Nonlinear Equalization Research Articles

Sequential Quadratic Optimization for Stochastic Optimization with Deterministic Nonlinear Inequality and Equality Constraints

Sequential Quadratic Optimization for Stochastic Optimization with Deterministic Nonlinear Inequality and Equality Constraints

Impact of Receiver Bandwidth on the Performance of Nonlinear Volterra Equalizer in IM/DD Systems

Nonlinear electrical equalization is an effective means to compensate for nonlinear waveform distortions in short-haul intensity-modulation (IM)/direct-detection (DD) optical transmission systems. Nonlinear distortions contain broader spectral components than the original signal. Thus, a wide receiver bandwidth and high sampling rate are beneficial to effective compensation of nonlinear distortions. However, this leads to poor signal-to-noise ratio at the receiver. In this paper, we investigate the impact of receiver bandwidth on the performance of 2nd-order nonlinear Volterra equalizer in IM/DD transmission systems. Through the computer simulation and the experimental verification performed with 4-ary pulse amplitude modulation signals, we show that an optimum receiver bandwidth for bit-error ratio performance increases with the amount of nonlinear distortions in IM/DD systems. We also show that the sampling rate should also be increased to avoid the aliasing of the nonlinear distortion components. The cost of the receiver increased by the wider receiver bandwidth and higher sampling rate could be offset in part by the non-integer fractionally sampled nonlinear Volterra equalizer.

Cognitive learning enabled agile optical network

Nonlinear equalization (NLE) is essential for guaranteeing the performance of an optical network (ON). Effective NLE implementation relies on key parameters of the transmission link, including the modulation format (MF) and the launch power. As ONs become more agile, the parameters of fiber optical transmission need to be adaptive and relevant to the routing condition. Therefore, successful NLE implementation relies on the realization of transmission awareness (TA). Although machine learning-enabled optical performance monitoring (OPM) has been extensively investigated in the past few years, current NLE algorithms cannot autonomously perceive transmission parameters. Furthermore, current TA implementation still needs human intervention to guide the NLE. In addition, existing ML-based OPM and NLE cannot be trained autonomously, leading to the incapability of environmental change and mislabeling. Here, we propose cognitive learning (CL) for TA-guided NLE in agile ONs. We perform an experiment involving 32 Gbaud polarization-division-multiplexed (PDM)-quadrature phase shift keying (QPSK)/16-quadrature amplitude modulation (QAM) transmission over 1500 km of standard single-mode fiber (SSMF) with a variable launch power from 0 to 3 dBm. When a deep neural network (DNN) with amplitude histograms (AHs) as inputs and one step per span-learned digital back-propagation (1stps-LDBP) are developed, the CL simultaneously enables both TA and NLE, with the capability of self-learning, mislabeling resistance, and dynamic adaptation. The proof-of-concept experimental results indicate that both the accuracy of TA and the Q-factor of PDM-16QAM can be improved by 34.8% and 0.84 dB, respectively, when the launch power is 3 dBm. Moreover, the accuracy of TA is enhanced by 35.3%, even when the used data has 30% mislabeling. Therefore, the CL framework can be customized to satisfy various NLE implementations, thereby supporting the adaptive transmission of agile ONs.

D-Band 4.6 km 2 × 2 MIMO Photonic-Assisted Terahertz Wireless Communication Utilizing Iterative Pruning Deep Neural Network-Based Nonlinear Equalization

In this paper, we explore the enhancement of a 4.6 km dual-polarization 2 × 2 MIMO D-band photonic-assisted terahertz communication system using iterative pruning-based deep neural network (DNN) nonlinear equalization techniques. The system employs advanced digital signal processing (DSP) methods, including down-conversion, resampling, matched filtering, and various equalization algorithms to combat signal distortions. We demonstrate the effectiveness of DNN and iterative pruning techniques in significantly reducing bit error rates (BERs) across a range of symbol rates (10 Gbaud to 30 Gbaud) and polarization states (vertical and horizontal). Before pruning, at 10 GBaud transmission, the lowest BER was 0.0362, and at 30 GBaud transmission, the lowest BER was 0.1826, both of which did not meet the 20% soft-decision forward error correction (SD-FEC) threshold. After pruning, the BER at different transmission rates was reduced to below the hard decision forward error correction (HD-FEC) threshold, indicating a substantial improvement in signal quality. Additionally, the pruning process contributed to a decrease in network complexity, with a maximum reduction of 85.9% for 10 GBaud signals and 63.0% for 30 GBaud signals. These findings indicate the potential of DNN and pruning techniques to enhance the performance and efficiency of terahertz communication systems, providing valuable insights for future high-capacity, long-distance wireless networks.

