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Abstract
Alamouti space-time block code (STBC) combined with a simple heterodyne coherent receiver can realize polarization-insensitive phase-diversity detection to reduce the cost. In the receiver, a joint equalizer has been used for STBC’s polarization demultiplexing and phase tracking. However, the joint equalizer requires two different step size parameters to update the tap weight coefficients for polarization demultiplexing and the phase noise estimation. This leads to the search process being complex so requiring more iterations for convergence. In this paper, we propose a partially decoupled equalizer that consists of a polarization and phase decoupled equalizer (PPDE) and a pilot-aided blind phase search (P-BPS) algorithm to accelerate the convergence and improve the phase noise tolerance. By theoretically calculating the phase noise, the PPDE can achieve polarization demultiplexing with only one single step size parameter, thus suppressing the searching space and greatly reducing the iterations required for convergence. In the carrier phase recovery stage, the P-BPS algorithm can effectively improve the phase noise tolerance and solve the cyclic slip problem of BPS. We conduct numerical simulations and an experiment to transmit a quadrature phase-shift keying (QPSK) signal. The results demonstrate that the number of iterations required for PPDE convergence is only half of that of the joint equalizer while maintaining polarization-insensitive characteristics in large phase noise. Meanwhile, the achievable linewidth tolerance of P-BPS is increased by three times compared with DD-LMS.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Numerous researchers and International standard organizations are investigating ways to increase the data rate of passive optical networks (PONs) in a cost-effective manner to meet the ever-increasing demand for optical access networks [1,2]. Although the current 50 G PON might cover the majority of application scenarios at the current stage, it is difficult to evolve to 200 G and beyond due to the operation principles of burst mode [3,4]. Despite all existing PONs utilizing direct-detection, simplified coherent detection by reducing the cost and digital signal processing (DSP) complexity has attracted increasing attention in short-reach applications, such as coherent lite [5,6], self-coherent [7,8] and etc. These are envisaged to target at 200 Gb/s or even higher bit rates per wavelength for future PONs with higher power budget. For the implementation of coherent PON, low cost, simplified DSP and lower power consumption are required for the end-user side. The self-coherent with single-ended photodetectors (PDs) can cope with the high cost caused by balanced photodetectors (BPDs) which require a high common mode rejection ratio [9,10]. However, the signal-signal beating interference (SSBI) resulting from single-end PDs has to be mitigated in the digital domain. A simple coherent receiver is demonstrated using symmetric 3 × 3 couplers in conjunction with three single-ended PDs instead of 90° optical hybrids with two pairs of BPDs [11]. This method can realize DSP-free detection. However, it is sensitive to dispersion and needs a guard band that reduces the spectral efficiency. The main cost and complexity of conventional coherent detection are from the four BPDs and analog-to-digital converters (ADCs) for polarization demultiplexing and carrier recovery. Self-homodyne coherent detection (SHCD) based transmissions can eliminate the impact of the carrier phase noise and frequency offset, permitting a remarkable tolerance of laser linewidth [12]. However, although the power consumption of DSP is greatly reduced, the complexity of the receiver is not decreased. The single-polarization coherent receiver is one of the common ways to reduce the receiver complexity, which only requires half the components of a conventional polarization-diverse coherent receiver. Meanwhile, single-polarization heterodyne coherent detection can further simplify the single-polarization coherent receiver with only half of PDs and ADCs [13,14]. However, the key problem of the single-polarization coherent receiver is that the receiver sensitivity is dependent on the state of polarization (SOP) of the incoming signal. In such a receiver, the SOP of the incoming signal needs to be tracked optically or should be aligned with the SOP of the LO laser to maintain the system performance which is impractical to implement on the ONU side. Therefore, the single-polarization coherent receiver that can simultaneously achieve polarization-insensitive reception is expected for ONUs in future PONs. Although polarization-insensitive reception can also be achieved by polarization beam splitters (PBSs). The presence of PBS brings challenges for full monolithic integration and results in increased production costs [15]. The proposal of a single-polarization heterodyne coherent receiver combined with the Alamouti space-time block code (STBC) eliminates the requirement for optical polarization controllers at the receiver [16]. Although the receiver sensitivity is reduced by 3 dB compared to the polarization division multiplexing (PDM) system operating at the same bit rate [17], the cost of the Alamouti coherent receiver combined with heterodyne detection is only one-quarter of the conventional coherent detection system. This technology has been simulated and experimentally verified in a variety of scenarios [1,5,18–20]. Some simulations and experiments also prove that this technology can achieve 200 G coherent PON transmission and obtain a 34.5 dB power budget [21,22]. In 2016, Faruk proposed a DSP algorithm as a joint equalizer in SC Alamouti STBC-assisted simplified heterodyne receiver [18]. This equalizer iteratively updates the tap weight coefficients and phase noise simultaneously with two different step size parameters. The optimization of two mutually influencing step size parameters at the same time is complicated. This means that an N × N (assuming each step parameter is searched N times) grid search is required to obtain the optimal step size parameter. It is difficult to find the global optimal point and consumes more time for convergence.
