Decision-directed Least Mean Square Research Articles

Coherent optical transmitter IQ imbalance mitigation with embedded carrier phase and frequency offset estimation.

We present a 4×4 real-valued channel equalizer with embedded phase estimator, designed for carrier phase and frequency offset estimation and compensation in coherent optical communications with in-phase/quadrature (IQ) impairments. These impairments include IQ timing skew, gain imbalance, and quadrature phase errors at the transmitter side. To achieve adaptive control of the equalizer's filter coefficients, we employ the decision-directed least mean square (DD-LMS) algorithm. This algorithm minimizes the error between the filter outputs and the desired signals in a symbol-by-symbol manner, resulting in faster channel coefficients adaptation speed. Simulation results for a 60 GBaud polarization-multiplexing 16 quadrature amplitude modulation (PM-16QAM) signal demonstrate that our proposed equalizer outperforms a 2×2 cascaded multi modulus algorithm-based (CMMA-based) equalizer, a 2×2 complex-valued method based on the DD-LMS algorithm, and the 4×4 real-valued equalizer without embedded frequency offset estimation (FOE), when the transmitter IQ impairments exist. Experimental validation is also conducted for the 60 GBaud PM-16QAM and 45 GBaud PM-64QAM signals. Similar to the simulation results, our experiments show that the proposed 4×4 real-valued equalizer with embedded FOE can achieve effective SNR penalties of less than 0.41 dB at 7 ps skew, 1.33 dB at 3.5 dB gain imbalance, and 1.64 dB at the bias voltage shift of 0.5 V for the 60 GBaud PM-16QAM signal. Finally, our experimental results confirm the effectiveness of our proposed method in carrier phase estimation as well as the frequency offset compensation, when compared with 4×4 real-valued equalizer without embedded FOE.

Fast polarization state tracking for dual-polarization direct detection with Jones-space field recovery.

Direct detection system is expected to possess the phase and polarization diversity in order to achieve high spectral efficiency and fiber impairment compensation such as chromatic dispersion and polarization rotation. In this Letter, we theoretically extend the concept of the proposed Jones-space field recovery (JSFR) to include a dynamic polarization rotation matrix and experimentally demonstrate the rapid polarization state tracking ability of the JSFR receiver based on a 3 × 3 optical coupler. Under a rotation of the state of polarization at a rate of 1 Mrad/s, we successfully transmit 59-GBd dual-polarization 16-ary quadrature-amplitude-modulation signals over an 80-km standard single-mode fiber based on a decision-directed least mean square (DD-LMS) or a recursive least square (DD-RLS), with a bit-error rate below the 14% hard-decision forward error correction threshold of 1 × 10-2. The experimental results indicate that the legacy polarization tracking algorithms designed for coherent optical communication are also applicable for this direct detection scheme. To our best knowledge, this work demonstrates the first polarization rotation-tolerant direct detection system with phase and polarization diversity, providing a low-cost and high-speed solution for short-reach communications.

FPGA implementation for 44.2368-Gbit/s PAM8 signal transmission with pruned pre-equalization.

In this experiment, we demonstrated an intensity modulation and direct-detection (IM/DD) system based on a field-programmable gate array (FPGA). The PAM8 signals are successfully delivered at 44.2368 Gbit/s over a 45-km standard single-mode fiber (SSMF), satisfying the soft-decision forward error correction (SD-FEC) criterion of 2.4 × 10-2, and the net bit rate may reach 36.864 Gbit/s without the need of optical amplifiers. At the transmitter, we used a pruned pre-equalization algorithm to process the PAM8 signals, and the high-speed parallel PAM8 signals were processed at the receiver using 64 parallel constant modulus algorithm (CMA) and decision-directed least mean square (DD-LMS) equalizers. Additionally, we analyzed the bit error rate (BER) performance of the DD-LMS equalizer in FPGA simulation with various data lengths, both before and after equalization.

Intelligent and adaptive intra-channel and inter-channel fiber nonlinearity compensation in coherent optical transmission systems.