MIMO Volterra polynomial equalizer for PDM ultrahigh-order QAM signals.

In this work, we proposed and experimentally demonstrated a multiple-input multiple-output (MIMO) Volterra polynomial equalizer (MVPE), in a probabilistic shaping (PS) polarization-division multiplexed (PDM) 4096-quadrature amplitude modulation (QAM) coherent optical transmission system. In comparison with the conventional single-input single-output (SISO) Volterra nonlinear equalizers (VNLEs), the proposed MVPE can equalize the in-phase (I) and quadrature (Q) components in both the X and Y polarizations simultaneously and effectively alleviate nonlinear distortions. In the PS-PDM-4096-QAM experiment, the normalized generalized mutual information (NGMI) reached the threshold of DVB-S2 low-density parity-check (LDPC) codes with a 20% overhead. Compared to the SISO VNLE scheme, the MVPE scheme improves receiver sensitivity by approximately 2.8 dB while maintaining nearly similar computational complexity.

Low-complexity end-to-end deep learning framework for 100G-PON

End-to-end learning allows communication systems to achieve optimal performance compared with conventional blockwise structure design. By modeling the channel with neural networks and training the transmitter and receiver on this differentiable channel, the whole system can be jointly optimized. However, in existing schemes, channel modeling methods, such as the generative adversarial network and long short-term memory network, have complex architectures and cannot track channel changes, leading to less effective end-to-end learning. Meanwhile, the complexity of neural networks deployed at the transmitter and receiver is too high for practical applications. In this work, we propose an efficient and low-complexity end-to-end deep learning framework and experimentally validate it on a 100G passive optical network. It uses a noise adaptation network to model channel response and noise distribution and employs offline pretraining and online tracking training to improve the efficiency and accuracy of channel modeling. For the transmitter, it consists of a pattern-dependent look-up table (PDLUT) based on a neural network (NN-PDLUT) with a single convolutional layer. Further, the receiver is also an NN with a single convolutional layer; thus, the end-to-end signal processing is extremely simple. The experimental results show that end-to-end learning improves the receiver sensitivity by 0.85 and 1.59 dB compared with receiver-only equalization based on Volterra nonlinear equalization (VNLE) and joint equalization based on a PDLUT and a feed-forward equalizer, respectively. Moreover, the number of multiply–accumulate operations consumed by the transmitter and receiver in the end-to-end learning scheme is reduced by 75.7% compared with VNLE-based receiver-only equalization.

MMSE-Driven Signal Constellation Scatterplot Using Neural Networks-Based Nonlinear Equalizers

MMSE-Driven Signal Constellation Scatterplot Using Neural Networks-Based Nonlinear Equalizers

Ultra-low-complexity weight-sharing trigonometric nonlinear equalizer for beyond net-200-Gb/s/λ short-reach optical interconnects.

An ultra-low-complexity third-order weight-sharing trigonometric nonlinear equalizer (WS-TNLE) is proposed to eliminate nonlinear signal distortions in short-reach optical interconnects exceeding net 200 Gb/s/λ. By replacing the second- and third-order nonlinear terms in a third-order weight-sharing diagonally pruned Volterra nonlinear equalizer (WS-DP-VNLE) with cosine and sine terms, respectively, the required number of real-valued multiplications per symbol of the proposed third-order WS-TNLE is significantly reduced to the same value as the number of weight-sharing kernels. When transmitting probabilistically shaped 16-level pulse amplitude modulation (PS-PAM-16) signals at net rates ranging from 200.4 Gb/s to 300.4 Gb/s over a 1-km standard single-mode fiber (SSMF), the proposed third-order WS-TNLE requires the lowest number of real-valued multiplications per symbol, ranging from 10 to 44, which reduces the computational complexity by up to 96.2% and 52.4% compared to the third-order WS-DP-VNLE and WS-DP-absolute-term nonlinear equalizer (WS-DP-ATNLE), respectively.