In this paper, we propose a partially decoupled equalizer consisting of a polarization and phase decoupled equalizer (PPDE) and a pilot-aided blind phase search (P-BPS) algorithm. The PPDE is used in the polarization demultiplexing stage to accelerate the convergence. The P-BPS is used in the carrier phase recovery stage to improve the tolerance to the laser linewidth. In the polarization demultiplexing stage, we deduce the phase noise based on the training symbols and tap weight coefficients. This enables the PPDE only use one step size parameter for polarization demultiplexing. The advantage is that the joint optimization of the two step size parameters becomes a 1-dimensional convex optimization. This accelerates the finding of the global optimization for convergence and reduces the number of searches from N2 to only N times. In the carrier phase recovery stage, we use the P-BPS algorithm to achieve the carrier phase recover, where the phase noise is also estimated based on the calculation in the polarization demultiplexing stage. Compared with the DD-LMS, the P-BPS greatly improves the linewidth tolerance. We conduct 8-GBaud and 12-Gbaud QPSK numerical simulations, and an experimental platform with Alamouti coding, respectively. The simulation results demonstrate that the proposed equalizer is also polarization-insensitive. Moreover, the proposed equalizer significantly reduces the number of iterations for convergence. The tolerance of laser linewidth is increased by about three times compared with the joint equalizer. The experiment results further demonstrate that the required number of iterations for convergence of the partially decoupled equalizer is only half of the joint equalizer in large phase noise.
2. Principle of partially decoupled equalizer
The Alamouti STBC introduces a 50% redundancy to use the channel twice during two symbol durations as shown in Fig. 1(a) [23,24]. Assuming that the phase noises experienced by two consecutive time slots are the same, the signal received by the simplified coherent receiver with a linear noise-free response can be expressed as (only consider the SOP$H = \left[ \begin{array}{cccc} {h_{xx}}&{h_{xy}}; {{h_{yx}}}&{{h_{yy}}} \end{array} \right]$ and phase noise):
Fig. 1. (a) Illustration of Alamouti coding in two polarization modes; (b) The structure of single polarization heterodyne coherent receiver.
In Eq. (1), the contribution of phase noise from two MIMO inputs is conjugated for a particular MIMO output, so the conventional 2 × 2 MIMO equalizer [25] is not applicable in the Alamouti STBC system. Therefore, a joint equalizer has been used for simultaneous polarization tracking and carrier phase estimation (CPE) as shown in Fig. 2(a) [18,19]. In Fig. 2(a), the received signal first goes through the serial-to-parallel (S/P) module. Then, in the convergence stage, the joint equalizer implements the de-polarization and phase noise estimation. While in the carrier phase recovery stage, the DD-LMS algorithm is used to update the phase noise. Two step size parameters are required to iteratively update the tap weight coefficients and phase noise. The conjugate phase relationship and joint optimization of two parameters lead to the coupled effect of the SOP and phase noise estimation. The equalizer has to intensively adjust the step size parameters and find the global optimum. This means that the equalizer has large optimization complexity since we need to conduct a grid search on the 2-dimensional plane to obtain the optimal step size parameters. Besides, using two mutually influencing step size parameters to implement carrier recovery also requires a large number of training sequences to ensure convergence.