Fiber nonlinearity mitigation is a crucial technology for extending transmission reach and increasing channel capacity in high-baud rate wavelength division multiplexing (WDM) systems. In this work, we propose a novel, to the best of our knowledge, architecture that combines learned modified digital back-propagation (L-MDBP) to compensate for intra-channel nonlinearity and a two-stage decision-directed least mean square (DDLMS) adaptive equalizer to mitigate inter-channel nonlinearity. By leveraging globally optimized model parameters and adaptive channel estimation, the proposed scheme achieves superior performance and lower computation complexity compared with conventional DBP. Specifically, in an 8 × 64 Gbaud 16-ary quadrature amplitude modulation (16QAM) experimental system over 1600 km of standard single-mode fiber (SSMF), our approach shows a 0.30-dB Q2-factor improvement and a complexity reduction of 82.3% compared with DBP with 8 steps per span (SPS). Furthermore, we enhance the adaptability of the architecture by introducing an online transfer learning (TL) technique, which requires only 2% of initial training epochs.

FPGA-based 4 × 29.4912 Gbit/s PS-PAM4 signal transmission with a low-complexity probabilistic shaping scheme.

In this experiment, we demonstrate a real-time intensity modulation and direct detection (IM/DD) system based on a field programmable gate array (FPGA). For high-speed parallel signal processing, we propose and implement the simplified parallel-constant modulus algorithm (CMA) and decision-directed least mean square (DDLMS) equalizers with low complexity and low latency. Moreover, the bit-class probabilistic shaping (PS) scheme is adopted with very few hardware resources. The digital signal processing (DSP) steps are implemented in the XCVU9P-FLGB2104-2-I Xilinx FPGA with a clock frequency of 230.4 MHz. Based on the experimental results, 4 × 29.4912 Gbit/s PS-pulse amplitude modulation (PAM4) signals can be successfully transmitted over 25 km of standard single-mode fiber (SSMF) while satisfying the hard-decision forward error correction (HD-FEC) threshold at 3.8 × 10-3. Compared with the uniformly distributed PAM4 signal, the low-complexity PS scheme can improve the receiver sensitivity by more than 1 dB.

Digital inter-symbol interference mitigation method for Nyquist OTDM signal detection with coherent sampling and KNN classifier

We propose and experimentally demonstrate a digital method to mitigate inter-symbol interference (ISI) in demultiplexing Nyquist optical time-division multiplexing (N-OTDM) signal using a simple but powerfully serviceable k-nearest neighbors (KNN) classifier. The proposed method is evaluated under four types of sampling pulse-width corresponding to various extents of ISI distortion in 160 Gbaud down to 40 Gbaud demultiplexing experiments, and the mitigation performance is found to be better along with the increasement of sampling pulse-width. Compared with the traditional decision-directed least mean square (DD-LMS) equalization method, a 3.9 dB required-OSNR (at a BER of 10−4) improvement is observed under the pulse-width of 5.3 ps. Demultiplexing performance is investigated under the pulse-widths of 3.2 ps and 3.9 ps as well, in which the improvement of required-OSNR are found to be 1.33 dB and 2.95 dB, respectively. The performance improvement clearly indicates the strong ISI mitigation ability of the KNN classifier for N-OTDM signal demultiplexing. In addition, experimental results also confirm that the proposed digital ISI mitigation method could release the pulse-width requirement of the N-OTDM demultiplexing, which paves the way for future practical application of N-OTDM technique.

Receiver IQ Imbalance and Skew Compensation for High Order Modulation Formats by Frequency Domain 4 × 2 MIMO

A typical coherent front-end may encounter imperfections, such as amplitude imbalance, phase imbalance and skew, between four sampling channels. These receiver (Rx) imperfections, together with traditional chromatic dispersion (CD) compensation, would induce in-phase (I) and quadrature (Q) mixing, which greatly degrades the system performance. In this work, we propose a modulation format independent frequency domain (FD) 4 × 2 multi-inputs and multi-outputs (MIMO) based on decision directed least mean square (DD-LMS) algorithm to solve this problem. Compared with traditional 2 × 2 MIMO, it enables to compensate for all of Rx imperfections. Compared with 4 × 4 MIMO, it can eliminate performance penalty induced by the IQ mixing after optical fiber transmission. Compared with traditional time domain 4 × 2 MIMO, the proposed FD 4 × 2 MIMO has advantages in terms of complexity and parallel processing. We also evaluate the Rx imperfections compensation capability, noise tolerance tolerance, polarization tracking capability, residual CD and transmitter imperfections tolerance of the proposed equalizers. The proposed FD 4 × 2 MIMO algorithms are verified by numerical simulation with different modulation formats and experimentally validated with 40 Gbaud polarization division multiplexed 16-ary quadrature amplitude modulation signal.