SkipNet: an adaptive neural network equalization algorithm for future passive optical networking

In this paper, we propose an original adaptive neural network equalizer (NNE) algorithm named SkipNet, which is suitable for rapid training on a packet-by-packet basis for burst-mode non-linear equalization in upstream PON transmission. SkipNet uses the simple LMS algorithm and avoids complex neural network training algorithms such as backpropagation and mini-batch training. We demonstrate SkipNet on captured continuous mode 100 Gbit/s PAM4 signals using an SOA preamplifier to achieve the challenging 29 dB PON optical loss budget. The adaptive SkipNet equalizer is shown to overcome combinations of severe SOA patterning effects and fiber dispersion impairments to achieve >29dB dynamic range back-to-back and >22.9dB dynamic range for up to 81.6 ps/nm accumulated dispersion. It can adapt in as little as 250 training symbols to each impairment scenario, which is equivalent to existing FFE/DFE solutions, while matching the non-linear performance of previously proposed static NNE solutions. To the best of our knowledge, SkipNet is the first ever adaptive NNE framework that can realistically be trained and adapted on a packet-by-packet basis and within strict PON packet preamble lengths.

Photonics-aided exceeding 200-Gb/s wireless data transmission over outdoor long-range 2 × 2 MIMO THz links at 300 GHz

Outdoor long-range terahertz (THz) communications often come at the expense of transmission rate. Moreover, the practicability of the single polarization optical/THz link, which is commonly used in the previous long-range THz demonstrations based on photonics, is extremely limited by the following two fatal defects. One is relying on active polarization control, and the other is not supporting the transparent bridging of optical polarization division multiplexed (PDM) signals for mature coherent optical communication networks. In this work, a large-capacity photonics-aided THz wireless communication system based on the outdoor long-range 2 × 2 multiple-input multiple-output (MIMO) links has been successfully demonstrated. We first build the 200-m 2 × 2 MIMO THz wireless links at the 300 GHz band. The cascaded linear and nonlinear equalizers are proposed which can significantly improve the transmission performance of 100- and 200-Gb/s PDM quadrature phase shift keying (QPSK) signals. Then an interesting 2 × 2 MIMO structure which can provide certain diversity reception gain under 200-m long-range wireless delivery using the same polarization scheme is also presented and further compared with the orthogonal polarization scheme. Since each THz receiver simultaneously receives data from both the two THz transmitters for this MIMO links, an improvement over 6 dB in receiving sensitivity and one order of magnitude in bit error ratio performance under low signal-to-noise ratio conditions can be achieved. Finally, based on the proposed cascaded equalizers and novel 2 × 2 MIMO structure, we successfully demonstrate a record-breaking 58-Gbaud (232-Gb/s) PDM-QPSK signal transmission over 200-m 2 × 2 MIMO THz wireless links. This is an important attempt for the photonics-aided THz wireless communication systems to achieve 2 × 2 MIMO transmission over a long wireless distance in the outdoors. Furthermore, attaining over 200 Gb/s at a wireless distance of 200 m also represents a key milestone for the long-range and large-capacity THz wireless communication systems.

A fast smoothing newton method for bilevel hyperparameter optimization for SVC with logistic loss

Support vector classification (SVC) with logistic loss has excellent theoretical properties in classification problems where the label values are not continuous. In this paper, we reformulate the hyperparameter selection for SVC with logistic loss as a bilevel optimization problem in which the upper-level problem and the lower-level problem are both based on logistic loss. The resulting bilevel optimization model is converted to a single-level nonlinear programming (NLP) based on the KKT conditions of the lower-level problem. Such NLP contains a set of nonlinear equality constraints and a simple lower-bound constraint. The second-order sufficient condition is characterized, which guarantees that the strict local optimizers are obtained. To solve such NLP, we apply the smoothing Newton method proposed in [Liang L, Sun D., Toh KC. A squared smoothing Newton method for semidefinite programming, 2023] to solve the KKT conditions, which contain one pair of complementarity constraints. We show that the smoothing Newton method has a superlinear convergence rate. Extensive numerical results verify the efficiency of the proposed approach and strict local minimizers can be achieved both numerically and theoretically. In particular, compared with other methods, our algorithm can achieve competitive results while consuming less time than other methods.

