Carrier Phase Estimation Algorithm Research Articles

End-to-End Optimization of Constellation Shaping for Wiener Phase Noise Channels With a Differentiable Blind Phase Search

As the demand for higher data throughput in coherent optical communication systems increases, we need to find ways to increase capacity in existing and future optical communication links. To address the demand for higher spectral efficiencies, we apply end-to-end optimization for joint geometric and probabilistic constellation shaping in the presence of Wiener phase noise and carrier phase estimation. Our approach follows state-of-the-art bitwise auto-encoders, which require a differentiable implementation of all operations between transmitter and receiver, including the DSP algorithms. In this work, we show how to modify the ubiquitous blind phase search (BPS) algorithm, a popular carrier phase estimation algorithm, to make it differentiable and include it in the end-to-end constellation shaping. By leveraging joint geometric and probabilistic constellation shaping, we are able to obtain a robust and pilot-free modulation scheme improving the performance of 64-ary communication systems by at least 0.1 bit/symbol compared to square QAM constellations with neural demappers and by 0.05 bit/symbol compared to previously presented approaches applying only geometric constellation shaping.

Kalman filter-enabled parameter estimation for simultaneous quantum key distribution and classical communication scheme over a satellite-mediated link.

An accurate estimation of system parameters is of significance for the practical implementation of the simultaneous quantum key distribution and classical communication (SQCC) over a satellite-mediated link when considering the finite-size effect. In this paper, we propose a Kalman filter (KF)-enabled parameter estimation method for the SQCC over a satellite-mediated link. The fast and slow phase drift can be both estimated by using the improved vector KF carrier phase estimation algorithm, and thus the phase estimation error can be tracked in real time and be almost approximate to the theoretical mean square error limit. Taking advantage of the achieved phase estimation and the dual modulation of the SQCC scheme, the excess noise can be estimated with not only a higher precise but also a lower sacrificing rate of raw keys. Numerical simulations demonstrate the feasibility of the SQCC in both the downlink and uplink in terms of the finite-size effect. As a comparison of the Mth-power algorithm, we find that the secret key rate and achievable zenith angle perform better by using the vector KF algorithm. It paves the way of practical implementations for the SQCC system.

Joint Carrier-Phase Estimation for Digital Subcarrier Multiplexing Systems With Symbol-Rate Optimization

Digital subcarrier multiplexing (SCM) has recently emerged as a promising solution for next-generation ultra-high-baudrate coherent optical communication systems. Among its distinctive advantages over traditional single-carrier modulation, SCM enables the exploitation of symbol-rate optimization (SRO), which has been shown to enable the passive mitigation of the nonlinear interference noise (NLIN) that is generated during propagation over dispersion-unmanaged optical fiber systems. However, the full exploitation of SRO-based NLIN mitigation is severely hindered by the uncompensated distortion caused by laser phase noise (LPN) and non-linear phase noise (NLPN), whose impact is magnified by the use of low-baudrate subcarriers. Resorting to low-complexity carrier phase estimation (CPE) algorithms, in this paper we experimentally demonstrate that it is possible to overcome the hurdles posed by LPN and NLPN in SCM systems, provided that adequate joint-subcarrier CPE processing is employed. A dual-stage joint-processing approach composed of a pilot-based CPE optionally followed by a blind phase search (BPS)-based estimator is implemented and experimentally assessed, enabling to effectively optimize the symbol-rate per subcarrier down to 3 GBaud, in accordance with the theoretical SRO predictions for the system under test. In addition, we demonstrate that signal-to-noise ratio (SNR) gains of more than 1 dB can be achieved through joint-subcarrier CPE processing in shorter-reach links, while this gain tends to progressively reduce with increasing propagation distance, down to about 0.5 dB gain after 3000 km propagation.

Phase noise impact on the performance of contention resolution slotted random access schemes

SummaryCost‐effective satellite terminals (STs) that are designed for a large scale deployment in the Internet of things (IoT) applications are constrained by consumer‐grade local oscillators (LOs) to lower the costs, on the one hand, and by limited equivalent isotropic radiated power (EIRP) due to limited peak transmit power and/or low antenna gain, on the other hand. To close the link budget with the low EIRP constraint, STs can resort to adopting robust forward error correcting (FEC) schemes and/or limiting the transmission baud rate, hence increasing the link margin with respect to the required receiver threshold. In general, the LOs at the terminal side are responsible for injecting multiplicative phase distortions to the transmitted signal, known as phase noise (PN). The PN caused by low‐cost LOs could be significant, especially for low baud rates, thus affecting the demodulator performance. Random access (RA) techniques are particularly encouraged on the return link in networks with low‐cost terminals and bursty/low‐duty cycle types of traffic. The state‐of‐the‐art time‐slotted RA techniques are often evaluated considering high EIRP at the terminal side and high baud rates. This paper investigates the impact of the PN in time‐slotted RA techniques and optimizes different carrier phase estimation (CPE) algorithms. Furthermore, it evaluates the impact of CPE errors on RA performance for a wide range of baud rates and typical STs phase noise masks.