Enhanced Performance of a Cascaded Receiver Consisting of a DNN-Based Waveform-to-Symbol Converter and Modified NN-Based DD-LMS in CAP Underwater VLC System

Underwater visible light communication (UVLC) based on LEDs has become a competitive candidate, which is able to provide high data rates, low latency and low cost for next-generation wireless communication technologies. However, it is still challenging to achieve high-speed communication because of bottleneck problems such as bandwidth limitation and linear and nonlinear distortions. Traditional Deep-learning Neural Network (DNN)-based waveform-to-symbol converter is verified to be an effective method to alleviate them, but impractical due to high complexity. To achieve a better tradeoff between communication performance and computation complexity, a cascaded receiver consisting of a DNN-based waveform-to-symbol converter and modified Neural Network (NN)-based decision-directed least mean square (DD-LMS) is then innovatively proposed. With fewer taps and nodes than the traditional converter, the front-stage converter could mitigate the majority of Inter-Symbol Interference (ISI) and signal nonlinear distortions. Then modified NN-based DD-LMS is cascaded to improve communication performance by reducing phase offset, making received constellation points more concentrated and closer to standard constellation points. Compared with the traditional converter, the cascaded receiver could achieve 89.6% of signal Vpp dynamic range with 12.4% of complexity in the 64APSK UVLC system. Moreover, the ratio of signal Vpp dynamic range and total trainable parameters is 1.24 × 10−1 mV, while that of the traditional converter is 1.95 × 10−2 mV. The cascaded receiver used in 64APSK UVLC systems is experimentally verified to achieve enhanced performance, thus as a promising scheme for future high-speed underwater VLC.

Joint estimation of dynamic polarization and carrier phase with pilot-based adaptive equalizer in PDM-64 QAM transmission system

A pilot-based adaptive equalizer is investigated for high cardinality polarization-division-multiplexing quadrature amplitude modulation transmission systems. Pilot symbols are periodically inserted for joint estimation of the dynamic state of polarization (SOP) and carrier phase, in a least mean square (LMS) sense. Compared to decision-directed least mean square (DDLMS) equalization and radially-directed equalization, the proposed equalizer can achieve robust equalization and phase estimation, especially in low optical signal-to-noise ratio (OSNR) scenarios. In an experiment on 56 GBaud PDM-64 QAM transmission over 400 km standard single-mode fiber, we obtained at least 0.35 bit per symbol generalized mutual information (GMI) improvement compared with other training symbol-based equalization when tracking 600 krad/s dynamic SOP. With the joint estimation scheme, the equalization performance will not be compromised even if the SOP speed reaches 600 krad/s or the laser linewidth approaches 2 MHz. For the first time, it is demonstrated that the pilot-based equalizer can track dynamic SOP rotation and compensate for fiber linear impairments without any cycle slips under extreme conditions.

A Simple and Robust Equalization Algorithm for Variable Modulation Systems

This paper proposes a simple and robust variable modulation-decision-directed least mean square (VM-DDLMS) algorithm for reducing the complexity of conventional equalization algorithms and improving the stability of variable modulation (VM) systems. Compared to conventional adaptive equalization algorithms, known information was used as training sequences to reduce the bandwidth consumption caused by inserting training sequences; compared with conventional blind equalization algorithms, the parameters and decisions of the equalizer were determinate, which was conducive to a stable equalization performance. The simulation and implementation results show that the proposed algorithm has a better bit error rate (BER) performance than that of the constant modulus algorithm (CMA) and modified constant modulus algorithm (MCMA) while maintaining the same level of consumption of hardware resources. Compared to the conventional decision-directed least mean square (DDLMS) algorithm, the proposed algorithm only needs to make quadrature phase shift keying (QPSK) symbol decisions, which reduces the computational complexity. In parallel 11th-order equalization algorithms, the operating frequency of VM-DDLMS can reach up to 333.33 MHz.