Neural Network Equalisation for High-Speed Eye-Safe Optical Wireless Communication with 850 nm SM-VCSELs

In this paper, we experimentally illustrate the effectiveness of neural networks (NNs) as non-linear equalisers for multilevel pulse amplitude modulation (PAM-M) transmission over an optical wireless communication (OWC) link. In our study, we compare the bit-error-rate (BER) performances of two decision feedback equalisers (DFEs)—a multilayer-perceptron-based DFE (MLPDFE), which is the NN equaliser, and a transversal DFE (TRDFE)—under two degrees of non-linear distortion using an eye-safe 850 nm single-mode vertical-cavity surface-emitting laser (SM-VCSEL). Our results consistently show that the MLPDFE delivers superior performance in comparison to the TRDFE, particularly in scenarios involving high non-linear distortion and PAM constellations with eight or more levels. At a forward error correction (FEC) threshold BER of 0.0038, we achieve bit rates of ~28 Gbps, ~29 Gbps, ~22.5 Gbps, and ~5 Gbps using PAM schemes with 2, 4, 8, and 16 levels, respectively, with the MLPDFE. Comparably, the TRDFE yields bit rates of ~28 Gbps and ~29 Gbps with PAM-2 and PAM-4, respectively. Higher PAM levels with the TRDFE result in BERs greater than 0.0038 for bit rates above 2 Gbps. These results highlight the effectiveness of the MLPDFE in optimising the performance of SM-VCSEL-based OWC systems across different modulation schemes and non-linear distortion levels.

Nonlinear equalization based on feature crosses neural networks for High-speed PAM4 transmission

In this paper, a low complexity feature crosses nonlinear equalization(FC-NLE) scheme using an additional feature crosses layer based on deep neural network is proposed. A 120 Gb/s quadratic pulse amplitude modulation (PAM4) transmission system based on intensity modulation and direct detection (IM/DD) is experimentally demonstrated. After a 5 km transmission, the FC-NLE scheme achieves a bit error rate (BER) below the hard-decision forward-error-correction threshold of 7% when the received optical power exceeds 4.5 dBm. This performance surpasses that of the Volterra equalizer(VE) and network-based equalization schemes. k-means clustering and model compression techniques were adopted to effectively diminish the complexity of the FC-NLE. Experimental results show that the performance of the FC-NLE is about 0.5–1.0 dB better than that of the VE, DNN, LSTM equalizers and its complexity is less than 50% of them. Compared with other low complexity network schemes in recent years, FC-NLE still exhibits a performance improvement of more than 1 dB at lower complexity.

FPGA implementation of power-lite Volterra-inspired neural network equalizer in 180-Gb/s net bitrate IMDD short-reach optical system.

A white-box power-lite Volterra-inspired neural network (VINN) equalizer is proposed to solve the problem of complexity discontinuity in a Volterra nonlinear equalizer (VNLE). By adjusting the granularity of the solution space, it conserves computational resources while maintaining nonlinear compensation capability. The performance of VINN is verified on a field-programmable gate array (FPGA) in a short-reach intensity modulation and direct detection (IMDD) system, and a 240-Gb/s real-time signal processing rate is achieved. Under the 25% overhead soft-decision forward error correction (SD-FEC) bit error rate (BER) threshold, we realize a record net rate of up to 180 Gb/s based on the FPGA.

High-Speed Dispersion-Unmanaged DML-Based IM-DD Optics at C-band With Advanced Nonlinear Equalization and Noise Whitening

High-Speed Dispersion-Unmanaged DML-Based IM-DD Optics at C-band With Advanced Nonlinear Equalization and Noise Whitening

Photonics-aided D-band 64-QAM MMW transmission utilizing modified multi-symbol output neural network equalization

We propose a modified multi-symbol output (MSO) neural network (NN)-based equalization scheme. In this scheme, the multi-label realization is optimized to support high-order 64-QAM MSO, and the performance is enhanced by adding residual connections, which split linear and nonlinear equalization in one model. The proposed scheme is utilized in a 70-m D-band photonics-aided millimeter wave (MMW) transmission system. We successfully realized 12-Gbaud 64-QAM transmission with Q-factor satisfying SD-FEC.