Blind carrier phase estimation based on geometric projection for the multi-band VLC system

The carrier phase offset induced by the mismatch between the signal subcarrier frequency and arbitrary waveform generator (AWG) reference clock is a key problem for high-speed multi-band visible light communication (VLC) system. In this paper, we propose a blind carrier phase estimation (CPE) algorithm based on geometric projection for multi-band Nyquist pulse amplitude modulation (PAM) system. Compared with traditional CPE algorithms, the proposed method does not contain any feedback loop and the Mth power operation. Therefore, the lower computational complexity and simpler system structure are both achieved. Furthermore, we experimentally validate the proposed CPE algorithm in a three-band PAM-8 VLC system. The measured results show the system performance can be greatly improved. And compared with the Viterbi and Viterbi (V&V) algorithm using training symbols (20% of the total symbol length), the proposed CPE algorithm can slightly outperform. However, the scheme is a blind estimation without any assistant training sequences. Thus higher bandwidth efficiency is achieved.

Carrier-phase estimation for simultaneous quantum key distribution and classical communication using a real local oscillator

The simultaneous quantum and classical communication (SQCC) protocol is an appealing protocol since it allows continuous-variable quantum key distribution (CV-QKD) and classical communication to be implemented simultaneously by using the same communication infrastructure and on a single wavelength. However, the main challenge of the SQCC protocol is its low tolerance of phase noise, limiting the transmission distance under the real local oscillator (LO) design. To eliminate the phase drift without the aid of the reference signal, the classical carrier phase estimation (CPE) algorithm in the field of coherent optical communication is employed for phase recovery and data extraction. Besides, the power of the residual phase noise in classical communication and the power of the excess phase noise in CV-QKD are derived, including the effects of the employed laser linewidth, the transversal filter length, and the classical signal power. On this basis, the performances of the SQCC protocol under the trusted and the untrusted phase noise model are discussed, respectively. Our work confirms the feasibility of the CPE algorithm in the SQCC protocol and demonstrates the conditions required for this protocol to operate on a coherent optical communication system.

Low complexity carrier phase estimation for m-QAM optical communication systems

A low complexity carrier phase estimation (CPE) algorithm for M-ary quadrature amplitude modulation (m-QAM) optical communication systems is investigated in this paper. In the proposed CPE algorithm, a two-stage CPE method is adopted. In the first stage, the QPSK points of the constellation are picked out to achieve a coarse phase estimation using the traditional Viterbi and Viterbi algorithm. In the second stage, all the points of the constellation are used for a fine phase estimation. In addition, the fourth-power operation is replaced by the 4-level absolute operation for the removal of modulated data phase, which greatly reduced the complexity. The proposed method was investigated through simulation, with 16-QAM, 32-QAM and 64-QAM modulation formats, respectively. The simulation results show that the proposed algorithm has both good linewidth tolerance and amplified spontaneous emission noise tolerance as well as low complexity. Moreover, when the equalization enhanced phase noise is considered, the proposed method also has better performance than traditional algorithm.

Pilot-Aided Joint-Channel Carrier-Phase Estimation in Space-Division Multiplexed Multicore Fiber Transmission

The performance of pilot-aided joint-channel carrier-phase estimation (CPE) in space-division multiplexed multicore fiber (MCF) transmission with correlated phase noise is studied. To that end, a system model describing uncoded MCF transmission where the phase noise comprises a common laser phase noise, in addition to core- and polarization-specific phase drifts, is introduced. It is then shown that the system model can be regarded as a special case of a multidimensional random-walk phase-noise model. A pilot-aided CPE algorithm developed for this model is used to evaluate two strategies, namely joint-channel and per-channel CPE. To quantify the performance differences between the two strategies, their respective phase-noise tolerances are assessed through Monte Carlo simulations of uncoded transmission for different modulation formats, pilot overheads, laser linewidths, numbers of spatial channels, and degrees of phase-noise correlation across the channels. For 20 GBd transmission with 200 kHz combined laser linewidth and 1% pilot overhead, joint-channel CPE yields up to 3.4 dB improvement in power efficiency or 25.5% increased information rate. Moreover, through MCF transmission experiments, the system model is validated and the strategies are compared in terms of bit-error-rate performance versus transmission distance for uncoded transmission of different modulation formats. Up to 21% increase in transmission reach is observed for 1% pilot overhead through the use of joint-channel CPE.

