Carrier Phase Recovery Algorithm Research Articles

Geometrically Shaped Odd-Bit QAM Constellations Suitable for Principal Component-Based Phase Estimation

For high-speed optical communication systems, laser phase noise (LPN) stands as a pivotal factor influencing the quality of the received signal. Therefore, the employment of a highly accurate carrier phase recovery (CPR) algorithm at the receiving end is indispensable to ensure the reliability of transmission. While a CPR algorithm called principal component-based phase estimation (PCPE) has been proven to be capable of achieving low-complexity and high-performance phase recovery for even-bit quadrature amplitude modulation (QAM) (i.e., square QAM) signals, it is not compatible with traditional cross-shaped odd-bit QAM signals. To circumvent this problem, a signal constellation design scheme based on geometric shaping (GS) is proposed. The pair-wise optimization (PO) algorithm is used to optimize the constellation structure of 32QAM and 128QAM signals in order to obtain results that are compatible with the PCPE algorithm. Monte Carlo simulation results reveal that for odd-bit QAM signals utilizing PCPE for phase recovery, the proposed GS constellations enhance the mutual information (MI) performance across the entire measured signal-to-noise (SNR) range. Moreover, compared with regular 32QAM and 128QAM constellations using the well-known blind phase search (BPS) algorithm, the proposed GS and PCPE scheme can achieve SNR gains of 1.10 dB and 2.59 dB, respectively, when considering the 20% soft-decision forward error correction (SD-FEC) overhead. Verification through commercial simulation software corroborates these findings, demonstrating that the proposed GS constellations are particularly suitable for the PCPE algorithm, especially under conditions of high optical signal-to-noise ratio (OSNR). To the best of our knowledge, this is the first time that the incompatibility between the PCPE algorithm and odd-bit QAM signals has been investigated, and the proposed GS scheme has broadened the application scope of the low-complexity CPR algorithm.

Deep Learning Equalizer Connected with Viterbi-Viterbi Algorithm for PAM D-Band Radio over Fiber Link.

D-band (110-170 GHz) has been regarded as a potential candidate for the future 6G wireless network because of its large available bandwidth. At present, the lack of electrical amplifiers operating in the high frequency band and the strong nonlinear effect, i.e., the D-band, are still important problems. Therefore, effective methods to mitigate the nonlinear issue resulting from the ROF link are indispensable, among of which machine learning is considered the most effective paradigm to model the nonlinear behavior due to its nonlinear active function and structure. In order to reduce the computation amount and burden, a novel deep learning neural network equalizer connected with typical mathematical frequency offset estimation (FOE) and carrier phase recovery (CPR) algorithms is proposed. We implement D-band 45 Gbaud PAM-4 and 20 Gbaud PAM-8 ROF transmission simulations, and the simulation results show that the real value neural network (RVNN) equalizer connected with the Viterbi-Viterbi algorithm exhibits better compensation ability for nonlinear impairment, especially when dealing with serious inter-symbol interference and nonlinear effects. In our experiment, we employ coherent detection to further improve the receiver sensitivity, so a complex baseband signal after down conversion at the receiver is inherently produced. In this scenario, the complex value neural network (CVNN) and RVNN equalizer connected with the Viterbi-Viterbi algorithm have better BER performance with an error rate lower than the HD-FEC threshold of 3.8 × 10-3.

Low-complexity algorithms for coherent optical systems with transceiver IQ imbalance.

The next-generation coherent optical communication relies on higher symbol transmission rates with higher modulation cardinality. The imbalance between the in-phase (I) and the quadrature (Q) arms in such systems leads to deteriorated performance if not corrected. In this work, we propose a reference constellation adaptation-based single-tap, novel carrier phase recovery algorithm, tolerant to the transmitter IQ imbalance and a pilot-free geometric parameter extraction-based receiver IQ imbalance correction method for MQAM signals. The proposed scheme is numerically investigated through extensive simulations for 32 GBaud polarization multiplexed (PM)-16QAM, and 64QAM modulation formats and experimentally verified for 32 GBaud PM-16QAM signal. The proposed pilot-free scheme is applicable to any modulation format and offers lower computational complexity than traditional algorithms.