466.4-Gbit/s PCS-64QAM wireless transmission enhanced by dual-polarized SISO link and vector-quantized variational autoencoders

We successfully demonstrate photonics-assisted single-carrier 466.4 Gbit/s wireless transmission over 20 km SSMF and 6 m single-input single-output (SISO) wireless delivery at 92.5 GHz. This is achieved by employing a likelihood-aware vector-quantized variational autoencoder (VQ-VAE) equalizer for polarization demultiplexing and an advanced multiple-input multiple-output (MIMO) Volterra nonlinear equalizer (MIMO-VNE) further to compensate linear and nonlinear crosstalk between two polarizations. The capacity and robustness of the fiber-wireless system are extensively investigated, verifying that our proposed VQ-VAE with shared-layer autoencoder (SLAE) model outperforms state-of-the-art blind equalizers such as those based on constant modulus algorithm (CMA), VAE, and VQ-VAE algorithms. Additionally, the dual-polarized 40 Gbaud PCS-64QAM signal achieves a 2 dB power margin while meeting the 0.83 NGMI threshold, and a 0.05 gain in NGMI compared to traditional methods. This low-complexity training framework paves the way for enhancing wireless air interface capacities to approach those of fiber-optic transmission within fiber-wireless integration systems.

Photonic deep residual time-delay reservoir computing

Time-delay reservoir computing (TDRC) represents a simplified variant of recurrent neural networks, employing a nonlinear node with a feedback mechanism to construct virtual nodes. The capabilities of TDRC can be enhanced by transitioning to a deep architecture. In this work, we propose a novel photonic deep residual TDRC (DR-TDRC) with augmented capabilities. The additional time delay added to the residual structure enables DR-TDRC superior to traditional deep structures across various benchmark tasks, especially in memory capability and almost an order of magnitude improvement in nonlinear channel equalization. Additionally, a specifically designed clipping algorithm is utilized to counteract the damage of redundant layers in deep structures, enabling the extension of the deep TDRC to dozens rather than just a few layers, with higher performance. We experimentally demonstrate the proof-of-concept with a 4-layer DR-TDRC containing 960 interrelated neurons (240 neurons per layer), based on four injection-locked distributed feedback lasers. We confirm the potential for scalable deep RC with elevated performance. Our results provide a feasible approach for expanding deep photonic computing to satisfy the boosting demand for artificial intelligence.

Conformal structure-preserving SVM methods for the nonlinear Schrödinger equation with weakly linear damping term

In this paper, by applying the supplementary variable method (SVM), some high-order, conformal structure-preserving, linearized algorithms are developed for the damped nonlinear Schrödinger equation. We derive the well-determined SVM systems with the conformal properties and they are then equivalent to nonlinear equality constrained optimization problems for computation. The deduced optimization models are discretized by using the Gauss type Runge-Kutta method and the prediction-correction technique in time as well as the Fourier pseudo-spectral method in space. Numerical results and some comparisons between this method and other reported methods are given to favor the suggested method in the overall performance. It is worthwhile to emphasize that the numerical strategy in this work could be extended to other conservative or dissipative system for designing high-order structure-preserving algorithms.

Sub-rate sampled, non-integer fractionally spaced Volterra nonlinear equalizer for IM/DD systems.

As high-speed analog-to-digital converters (ADC) account for a significant portion of the receiver's cost in intensity-modulation (IM)/direct-detection (DD) systems, there have been substantial efforts to employ an ADC operating at a relatively low sampling rate. However, half-symbol-spaced electronic nonlinear equalizers used to compensate for nonlinear waveform distortions commonly operate at 2 samples/symbols, and thus require digital upsampling before the equalization. This implies that the digital signal processing (DSP) at the receiver should also operate at 2 sample/symbol even though the ADC operates at the sub-rate (i.e., < 2 sample/symbol). Hence, we propose and experimentally demonstrate the sub-rate sampled, non-integer fractionally spaced Volterra nonlinear equalizer (VNLE). This equalizer does not require the digital upsampling, and thus makes the entire receiver DSP block operating at the sub-rate, the same as the sampling rate of ADC. We estimate the complexity of the proposed equalizer by the number of multipliers required for its implementation. We also evaluate the performance of the proposed VNLE over a 64-Gb/s 4-ary pulse amplitude modulation link and compare the performance with the conventional VNLE (requiring digital upsampling). The results show that the proposed VNLE incurs a very slight performance degradation compared with the conventional VNLE, but reduces the implementation complexity considerably since it obviates the need for digital upsampling at the receiver.