Fig. 2. (a) The structure of the joint equalizer; (b) The structure of the PPDE. S/P: serial-to-parallel, P/S: parallel to serial, ${[{\cdot} ]^\ast }$: complex conjugate.
In order to reduce the complexity of the search for the optimal step size parameters and the number of training symbols required in the initial convergence stage, we propose a PPDE in the Alamouti STBC system that only requires a single step size parameter as shown in Fig. 2(b). Assuming that the transmission link is linear and noise-free, it can be seen from Eq. (1) that the phase of the received signal is jointly affected by the phase of the transmitted signal, the phase noise from the laser linewidth and the phase of the channel matrix. To mitigate the interaction of different effects, we first remove the influence from the phase of the transmitted signal by a certain transformation of the training and received signal as Eq. (2),
In Eq. (2), the phase after a certain transformation on the left side of the equation is only related to the phase of the channel matrix and laser phase noise. Considering that the matrix of tap weight coefficients $W = \left[ {\begin{array}{{cc}} {{w_{xx}}}&{{w_{xy}};\begin{array}{{cc}} {{w_{yx}}}&{{w_{yy}}} \end{array}} \end{array}} \right]$ estimated by the equalizer after convergence and the channel matrix H are inverses to each other, the parameters in the channel matrix can be expressed as,
When the equalizer converges, ${\theta _1} \approx {\theta _2}$. Therefore, the laser phase noise can be estimated directly without iterations so that we can use only one step parameter to achieve polarization tracking and mitigate the phase noise. In this way, the complexity of the search process for the optimal step size parameter is reduced from N2 times for a 2-dimension grid search to N times for a 1-dimension linear search.
The iterative process of PPDE in the convergence stage is the same as that of the joint equalizer. The error signal is calculated using the LMS algorithm as
where ${d_{o/e}}$ are the training symbols. The tap weight coefficients of FIR filters are then updated phase-insensitively using the LMS algorithm with only one step size parameter µ as,To help convergence, some constraints should be added to the update of the tap weight coefficients, as shown in Eq. (7).
In the carrier phase recovery stage, the BPS algorithm, DD-LMS algorithm and phase-locked loop (PLL) are both well-chosen [18,27]. Although the phase noise tolerance of the BPS algorithm is larger than that of DD-LMS, the BPS algorithm is affected by the cycle slip problem leading to system instability. The performance of BPS under large phase noise can be greatly improved by inserting pilot symbols (Fig. 3). However, due to the conjugate property of Alamouti STBC, it is difficult to directly calculate the phase noise by pilot symbols. In this paper, we reuse the deduced phase of Eq. (4) to solve this problem and enable the pilot-based BPS method. The pilot insertion rate is 1/Ns (i.e. one symbol out of every Ns symbols). The phase noise of the information sequence is estimated by the BPS algorithm. To reduce the impact of cycle slip on system performance, we need to obtain a relatively accurate phase noise in the estimation process of the BPS algorithm for correction of the estimation results. Equation (4) enables us to obtain relatively accurate phase noise by pilot symbols even when polarization and phase noise are coupled to each other. These pilot symbols can also be used to update the tap weight coefficients when the rotation of SOP (RSOP) is large. In this case, the iterative update of the polarization is only performed at the pilot symbol, and the iterative process is the same as that of the initial convergent stage.