Experimental Comparison of Carrier Phase Recovery Algorithms for Uniform and Probabilistically Shaped QAM in a 324.1 Gb/S Fiber-mm-Wave Integration System at W-Band

We have experimentally implemented a photonics-aided large-capacity fiber-mm-wave wireless communication system employing a simple dual-polarized single-input single-output (SISO) wireless based on polarization multiplexing at the W-band. To compare the performance of different algorithms, 18G-baud, and 35G-baud 16-level quadrature-amplitude-modulation (16QAM), probabilistically shaped 16QAM (PS-16QAM), 64QAM and PS-64QAM signal using different carrier phase recovery (CPR) algorithms are transmitted in the system. Moreover, we compare the Viterbi–Viterbi (VV), improved new algorithm based on VV (NVV), blind phase search (BPS), and two-stage BPS algorithms’ computational complexity to better compare different algorithms. Using the experiment result, we can demonstrate that the BPS algorithm is about half a magnitude better than the NVV algorithm for PS-QAM signals, while the NVV algorithm has the lowest computational complexity. Additionally, we also achieve error-free wireless transmission at a net data rate of 324.1 Gb/s with the bit error ratio (BER) below the forward-error correction (FEC) threshold of 1 × 10−2 assuming soft-decision forward-error correction (SD-FEC) when using the BPS algorithm.

On the Nonlinear Shaping Gain With Probabilistic Shaping and Carrier Phase Recovery

The performance of different probabilistic amplitude shaping (PAS)techniques in the nonlinear regime is investigated, highlighting its dependence on the PAS block length and the interaction with carrier phase recovery (CPR). Different PAS implementations are considered, based on different distribution matching (DM) techniques—namely, sphere shaping, shell mapping with different number of shells, and constant composition DM—and amplitude-to-symbol maps. When CPR is not included, PAS with optimal block length provides a nonlinear shaping gain with respect to a linearly optimized PAS (with infinite block length); among the considered DM techniques, the largest gain is obtained with sphere shaping. On the other hand, the nonlinear shaping gain becomes smaller, or completely vanishes, when CPR is included, meaning that in this case all the considered implementations achieve a similar performance for a sufficiently long block length. Similar results are obtained in different link configurations ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$1\times 180$</tex-math></inline-formula> km, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$15\times 80$</tex-math></inline-formula> km, and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$27\times 80$</tex-math></inline-formula> km single-mode-fiber links), and also including laser phase noise, except when in-line dispersion compensation is used. Furthermore, we define a new metric, the nonlinear phase noise (NPN) metric, which is based on the frequency resolved logarithmic perturbation models and explains the interaction of CPR and PAS. We show that the NPN metric is highly correlated with the performance of the system. Our results suggest that, in general, the optimization of PAS in the nonlinear regime should always account for the presence of a CPR algorithm. In this case, the reduction of the rate loss (obtained by using sphere shaping and increasing the DM block length) turns out to be more important than the mitigation of the nonlinear phase noise (obtained by using constant-energy DMs and reducing the block length), the latter being already granted by the CPR algorithm.

Recent Advances in Carrier Phase Recovery Algorithms

Recent Advances in Carrier Phase Recovery Algorithms

Two-Stage Pilot-Aided Carrier Phase Recovery Algorithm for FTN System

Two-Stage Pilot-Aided Carrier Phase Recovery Algorithm for FTN System

Extended Study on Equalization-Enhanced Phase Noise for High-Order Modulation Formats

Coherent optical receivers with electronic dispersion compensation (EDC) become ubiquitous for high-capacity and long-distance transmissions. However, their performance can be decimated by equalization-enhanced phase noise (EEPN). The existing reports have shown some unique features that deserve further studies, especially for high-order modulation formats. In this paper, we firstly show that the intra-symbol EEPN is dominated by the phase noise. Therefore, the probability density function (PDF) due to intra-symbol EEPN and inter-symbol EEPN are quite different. We derive the variances of EEPN-induced phase and amplitude noise for QPSK signals, and fully explain the elliptical PDF. Secondly, we show that the intra-symbol EEPN is scaled with the power of the different modulus, but the inter-symbol one, the average power of the constellation. Subsequently, we elucidate that for higher-order constellations with multiple moduli, the total EEPN, the EEPN-induced phase noise and amplitude noise, all distribute differently on the different moduli. Finally, we analyze the EEPN after the practical carrier phase recovery algorithms, also with a closed-form prediction. Our analysis of EEPN is readily extended to arbitrary modulation formats.