Wide range rate adaptation of QAM-based probabilistic constellation shaping using a fixed FEC with blind adaptive equalization.

We investigate the rate adaptability of quadrature amplitude modulation (QAM)-based probabilistic constellation shaping (PCS) using a fixed forward error correction (FEC) scheme over a wide range of information rates (IRs). Blind adaptive equalization that does not sacrifice any of the IRs was adopted. We show that the conventional decision directed least mean square (DDLMS) algorithm can cause a problem of mis-convergence when it is applied to the PCS of a low IR. To avoid the mis-convergence of DDLMS, we propose a DDLMS-based algorithm that simultaneously minimizes the error between the average symbol power of filter outputs and that of a transmitted PCS signal. Using this technique, we conducted a wavelength-division multiplexed transmission experiment with 32-Gbaud 16/64QAM-based PCS and a fixed FEC of a low-density parity-check code for DVBS-2, where the IR of PCS was optimized at each transmission distance. We confirmed that the data rate of PCS with a fixed FEC and DSP configuration could be improved up to 1.9 times compared with that of QAM-only rate adaptation and that 64QAM-based PCS could provide a wider range of transmission distance and IR while almost covering that of the 16QAM-based one.

180 Gb/s PAM8 Signal Transmission in Bandwidth-Limited IMDD System Enabled by Tap Coefficient Decision Directed Volterra Equalizer

In this paper, we have explored the feasibility of 8-level pulse amplitude modulation (PAM8) format for realizing beyond 100 Gb/s transmission in a bandwidth-limited intensity modulation and direct-detection (IMDD) system. Generally, Volterra equalizer is a practical algorithm which can be used in bandwidth-limited system to combat the linear and nonlinear impairments. However, we observe that the performance of Volterra equalizer would be restricted by a phenomenon of level-dependent equalized effect, especially in low optical signal-to-noise ratio case. To cope with this problem, a tap coefficient decision directed Volterra equalizer (TDD-Volterra) with multiple sets of tap coefficients is proposed. The optimal set of tap coefficients is selected according to the decision results of input symbols before equalization. By this means, the TDD-Volterra is effective for overcoming the level-dependent equalized effect. By using the proposed TDD-Volterra, 180 Gb/s PAM8 signal is successfully transmitted over 2 km standard single mode fiber in an IMDD system with the 10-dB bandwidth of 17.5 GHz for the first time. The achieved bit error rate (BER) value is below the 7% overhead hard-decision forward error correction threshold of $3.8\times 10 ^{-3}$ . Moreover, the performance comparison among decision-directed least-mean square equalizer, Volterra Equalizer, and TDD-Volterra are analyzed in this paper. The experimental results show that compared to the conventional Volterra equalizer, the algorithm complexity of TDD-Volterra is decreased by over 50% for achieving a given BER value. Our research may provide a solution for beyond 100 Gb/s short-reach applications using bandwidth-limited electro-optical components.

Enhancement of Spectral Efficiency and Power Budget in WDN-PON Employing LDPC-Coded Probabilistic Shaping PAM8

A $5\times60$ Gb/s low-density parity-check (LDPC) coded WDM-PON using 8-level pulse amplitude modulation (PAM8) signals with probabilistic shaping (PS) over 20-km standard single-mode fiber (SSMF) with direct detection is proposed and experimentally demonstrated. Each optical carrier at the optical line terminal (OLT) carries a 20-Gbaud probabilistically-shaped LDPC-coded PS-PAM8 signal from a conventional uniform PAM16 signal following a designed approximate Gaussian distribution (AGD). A LDPC-based bit-interleaved coded modulation scheme with iterative decoding (BICM-ID) is utilized to recover the inherently overlapped symbols due to the proposed PS mapping. Combining decision-directed least-mean-squares (DD-LMS) and constant modulus algorithm (CMA), a blind feedback pre-equalizer is adopted to reduce the frequency loss of the signal owing to imperfect electro-optical components in experiment. As for experimental verification and application, 1.48-dB superior receiver power sensitivity and 1.7-dB launch fiber power optimization are gained respectively by using LDPC-coded PS-PAM8 signals in WDM-PON, which can bring higher spectral efficiency and power loss budget for the optical distribution networks.