3. Simulation results and discussions
To verify the principle of the proposed equalizer, we conduct numerical simulations for 8-GBaud QPSK transmission systems over 100 km of standard SMF, where $3.2 \times {10^5}$ QPSK symbols are encoded according to the Alamouti STBC format shown in Fig. 1(a). The SC Alamouti coded signal is generated by a dual-polarization (DP) transmitter. We use an oversampling factor of 4 and a root-raise cosine (RRC) filter with a roll-off factor of 0.1 for pulse shaping. The signal is then transmitted over the fiber whose transfer function is assumed to be as,
To verify the tolerance of the proposed equalizer to polarization changes over the entire Poincaré sphere, the parameters $\theta$ and $\phi$ in Eq. (8) are swept between $- {\pi / 2}$ to ${\pi / 2}$. Figure 4(a) shows the Q2-factor for such a 2-dimensional sweep and the SOP is static. The laser phase noise corresponds to the combined linewidths symbol duration product $\Delta v{T_s}$ of $1 \times {10^{ - 4}}$. As shown in Fig. 4, the partially decoupled equalizer can also make the system performance robust to any polarization rotation.
Fig. 4. (a) Q2-factor performance results for a 2-dimensional sweep of polarization states when the SOP is static; (b) The tolerance of RSOP rate of the two equalizers when the SOP is dynamic.
We also further analyze the tolerance of the joint equalizer and the partially coupled equalizer to dynamic SOP. The dynamic SOP can be characterized with a three-parameter model denoted by the amplitude ratio angle and the phase angles [28,29]. The update of the polarization weight in the carrier recovery stage of the two equalizers is only performed at the pilot symbols. Figure 4(b) shows the transmission performance versus the RSOP changes. In Fig. 4(b), the partially coupled equalizer exhibits better RSOP variation tolerance compared to the joint equalizer.
The BER performance of different equalizers for $\Delta v{T_s} = 1 \times {10^{ - 4}}$ is evaluated as a function of SNR as shown in Fig. 5. To ensure sufficient convergence, we use 5000 symbols as the training symbols for the convergence stage. The P-BPS algorithm and DD-LMS algorithm are compared in the process of carrier phase recovery. The pilot insertion rate of P-BPS is 1/32. Note that we also insert pilot symbols with the same insertion rate into the DD-LMS algorithm to ensure stability. In the theoretical performance calculation, the inherent 3-dB sensitivity degradation for heterodyne detection and 6-dB sensitivity gain due to 4× up-sampling is considered [17,26]. The SNR penalty of the joint equalizer or the partially decoupled equalizer compared to the theoretical result is found to be around 0.3 dB at BER of $1.5 \times {10^{ - 2}}$. In addition, we find that when using the same phase estimation method and ensuring a sufficient number of iterations, the convergence performance of PPDE and joint equalizer is consistent. Due to the small phase noise, the P-BPS algorithm shows only a small performance gain compared to DD-LMS. As the decrease in SNR, the advantage of P-BPS over DD-LMS also gradually decreases. The reason is that the increase in noise decreases the accuracy of the phase noise calculation.
We further compare the phase noise tolerance of different equalizers under different modulation formats. Figure 6 shows the phase noise tolerance of different carrier phase recovery algorithms in the carrier recovery stage including the DD-LMS, BPS and P-BPS. Figure 6(a) shows the measured SNR penalty at a BER of $1.5 \times {10^{ - 2}}$ as a function of under $\Delta v{T_s}$ QPSK modulation format at a symbol rate of 8GBaud. It is found that in the case of the same carrier phase recovery algorithm (DD-LMS), the performance of joint equalization and PPDE is consistent. This means that the PPDE can effectively extract the channel matrix from the coupled polarization and phase noise even with only one step size parameter. In the carrier recovery stage, the BPS algorithm has the best performance. However, because of the cyclic slip problem, the BPS algorithm is not suitable for the system with larger linewidth lasers. In Fig. 6(a), when the $\Delta v{T_s}$ is larger than $1.5 \times {10^{ - 4}}$, the BPS performance becomes extremely unstable and the phase noise cannot be accurately estimated. By comparing the phase noise tolerance of DD-LMS and P-BPS, we find that the allowed $\Delta v{T_s}$ based on DD-LMS is $2.5 \times {10^{ - 4}}$ which corresponds to 2 MHz at a symbol rate of 8 GBaud when the SNR penalty is 1 dB. By contrast, the $\Delta v{T_s}$ is $6.7 \times {10^{ - 4}}$ which corresponds to about 5.4 MHz for P-BPS. The tolerance of laser linewidth of the P-BPS algorithm is more than two times that of DD-LMS. In addition, we find that the SNR penalty of P-BPS is slightly larger than that of BPS and DD-LMS when the linewidth is small. It is because there is an error between the calculated phase noise and the accurate one due to the influence of noise. We further simulate the phase noise tolerance of different equalizers for a symbol rate of 12 GBaud. Since we take the ratio of linewidth to symbol rate as the horizontal axis, the SNR penalty corresponding to the same ratio is independent of the symbol rate without considering the influence of other channel effects. Therefore, the performance of the 12-GBaud transmission is consistent with the 8-GBaud transmission but its linewidth tolerance is improved.