Design of Probabilistic Shaping 4D Ultra High Order Modulation Format With 8APSK Pilot Aided Carrier Phase Recovery

A new four-dimensional (4D) probabilistically shaped (PS) ultra-high order modulation format is proposed with pilot aided carrier phase recovery (PA-CPR) algorithm to improve the general mutual information (GMI) performance. While the structure of the 4D-PS ultra-high order modulation format uses constellation set partitioning (SP) based on amplitude translation, and the PA-CPR applies 8-level amplitude phase shift keying (8APSK) pilot. To demonstrate the feasibility of the scheme, the PS-1024QAM system is designed on the simulation platform. The results show that the proposed 8APSK-PA-CPR algorithm outperforms the QPSK-PA-CPR algorithm. The gain of the optimal 8APSK-PA-CPR is &#x223C;0.4 dB compared with the quadrature phase-shift keying (QPSK) PA-CPR algorithm. The steep drop of blind phase search (BPS) with different PS factors is overcome by 8APSK-PA-CPR when the signal-to-noise (SNR) ratio is lower than 24 dB. The maximum improvement of GMI is 1.3 bit&#x002F;symbol when the SNR is higher than 26 dB and the complexity of the 8APSK-PA-CPR algorithm is reduced by 99.9&#x0025; compared with the BPS algorithm. The gain obtained from the proposed 4D modulation scheme is &#x223C;2.8 dB under the condition of GMI &#x003D; 17.7971 bit&#x002F;symbol. It demonstrated that this scheme has a strong tolerance to noise, low computational complexity, and high coding gain. It is a feasible and flexible scheme for PS high-order modulation format transmission systems.

End-to-End Learning of a Constellation Shape Robust to Channel Condition Uncertainties

Vendor interoperability is one of the desired future characteristics of optical networks. This means that the transmission system needs to support a variety of hardware with different components, leading to system uncertainties throughout the network. For example, uncertainties in signal-to-noise ratio and laser linewidth can negatively affect the quality of transmission within an optical network due to e.g. mis-parametrization of the transceiver signal processing algorithms. In this paper, we propose to geometrically optimize a constellation shape that is robust to uncertainties in the channel conditions by utilizing end-to-end learning. In the optimization step, the channel model includes additive noise and residual phase noise. In the testing step, the channel model consists of laser phase noise, additive noise and blind phase search as the carrier phase recovery algorithm. Two noise models are considered for the additive noise: white Gaussian noise and nonlinear interference noise model for fiber nonlinearities. The latter models the behavior of an optical fiber channel more accurately because it considers the nonlinear effects of the optical fiber. For this model, the uncertainty in the signal-to-noise ratio can be divided between amplifier noise figures and launch power variations. For both noise models, our results indicate that the learned constellations are more robust to the uncertainties in channel conditions compared to a standard constellation scheme such as quadrature amplitude modulation and standard geometric constellation shaping techniques.

Modified QPSK partition algorithm based on MAP estimation for probabilistically-shaped 16-QAM

Probabilistic shaping (PS) is investigated as a potential technique to approach the Shannon limit. However, it has been proved that conventional carrier phase recovery (CPR) algorithm designed for uniform distribution may have extra penalty in PS systems. In this paper, we find that the performance of QPSK partition algorithm is degenerated when PS is implemented. To solve this issue, a modified QPSK partition algorithm that jointly optimizes the amplitude decision threshold and filter weight is proposed, where the optimization of decision threshold is based on maximum a posterior probability (MAP) estimation. Different from the conventional decision methods which commonly use Euclidean distance metric, the MAP-based decision introduces the statistical characteristics of the received signals to obtain an accurate amplitude partition. In addition, the filter weight is optimized for different decision thresholds to enhance the tolerance of ASE-induced phase noise. We verify the feasibility of the proposed algorithm in a 56 GBaud PS 16-ary quadrature amplitude modulation (16-QAM) system. The proposed algorithm reduces the error of phase noise estimation by nearly half. Compared with conventional QPSK partition, the proposed algorithm could narrow the gap with theoretical mutual information (MI) by more than 0.1 bit/symbol. The channel capacity is increased by 4.2%, 4.3% and 3.6% with signal-to-noise ratio (SNR) from 8 dB to 10 dB respectively. These observations show that the proposed algorithm is a promising method to relieve the penalty of QPSK partition algorithm in PS systems.

Carrier Phase Estimation Softwarized on GPU Using Decision-Aided Phase Unwrapping for Flexible Optical Coherent Access Systems

We present a unique carrier phase recovery algorithm which permits the softwarization of optical coherent receiver digital signal processing (DSP). The proposed algorithm with decision-aided phase unwrapping overcomes the serial dependency issue that has, up to now, prevented the softwarization of optical coherent systems. Its performance is verified through extensive Monte Carlo (MC) simulations and hardware experiments on quadrature phase-shift keying (QPSK) signaling. The proposed algorithm is implemented on a general-purpose graphic processing unit (GPU) and achieves 5-Gbit/s throughput for real-time streaming signals, as well as −44.5-dBm receiver sensitivity. The softwarization of carrier phase recovery function is demonstrated for the first time; our enhancement of disaggregated optical coherent receivers will significantly improve system flexibility and scalability, towards the realization of a new access networking that exploits general purpose hardware and virtualized transport layer functions.