Discrete FRFT-Based Frame and Frequency Synchronization for Coherent Optical Systems

A joint frame and carrier frequency synchronization algorithm for coherent optical systems, based on the digital computation of the fractional Fourier transform (FRFT), is proposed. The algorithm utilizes the characteristics of energy centralization of chirp signals in the FRFT domain, together with the time and phase shift properties of the FRFT. Chirp signals are used to construct a training sequence (TS), and fractional cross-correlation is employed to define a detection metric for the TS, from which a set of equations can be obtained. Estimates of both the timing offset and carrier frequency offset (CFO) are obtained by solving these equations. This TS is later employed in a phase-dependent decision-directed least-mean square algorithm for adaptive equalization. Simulation results of a 32-Gbaud coherent polarization division multiplexed Nyquist system show that the proposed scheme has a wide CFO estimation range and accurate synchronization performance even in poor optical signal-to-noise ratio conditions.

ICI Mitigation for Dual-Carrier Superchannel Transmission Based on m-PSK and m-QAM Formats

We propose an intercarrier interference (ICI) mitigation scheme for dual-carrier superchannel transmission. Using the modified decision directed least mean square (DD-LMS) based multi-input multi-output (MIMO) processing, we can simultaneously realize both polarization division demultiplexing and ICI mitigation for PDM-m-PSK and PDM-m-QAM formats. Thanks to the innovative modification of frequency-shift operation, the MIMO equalization enables synchronous output of dual-carrier signals with robust performance against frequency offset, laser phase noise, as well as polarization mode dispersion (PMD). The experimental back-to-back transmission of PDM-16QAM signal has 8.3-dB OSNR improvement using the proposed MIMO processing. Three dual-carrier 160-Gb/s PDM-16QAM superchannels within 25-GHz grid are experimentally transmitted over 640-km standard single-mode fiber (SSMF), leading to net spectral efficiency of 5.7b/s/Hz.

64QAM Vector Radio-Frequency Signal Generation Based on Phase Precoding and Optical Carrier Suppression Modulation

We propose a 64-quadrature-amplitude-modulation (QAM) vector radio-frequency (RF) signal generation and detection via one intensity Mach–Zehnder modulator (MZM) by using optical carrier suppression modulation. In order to obtain an ordinary QAM modulated RF signal after the square-law detection of the photodiode, amplitude preequalization and phase precoding technique are employed. The 1Gbaud 64QAM vector signals performed on various RF carrier frequencies from 20 to 444 GHz are experimentally demonstrated. The bit-error-rate (BER) performance of unbalanced phase precoding adopting 64QAM modulation is investigated. The digital filter based on blind Decision-Directed Least-Mean-Squares (DD-LMS) algorithm is the key digital-signal-processing, which is used to improve system performance. This concept of the generation of 64QAM-modulated vector signal can be employed in the radio-over-fiber (ROF) system.

Modulation format identification aided hitless flexible coherent transceiver

We propose a hitless flexible coherent transceiver enabled by a novel modulation format identification (MFI) scheme for dynamic agile optical networks. The modulation format transparent digital signal processing (DSP) is realized by a block-wise decision-directed least-mean-square (DD-LMS) equalizer for channel tracking, and a pilot symbol aided superscalar phase locked loop (PLL) for carrier phase estimation (CPE). For the MFI, the modulation format information is encoded onto the pilot symbols initially used for CPE. Therefore, the proposed MFI method does not require extra overhead. Moreover, it can identify arbitrary modulation formats including multi-dimensional formats, and it enables tracking of the format change for short data blocks. The performance of the proposed hitless flexible coherent transceiver is successfully evaluated with five modulation formats including QPSK, 16QAM, 64QAM, Hybrid QPSK/8QAM and set-partitioning (SP)-512-QAM. We show that the proposed MFI method induces a negligible performance penalty. Moreover, we experimentally demonstrate that such a hitless transceiver can adapt to fast block-by-block modulation format switching. Finally, the performance improvement of the proposed MFI method is experimentally verified with respect to other commonly used MFI methods.