Fig. 6. (a) SNR penalty at BER of 1.5% versus combined laser linewidths symbol period product under QPSK modulation format; (b) SNR penalty at BER of 1.5% versus combined laser linewidths symbol period product under 16QAM modulation format at a symbol rate of 8GBaud.
Figure 6(b) shows the measured SNR penalty as a function of $\Delta v{T_s}$ under 16-QAM modulation format at a BER of $1.5 \times {10^{ - 2}}$. Similar to QPSK, the joint equalization and PPDE have almost the same performance with the same carrier phase recovery algorithm. If the same PPDE is used in the first stage, the allowed $\Delta v{T_s}$ based on DD-LMS is $5.5 \times {10^{ - 5}}$ corresponding to 440 KHz at a symbol rate of 8GBaud when the SNR penalty is 1 dB. While the $\Delta v{T_s}$ is $2.5 \times {10^{ - 4}}$ corresponding to 2 MHz for P-BPS. The tolerance of laser linewidth of the P-BPS algorithm is increased by about three times. This means a further reduction in laser costs.
From Fig. 5 and Fig. 6, we find that PPDE has no performance penalty compared with the joint equalizer when the carrier phase recovery algorithms are the same. We further compare the convergence performance between the partially decoupled equalizer and joint equalizer. We compare the number of training symbols for 100-km fiber transmission. Figures 7(a) and (b) show the convergence performance of QPSK and 16QAM, respectively. The number of taps is 1 for both equalizers. From Fig. 7, it is obvious that the partially coupled equalizer shows better convergence performance compared to the joint equalizer. For QPSK with $\Delta v{T_s} = 2.5 \times {10^{ - 4}}$, the partially decoupled equalizer only needs 40 symbols to achieve convergence, while the joint equalizer needs more than 450 symbols to achieve convergence. For 16 QAM with $\Delta v{T_s} = 5 \times {10^{ - 5}}$, the PPDE reduces the required training sequences from 500 to 100 compared with the joint equalizer. This means that the training sequences required by the partially decoupled equalizer are greatly reduced compared to the joint equalizer. In addition, compared with the joint equalizer, the partially coupled equalizer also shows better convergence stability.
Fig. 7. (a) The number of training symbols required for the two equalizers to converge after 100 km fiber link transmission under QPSK modulation format with $\Delta v{T_s} = 2.5 \times {10^{ - 4}}$; (b) The number of training symbols required for the two equalizers to converge after 100 km fiber link transmission under 16QAM modulation format with $\Delta v{T_s} = 5 \times {10^{ - 5}}$.
4. Experimental verifications and discussions
We conduct an experiment of 12-GBaud dual polarization (DP) Alamouti-coded QPSK transmission over 25-km fiber distance. The lower symbol rate is more sensitive to laser linewidth. Figure 9 shows the experimental setup. The $6 \times {10^4}$ QPSK signals are pulse-shaped with a root-raised cosine (RRC) filter with a roll-off factor of 0.01. They are uploaded to an 8-bit, four-channel, 92-GSa/s AWG to generate the driving signals. A tunable 100-KHz linewidth (ECL) operating at 1550 nm generates the optical signal at a power of 10 dBm. Due to the lack of large linewidth laser components, large phase noise corresponding to 3-MHz linewidth (corresponds to $\Delta v{T_s} = 2.5 \times {10^{ - 4}}$) is added by performing numerical simulations at the transmitter. Then, a commercially available LiNbO3 dual-polarization IQ modulator with a 3-dB bandwidth of 23 GHz is used for electro-optic conversion.