Theoretical Analysis of Phase Noise Induced by Laser Linewidth and Mismatch Length in Self-Homodyne Coherent Systems

The intra- and inter-data center links face continuously increasing pressure on the transmission capacity while having to meet strict constraints in cost and power consumption. For such cost-sensitive data center applications, self-homodyne coherent (SHC) system as a promising solution has attracted increasing attention. In this article, the influence of phase noise (PN) for the SHC system is thoroughly analyzed by studying the PN probability distribution characteristic and bit-error rate (BER) performance in theory. The correctness of the theorical PN analysis is also verified by simulation. Further, the PN characteristics in SHC systems are investigated and discussed with and without the implementation of carrier phase recovery (CPR). A unique property is found for SHC systems that the system performance is only related to the product of linewidth and mismatch length, independent of the symbol rate in the absence of CPR. For SHC-16QAM systems, in order to limit the OSNR penalty to below 1-dB, the product of laser linewidth and delay mismatch should be kept below 0.18 MHz · m. Under such a conduciton, no CPR algorithm is reuiqred. Alternatively, in presence of CPR, the phase noise tolerance of SHC and conventional coherent systems are similar.

Carrier phase recovery friendly probabilistic shaping scheme based on a quasi-Maxwell-Boltzmann distribution model.

A novel probabilistic shaping (PS) scheme based on the quasi-Maxwell-Boltzmann (quasi-MB) distribution model is proposed in order to solve the incompatibility between PS and carrier phase recovery (CPR) algorithms, such as blind phase search (BPS) and principal component-based phase estimation (PCPE). In the proposed quasi-MB model, the same occurrence probability is assigned to each constellation point on the same square-ring, rather than on the same circle. Signals obeying the quasi-MB model have superior CPR friendliness compared to traditional PS signals. For a PS 64 quadrature amplitude modulation system, the simulation results indicate up to 51% and 21% normalized generalized mutual information (NGMI) improvements for PCPE and BPS, respectively. Experimental verification of the proposed quasi-MB scheme was demonstrated in a 10 Gbaud coherent detection system. The results show that when the optical signal-to-noise ratio (OSNR) is low, the quasi-MB model can help the BPS algorithm to achieve better NGMI performance and, when the OSNR is high, the proposed model can also solve the incompatibility between the PCPE algorithm and PS.

Phase and Frequency Recovery Algorithms for Probabilistically Shaped Transmission

In additive white Gaussian noise (AWGN) channels, probabilistic shaping (PS) approaches the information-theoretical capacity using typical square $M$ -ary quadrature amplitude modulation ( $M$ -QAM) constellations with nonuniform a priori probabilities. However, shaped constellations may have a detrimental impact on the chain of digital signal processing algorithms of typical coherent receivers, reducing or even eliminating the expected shaping gains. In this article, we evaluate carrier phase recovery (CPR) and carrier frequency recovery (CFR) algorithms for probabilistically shaped transmission. The results are obtained by offline processing of experimental data with laser imperfections and additive noise loading. We investigate an existing two-stage CPR, where the first stage is based on pilots with 3.1% overhead, and the second stage uses the blind phase search (BPS) algorithm. We show that the BPS stage is severely impaired by PS at moderate and low SNRs. These impairments can be partially compensated using extremely long noise rejection windows, which may be undesirable from a computational complexity standpoint. Conversely, using only the pilot-based stage exhibits penalties at high SNRs, as phase noise dominates over additive noise. In order to obtain a suitable performance over the full range of SNRs, we propose to detune the PS factor (with respect to the AWGN-optimized value) to optimize the BPS stage. In scenarios with relatively low phase noise and low modulation orders, we show that an equivalent performance can be obtained without shaping factor detuning by operating with the two-stage approach at high SNRs and switching off the BPS stage at moderate and low SNRs. We also show that typical blind CFR based on the 4 $^{th}$ power algorithm is severely affected by PS at moderate block lengths, but its expected performance can be recovered using very long block lengths. Alternatively, a pilot-based algorithm with 3.1% overhead (employing the same pilots used for the first stage of the two-stage CPR algorithm) achieves a suitable performance in all the investigated scenarios.