Comparison of Probabilistically Shaped 64QAM With Lower Cardinality Uniform Constellations in Long-Haul Optical Systems

In this paper, we compare the performance of probabilistically shaped 64QAM with uniform 16QAM and 32QAM modulation formats at the same net data rate in long-haul coherent optical communications systems. Experimental results at 16 GBaud are shown, with offline postprocessing of the data performed using either an ideal or a realistic carrier phase estimation (CPE) scheme. We show that the choice of the CPE algorithm at the receiver is crucial, since, as predicted by current models, most of the additional nonlinear noise introduced by the shaping is nonlinear phase noise (NLPN). Thanks to the use of probabilistic shaping (PS), maximum reach gains ranging from 15.5% and 34% are obtained over pure silica-core fiber, where the NLPN can be efficiently compensated for by standard CPE algorithms, while over nonzero dispersion-shifted fiber, the gain of PS is drastically reduced, due to residual short-correlated NLPN.

Carrier-phase-estimation algorithm featuring fast trackability for high-speed coherent WDM PON based on RSOA.

There have been substantial efforts to implement high-speed (>10 Gb/s) upstream transmission using reflective semiconductor optical amplifiers (RSOAs) in a coherent wavelength-division-multiplexed (WDM) passive optical network (PON). In such a network, it is necessary to estimate the carrier phase of upstream optical signal to retrieve the phase-modulated information created by RSOA. However, due to the severe waveform distortions caused by the limited modulation bandwidth of RSOA (typically less than 3 GHz), previously reported carrier phase estimation (CPE) algorithms cannot accurately estimate the carrier phase of high-speed quadrature phase-shift keying (QPSK) signal generated from the RSOA seeded by a distributed-feedback (DFB) laser. We propose a novel CPE method capable of tracking the carrier phase rapidly by using a small number of symbols (e.g., 15 symbols) even when the waveforms are severely distorted by the limited modulation bandwidth of RSOA. The proposed CPE method utilizes the linear relationship between the intensity modulation and phase modulation indices inherent in the semiconductor opto-electronic device. By using the proposed method, we demonstrate the transmission of 25.78-Gb/s QPSK signal in a 20-km long loopback fiber link. In this experiment, a commercial DFB laser (linewidth: 3 MHz) is used as the seed light instead of an expensive narrow-linewidth laser. Also shown through the experiment is that the proposed CPE method is highly unsusceptible to variations of parameters required in the proposed method, such as the number of test phases, the accuracy of linewidth enhancement factor, and the accuracy of the normalized amplitude of DC component.

Multiplier-Free Carrier Phase Estimation for Optical Coherent Systems

A multiplier-free carrier phase estimation (CPE) algorithm is investigated in this letter. The Viterbi and Viterbi (V&V) algorithm is widely used in CPE, because the V&V algorithm is less complex than many other feed-forward CPE algorithms, including the blind phase search algorithm. In the V&V algorithm, an M th power operation is used to remove the modulated data phase. However, one disadvantage of the $M$ th power operation is its large computational complexity. In this letter, we extend our prior simplified M th power method from frequency offset estimation to CPE and propose an improved V&V algorithm, adopting the absolute operation as an approximation and making the V&V algorithm a multiplier-free CPE algorithm. Using simulations, the improved V&V algorithm is compared with the traditional V&V algorithm in the 32-GBd quadrature phase-shift keying, 8PSK, star-8QAM, and square-16QAM systems for CPE. The results show that the improved V&V algorithm outperforms the traditional V&V algorithm. In addition, the complexity is reduced effectively and no multiplier is required.

Extended Kalman filtering for joint mitigation of phase and amplitude noise in coherent QAM systems.

