (2) the first term in brackets represents the complexity of the 4-th power operation in VVPE, and the second term represents the multipliers used for phase rotation. In Eq. (3), the first term and the second term give the complexity of weighted sum and weight update, respectively. In Eq. (4), the complexity is doubled due to the inclusion of both the real and imaginary components in CV-DD-LMS.
In a typical 3-lane diversity system, m is set to 3. In our proposed FPGA prototype, l corresponds to the number of parallel lanes employed NP, thereby aggregating NP weight changes to update each signal's weight in one clock period. Thus, the final complexity is listed in Table. 1.
Table 1. Number of real-value multipliers in different schemes
4. Real-time diversity receiving experiment
To investigate the performance of the CV-DD-LMS in a practical diversity system, we conduct a real-time experiment utilizing an FPGA chip in a 1.25GBaud QPSK back-to-back transmission system. The FPGA model is Stratix V 5SGSMD8K2F40C3 with a clock frequency of 156.25 MHz. As is shown in Fig. 8, a pulse pattern generator (PPG) is used at the transmitter side to generate two 1.25Gbps independent pseudo-random bit sequences (PRBS), which are modulated to the in-phase and quadrature branches separately. The variation in coupling efficiency is simulated by attenuating three copies of the modulated optical signal using three separate VOAs. The VOAs’ attenuation is controlled by a pulse generator. To simulate the receiving end, the three signals are initially amplified using Erbium-Doped Fiber Amplifiers (EDFAs). To eliminate out-band noise introduced by the EDFAs, Santec OTF-320 optical bandpass filters (OBPFs) are employed. Subsequently, we use three sets of receiving equipment comprising a coherent receiver (Fujitsu FIM24706) and an ADC (EV8AQ160) working in two-channel mode with a sampling rate of 2.5GSa/s.
Fig. 8. (a) The real-time experiment setup and the real-time DSP flow of the 1.25GBaud QPSK transmission system with the three inputs diversity receiving. VOA: variable optical attenuator. EDFA: erbium-doped fiber amplifier. OBPF: optical bandpass filter. GCDR: Gardner clock and data recovery. (b) The picture of the VOA. (c) The picture of the FPGA board.
The entire DSP flow is implemented in the FPGA, where three Gardner clock and data recovery (GCDR) modules are used for the timing recovery. Subsequently, the CV-DD-LMS BC module compensates for carrier phase noise induced by the discrepancy in optical path arrival at three local oscillators, while simultaneously combining three signal copies. The total number of Adaptive Logic Modules (ALMs), DSP blocks, and RAM blocks used by a design in an Intel FPGA device is frequently utilized as an indicator for assessing the hardware resources consumed by the design [24]. Based on this indicator, this DSP flow occupies a relatively small portion of FPGA hardware resources, and the detailed utilization of FPGA resources is presented in Table 2. As mentioned in Table 2, the Signal Tap module is used for capturing and displaying the real-time signal behavior in an Intel FPGA design, which may be disregarded during resource measurement. Without three additional VVPE modules, only 15% of the logic utilization (in ALMs) and 34% of the total DSP blocks are used.
Table 2. Resource utilization of the Rx DSP
Firstly, we evaluate the back-to-back sensitivity. In this experiment, each signal is of the same received optical power (ROP) measured at the output port of VOAs. The output data from both the GCDR modules and the DD-LMS BC module are preserved for comparative analysis. The result is presented in Fig. 9, where the BER of each input lane is obtained by processing the post-GCDR data using offline VVPE, while the BER of the real-time processed data is directly calculated in FPGA without any offline processing. It should be noticed that the performance and configuration of these integrated coherent receivers vary, resulting in differences in BER among these received signals. Additionally, for comparison purposes, we also processed the post-GCDR data using the offline CV-DD-LMS BC algorithm. All the BER results are calculated using more than 2.5 × 105 bits.
The results in Fig. 9 show that, under a static ROP, the real-time DD-LMS BC algorithm successfully performs CPR and BC, without any reliance on an additional CPR module. Additionally, its output outperforms that of all three input lanes. Furthermore, the real-time version exhibits comparable performance to its offline counterpart.
Then the dynamic performance is evaluated by employing three VOAs to simulate the fluctuations in coupling efficiency. Given that the cutoff frequency for the temporal variations of ROP under turbulence remains approximately to the order of one kHz [9], we employ a 100 kHz power fluctuation to illustrate the power tracking capability of the proposed Scheme III. This is aimed at demonstrating its ability to respond to sudden situations. The three VOAs are driven by a pulse generator, which generates a sine wave with a frequency of 100KHz and the optical power at the output port of VOA ranges from −43dBm to −53dBm. It should be noticed that the variation of phase is caused by the transmission fiber length and the laser center frequency drift. Because of the disparity in fiber length among the three receiving links, the phase variation is non-synchronous. The reset signal is used as the capture trigger during data collection from the FPGA. The values of the weights will return to 1/3 upon being reset.
The experimental results are shown in Fig. 10. It can be observed that the variation of the weights’ amplitude can keep pace with the 100KHz fluctuation of the optical power. Besides, in inset (c) and inset (d), after the reset signal is released, the amplitude and phase of weights soon converge in less than 0.5us. Finally, we calculate the BER with a power fluctuation at a frequency of 100kHz: the BER of the combined signal is 1E-3 while the three input signal copies cannot be demodulated as they are suffering from cycle slips several times in this period of 100us.
Fig. 10. The variation curve of the weights in the real-time experiment. (a) The amplitude of weights. (b) The phase of weights. (c) The amplitude of weights after reset (enlarged). (d) The phase of weights after reset (enlarged).
The result proves that, without any help of extra CPR modules, our proposed real-time diversity receiver with CV-DD-LMS algorithm can realize a combination of 3 QPSK signals with different carrier phases.
In conclusion, the prototype based on our proposed scheme demonstrates superior performance compared to input signals processed separately in both static and dynamic attenuation channels, thus highlighting its potential for practical applications. Additionally, it can mitigate the power fluctuation at a frequency of 100kHz, which surpasses the coupling efficiency change rate observed in real FSO channels.
5. Conclusion
In this study, through numerical simulations, we prove that our proposed BC scheme based on the CV-DD-LMS algorithm can mitigate the performance degradation caused by sudden CS, while also providing improved performance under long-term turbulence simulations with reduced complexity compared to traditional BC schemes. Moreover, the FPGA prototype of a 3-input 2.5Gbps QPSK diversity receiver with the real-time experiment demonstrates that, with either static or dynamic (up to 100KHz) coupling efficiency, the receiver can provide stable performance gain, which makes it a potential solution for a practical mode-diversity scheme in the FSO system.
Funding
National Key Research and Development Program of China (2022YFB2903201).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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