VLSI Implementations of Carrier Phase Recovery Algorithms for M-QAM Fiber-Optic Systems

We present circuit implementations of blind phase search (BPS) carrier phase recovery (CPR) for M-QAM coherent optical receivers and highlight some BPS algorithm modifications necessary to obtain efficient VLSI circuits. In addition, we show how three key design parameters (input word length, number of test phases, and type and size of averaging window) affect the resulting implementation. To study design tradeoffs, we develop BPS CPR circuit netlists for a 32-GBaud system, using a 22-nm CMOS process technology: our implementations reach energy efficiencies of around 1 pJ/bit for 16QAM up to 3 pJ/bit for 256QAM, at an SNR penalty of approximately 0.25 dB at a BER of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-2</sup> . Furthermore, we present a circuit implementation of pilot-symbol-aided CPR, reaching 0.38 pJ/bit and 0.34 pJ/bit for 16QAM and 256QAM, respectively, at a slightly higher SNR penalty. The two CPR methods are also evaluated in terms of silicon area and scaling to higher-order modulation formats.

用于概率整形信号的自适应载波相位恢复算法

用于概率整形信号的自适应载波相位恢复算法

Phase noise effects on the performance of joint carrier phase recovery algorithms in phase-locked WDM superchannel transmission systems

Phase-locked WDM superchannel transmission systems have been demonstrated to effectively provide reduced DSP complexity. Nonetheless, the subchannels may be affected by laser and nonlinear phase noises, which can be effectively compensated using joint carrier phase recovery (JCPR) techniques. In this work, for the first time, we demonstrated via extensive numerical simulations that the subchannels experience intra-channel dispersion-induced phase-offset (DIPO), which significantly degrade the phase coherence and hence performance of JCPR schemes. In addition, we described in detail and identified the origin of DIPO as the interaction between the LO laser linewidth and the dispersive walk-off effect among the subchannels after long-haul dispersion-unmanaged fiber transmissions. Comprehensive simulation results revealed that the channel spacing, fiber length and the number of subchannels within the superchannel enhances the DIPO. Consequently, the effect of DIPO on JCPR performance is severe for all outer subchannels than the inner subchannels. Finally, we proposed and demonstrated a modified JCPR where simulation results showed that the BER limit for soft-decision FEC could still be achieved in a 50 GHz-spaced 5-channel 45 Gbaud 16-QAM phase-locked WDM superchannel transmission system over a 3000 km fiber link with phase noise effects from both laser linewidth and fiber nonlinearities.

Wide-linewidth coherent systems with parallelized extended Kalman filter for data center inter-connection

Adopting wide-linewidth lasers is viewed as a promising way to reduce cost in coherent systems for data center inter-connection. However, in this application, low-complexity and effective carrier phase recovery algorithm is indispensable for compensating phase noise of lasers. In this paper, a data-assisted based parallelized extended Kalman filter (EKF) algorithm is firstly proposed for carrier phase recovery in coherent systems with low-cost and wide-linewidth lasers, as the proposed algorithm has large linewidth tolerance and realizes a parallelized processing architecture. We verify the feasibility of our proposed algorithm in a wide-linewidth coherent system with 448 Gb/s polarization division multiplexing 16-ary quadrature-amplitude-modulation (PDM-16QAM). The results show that the proposed algorithm has an enhanced linewidth tolerance than the blind phase search (BPS) and QPSK partition. For lasers linewidth up to 20 MHz, at 20% forward error correction (FEC) threshold, the OSNR sensitivity of the proposed algorithm is 0.39 dB and 0.68 dB better than BPS and QPSK partition, respectively. At 7% FEC threshold, the OSNR sensitivity of the proposed algorithm is 0.68 dB and 1.38 dB better than BPS and QPSK partition, respectively. Finally, the computational complexity is analyzed, where the proposed algorithm reduces the number of multiplication by 72% compared to BPS.

Research on Carrier Recovery Algorithm in Coherent Optical Communication System

In coherent optical communication system, the random deviation of optical phase which is caused by the spontaneous emission of laser leads to phase noise, resulting in a random rotation of signal constellation. In order to ensure the reliability of the communication system, carrier phase estimation and recovery are required at the receiver. In this paper, we investigate two traditional carrier phase recovery (CPR) algorithms, including Viterbi&Viterbi phase estimation (VVPE) and blind phase search (BPS) algorithm. On this basis, we also discuss seven improved algorithms. Then we compare and analyze the hardware complexities of eight different CPR algorithms. In addition, the cause of equalization enhanced phase noise (EEPN) and its impact on the carrier phase recovery algorithm are also discussed.