We numerically investigate our proposed carrier phase and amplitude noise estimation (CPANE) algorithm using extend Kalman filter (EKF) for joint mitigation of linear and non-linear phase noise as well as amplitude noise on 4, 16 and 64 polarization multiplexed (PM) quadrature amplitude modulation (QAM) 224 Gb/s systems. The results are compared to decision directed (DD) carrier phase estimation (CPE), DD phase locked loop (PLL) and universal CPE (U-CPE) algorithms. Besides eliminating the necessity of phase unwrapping function, EKF-CPANE shows improved performance for both back-to-back (BTB) and transmission scenarios compared to the aforementioned algorithms. We further propose a weighted innovation approach (WIA) of the EKF-CPANE which gives an improvement of 0.3 dB in the Q-factor, compared to the original algorithm.

Carrier Phase Estimation for 32-QAM Optical Systems Using Quasi-QPSK-Partitioning Algorithm

In this letter, we propose a quasi-quadrature phase shift keying (QPSK) partitioning to replace the conventional QPSK partitioning in the first stage of the carrier phase estimation (CPE) algorithms for 32-QAM optical systems. By using the proposed quasi-QPSK-partitioning method, the ability of the CPE algorithms to track changes of phase is greatly enhanced. For 1-dB SNR penalty at bit error rate of 1E-2, the proposed method can achieve an improvement of 85% in combined linewidth symbol duration product tolerance when compared with the conventional method for single stage. With the help of the fine estimation stages, the maximum tolerable combined linewidth symbol duration product can be up to 5.1E-5.

Impact of Sampling Source Repetition Frequency in Linear Optical Sampling

We report the repetition frequency constraint of a passively mode-locked fiber laser (PMFL) for linear optical sampling based M-ary phase-shift keying (PSK) constellation characterization, based on the commonly used Viterbi–Viterbi carrier phase estimation algorithm. Referred to the error vector magnitude (EVM) value by the conventional high-speed electronic coherent detection, we put forward a benchmark of EVM $_{\rm {RMS\_{}error}}$ to evaluate the accuracy of constellation characterization by linear optical sampling. Due to the repetition frequency constraint of the PMFL, the EVM $_{\rm {RMS\_{}error}}$ will reach an ultimate value of 0.25% for the purpose of PSK constellation characterization, while it is insensitive to the signal symbol rate. The analytical results of a typical 28-GBaud QPSK signal well agree with those of the simulation and the experiment, therefore providing guidance for the repetition frequency optimization of PMFL-based linear optical sampling.

A flexible Mth power carrier phase estimation algorithm for coherent optical receiver

Although Mth power carrier phase estimation (PE) algorithm has been investigated intensively in coherent optical phase-shift-keying (PSK) system, block length effect impacts system performance and increases hardware complexity. In this paper, a flexible Mth power carrier PE algorithm is proposed to remove the effect of block length on system performance. We take into account the time-varying phase noise, instead of the assumption of constant phase noise over the entire observation interval. The weighted factors obtained from maximum likelihood method are introduced to eliminate the block length effect. Furthermore, we derive the analytical expression of phase error variance for the performance prediction of coherent receiver with the flexible Mth power. Bit error ratio (BER) performance of coherent receiver is evaluated and compared through both theoretical derivation and Monte Carlo simulation. The results show that by using the flexible Mth power algorithm, signal noise ratio (SNR) improvement over conventional Mth power algorithm with fixed block length of 20 and 40 at BER of 10−5 is 0.47dB and 2.18dB, when the linewidth per laser is 2MHz. By employing our flexible Mth power algorithm, SNR improvement over conventional Mth power algorithm with optimum block length will be larger as increasing laser linewidth . It means our flexible Mth algorithm is more effective for mitigating phase noise than the conventional Mth power algorithm.

Performance of ADC resolution-limited optical coherent receivers with DA-ML carrier phase estimation

We investigate the performance of decision-aided maximum likelihood (DA-ML) carrier phase estimation (CPE) algorithm in analog-to-digital converter (ADC) resolution-limited optical coherent receivers. The signal model of ADC resolution-limited receiver is established, and its bit error rate with DA-ML CPE in M-ary phase-shift keying (MPSK) system is analyzed. Simulation results show that $$N+2$$N+2 (N bits per symbol)-bit resolution is acceptable and $$N+3$$N+3 bit is sufficient for DA-ML receiver in MPSK $$(M=4, 8, 16)$$(M=4,8,16) system, and for 16-ary quadrature amplitude modulation system, 6-bit resolution is reasonable. Simulation and theoretical analyses both indicate that the optimal ADC quantization range equals to the average amplitude of received symbols which carry the information symbols with the highest energy on the constellation.