Demonstration of 4 128-Gb/s DFT-S OFDM Signal Transmission over 320-km SMF With IM/DD

Intensity modulation and direct detection (IM/DD) is a promising solution for short-reach links. In this paper, we experimentally demonstrate transmission of 128-Gb/s discrete Fourier transform (DFT)-spread 16-ary quadrature amplitude modulation (QAM) orthogonal frequency division multiplexing (OFDM) signals in an IM/DD optical system. Vestigial sideband (VSB) modulation with 20-dB suppression ratio enabled by a cost-effective interleave or a tunable optical filter (TOF) is employed to eliminate chromatic dispersion (CD) induced power fading. Meanwhile, a combination of advanced digital signal processing (DSP) algorithms, such as pre-equalization, nonlinearity equalization (NE), and decision-directed least mean square (DDLMS), are utilized to mitigate the system distortions and improve system performance. By using this scheme, the 32-GBd DFT-spread OFDM signals can be successfully transmitted over 320-km standard single mode fiber (SSMF) with the measured bit error ratio (BER) under the soft-decision forward-error-correction (SD-FEC) threshold of $2\times 10^{-2}$ . We also achieved 144-Gb/s for 80-km SSMF transmission. Additionally, 4 $\times$ 128-Gb/s wavelength division multiplexing (WDM) DFT-S OFDM signals transmission over 320-km SSMF is experimentally investigated. To the best of our knowledge, this work reports the longest transmission distance for beyond 100-Gb/s IM/DD system.

Performance Comparison of Dual-Carrier 400G With 8/16/32-QAM Modulation Formats

The performance of dual-carrier 400G solutions based on three high-order quadrature amplitude modulation (QAM) formats (8/16/32-QAM) is investigated on the same platform. We first study the benefit and penalty differences of gray and differential coding, and then experimentally compare OSNR sensitivities and transmission performance using flexible transceiver configuration and the same erbium-doped fiber amplifiers-only standard single-mode fiber link. Experimental results show the implementation penalties are 1.85 dB for 42-GBd polarization multiplexed (PM)-8QAM, which is 1 and 1.5 dB better than 32-GBd PM-16QAM and 25-GBd PM-32QAM at 5.92-dB Q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -factor (corresponding to a bit error rate of 2.4E-2), respectively. It is also found that maximum reaches of 2460 km for 42.6-GBd PM-8QAM, 1640 km for 32-GBd 16QAM, and 820 km for 25-GBd PM-32QAM are achieved. Furthermore, the performances of cascaded multimodulus algorithm and decision-directed least mean square (DD-LMS) algorithms are also compared. The improvement by addition of one sample/symbol-based DD-LMS digital filter is investigated as well.

Long-haul quasi-single-mode transmissions using few-mode fiber in presence of multi-path interference.

We study long-haul Quasi-Single-mode (QSM) systems in which signals are transmitted in the fundamental modes of a few-mode fiber (FMF) while keeping other system components such as amplifiers and receivers are kept single-moded. The large-effective-area nature of the FMF fundamental modes improves system nonlinear tolerance in the expense of mode coupling along FMF transmissions which induces multi-path interference (MPI) and needs to be compensated. We analytically investigate 6-spatial-polarization mode QSM transmission systems in presence of MPI and show that in the weak coupling regime, the QSM channel is a Gaussian random process in frequency. MPI compensation filters are derived and performance penalties due to MPI and signal loss from higher-order modes are characterized. We also experimentally demonstrate 256 Gb/s polarization multiplexed (PM)-16-QAM QSM transmissions over a record distance of 2600 km with 100-km span using decision directed least mean square (DD-LMS) algorithm for MPI compensation.