During transmission, an erbium-doped fiber amplifier (EDFA) with a noise figure of 5 dB is used to set the launch power. The signal is transmitted over 25-km-SSMF with an attenuation of 0.2 dB/km and a dispersion coefficient of 16.8 ps/nm/km. For receiver sensitivity measurements, the signal power is varied using a VOA, which is also used to emulate the splitter loss in a typical optical distribution network, as illustrated in Fig. 8. A 3-dB coupler and a 23-GHz PIN BPD compose of the low-complexity coherent receiver front end. The LO laser is an ECL with 100-KHz linewidth operating at 1549.947 nm at a power of 9.5 dBm. After coherent detection and electrical amplification, the signal is digitized by a digital storage oscilloscope (DSO) operating at 256 GSa/s for offline DSP.
Fig. 8. Experimental setup. ECL, external cavity laser; AWG, arbitrary waveform generator; DP-IQ MOD, dual-polarization in-phase quadrature modulator; EDFA, erbium-doped fiber amplifier; VOA, variable optical attenuator; PM, power meter; ADC, analog-to-digital converter; FO: frequency offset.
In the offline DSP, the intermediate frequency is first estimated for baseband down-conversion. The remaining DSP processes include CD compensation, residual frequency offset removal, frame synchronization, clock synchronization and matched filter. After resampling the signals to two samples per symbol, we compare the two equalizers for adaptive equalization, respectively. The number of filter taps is 13. After carrier phase recovery, the BER is measured by direct counting method.
Firstly, we have varied the RSOP to test the tolerance of the partially coupled equalizer at a laser linewidth of 100KHz. Due to the lack of an optical polarization scrambler, we verify the RSOP variation tolerance of the two equalizers by simulating the dynamic SOP in the transmitter DSP. As shown in Fig. 9, the partially coupled equalizer exhibits better RSOP variation tolerance.
We evaluate the receiver sensitivity of the two equalizers under different laser linewidth when the SOP is static. We both use 2500 training symbols. As shown in Fig. 10, when the total linewidth is small, the performance of the two equalizers is almost the same. However, as the laser phase noise increases, the proposed equalizer shows stronger phase noise tolerance than the joint equalizer. The proposed equalizer exhibits a 0.5-dB receiver sensitivity gain over the joint equalizer under 3-MHz laser linewidth at the BER threshold of $1.5 \times {10^{ - 2}}$. This gain is consistent with the performance gain for the corresponding $\Delta v{T_s}$ in Fig. 6(a) by simulations.
Fig. 10. Comparisons of the receiver sensitivity of the two equalizers; (a) the total linewidth of lasers is 200KHz; (b) the total linewidth of lasers is 3 MHz.
We further evaluate the convergence performance of the joint equalizer and the partially decoupled equalizer in Fig. 11. In Fig. 11(a), the required number of training symbols for both equalizers is close when the phase noise is small. This is because when the phase noise is small, both equalizers in the convergence stage approximate an identical LMS-based butterfly filter. However, when there is a large phase noise, the number of training symbols for the partially decoupled equalizer is only half of that of the joint equalizer and achieves better performance (The number of training symbols required has been reduced from 2000 to 1000).
Fig. 11. Comparison of the number of training symbols required for convergence of the two equalizers (a) the total linewidth of lasers is 200 KHz; (b) the total linewidth of lasers is 3 MHz.
5. Conclusion
We have proposed and experimentally demonstrated the partially decoupled equalizer to accelerate the convergence. Compared with the joint equalizer, the proposed equalizer simultaneously ensures the polarization-insensitive, reduces the search complexity and increases the tolerance of phase noise.
Funding
National Natural Science Foundation of China (61871082, 62111530150); Open Fund of IPOC (No. IPOC2020A011); Science and Technology Commission of Shanghai Municipality (SKLSFO2021-01); Fundamental Research Funds for the Central Universities (ZYGX2019J008, ZYGX2020ZB043).
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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