Efficient Maximum-Likelihood Based Clock and Phase Estimators for OQPSK Signals

In this paper we propose an algorithm for joint carrier phase and timing estimation with OQPSK modulations. The derivation is based on the maximum-likelihood criterion, and exploits a very efficient algorithm for the detection of differentially encoded M-PSK symbols already described in literature. Though we are mainly interested in measuring the phase and clock parameters, estimates of the transmitted symbols are also obtained as by-products. The resulting scheme has a feedforward structure and provides phase and timing information in a fixed time, differently from closed-loop architectures. It can be implemented in digital form and is particularly suitable for burst mode transmissions. Its performance is investigated by simulation and the results are compared with Cramér-Rao bounds. It turns out that the estimation accuracy is very close to the theoretical limits, even with short observation intervals and small values of the excess bandwidth. In such conditions, the proposed estimators largely outperform other schemes already known in literature. Their superiority becomes less significant as the signal bandwidth increases.

A Linewidth-Tolerant Two-Stage CPE Using a New QPSK-Partitioning Approach and an Enhanced Maximum Likelihood Detection for 64-QAM Coherent Optical Systems

We propose a novel two-stage carrier phase estimation (CPE) algorithm for 64-ary quadrature amplitude modulation (64-QAM) coherent optical systems. It makes use of a new quadrature phase shift keying partitioning (QPSK-partitioning) approach and an enhanced maximum likelihood (ML) detection. 36/64 of the 64-QAM symbols are statistically employed by the modified QPSK-partitioning approach to estimate phase noise in the first stage. The proposed enhanced ML detection is used in the second stage to improve linewidth tolerance performance. The simulation results show that, our proposed VVPEc1&c2 + EML algorithm can achieve much higher linewidth tolerance performance than the modified V&V + MLE algorithm at high laser linewidth level. A combined laser linewidth symbol duration product as high as $5.6 \times 10^{- 5} $ is achieved by the proposed two-stage CPE algorithm at a target BER of $1 \times 10^{- 2} $ with 1 dB SNR sensitivity penalty.

Carrier Phase Estimation Through the Rotation Algorithm for 64-QAM Optical Systems

A novel low-complexity two-stage digital feed-forward carrier phase estimation algorithm based on the rotation of constellation points to remove phase modulation for a 64-ary quadrature amplitude modulation (QAM) system is proposed and analyzed both experimentally and through numerical simulations. The first stage is composed of a Viterbi and Viterbi (V&V) block, based on either the standard quadrature phase shift keying (QPSK) partitioning algorithm using only Class-1 symbols or a modified QPSK partitioning scheme utilizing both Class-1 and outer most triangle-edge (TE) symbols. The second stage applies the V&V algorithm after the removal of phase modulation through rotation of constellation points. Comparison of the proposed scheme with constellation transformation, blind phase search (BPS) and BPS+MLE (maximum likelihood estimation) algorithm is also shown. For an OSNR penalty of 1 dB at bit error rate of $10^{-2}$ , the proposed scheme can tolerate a linewidth times symbol duration product ( $\Delta \hbox{$\nu $}\cdot T_{s}$ ) equal to $3.7\times 10^{-5}$ , making it possible to operate 32-GBd optical 64-QAM systems with current commercial tunable lasers.

A simple three-stage carrier phase estimation algorithm for 16-QAM systems based on QPSK partitioning and maximum likelihood detection

This paper proposes a simple three-stage carrier phase estimation (CPE) algorithm for 16-ary quadrature amplitude modulation (16-QAM) optical coherent systems, based on a simplified quadrature phase shift keying partition (QPSK-partitioning) scheme for the first stage and the maximum likelihood (ML) detection for the second and the third stage. Only 25% of the symbols of 16-QAM systems are employed for the first stage phase estimation, while only 50% are used for the second stage phase estimation. Therefore, the computational complexity of the proposed three-stage CPE algorithm for 16-QAM systems is similar to that of the QPSK-partitioning+ML algorithm. The performance of two different ML detections is compared and the simulation results show that the “constellation-assisted” ML detection can achieve better linewidth tolerance performance than the “conventional” ML detection for 16-QAM systems. A combined linewidth symbol duration product of 1×10−4 is tolerable for a signal noise ratio (SNR) sensitivity penalty of 0.8dB at a BER of 1×10−3, based on the block averaging instead of the sliding window averaging. A good bit error rate (BER) performance for the proposed three-stage CPE algorithm is achieved especially at high SNR levels in the simulation. The performance of the proposed three-stage CPE algorithm is similar to that of the BPS algorithm with 22 test phase angles, but with reducing the computational complexity by a factor of about 5.3.