Volterra Equalizer Research Articles

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.

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.

A Comprehensive Comparison of Simplified Volterra Equalization and Kramers–Kronig Schemes in 200 Gb/s/λ PON Downlink Transmission

The emerging high-bandwidth services of 6G, such as high-definition video transmission and real-time interaction, have promoted the progress of the fiber optic access network industry, driving its development towards the next-generation PON with higher speed and larger system capacity. In response to the future requirements of 200 Gb/s/λ PON for both 20 km and 30 km downlink transmission scenarios, this paper proposes a Simplified Volterra Equalization (SVLE) scheme based on Nyquist PAM4 single-sideband modulation direct detection (SSBM-DD) scheme. In order to verify its advantages, the IQ-modulated Kramers–Kronig reception (KK) scheme is introduced for comparison. Simulation validation platforms for two schemes are conducted, and the performance comparison of the SVLE and KK schemes is carried out. In both, the impact of the carrier signal power ratio (CSPR) on receiver sensitivity, the influence of input optical power on power budget and receiver sensitivity, and the tolerance of receiver sensitivity to the linewidth of the DFB laser are investigated in the simulation. Finally, a comprehensive comparison of the two schemes is presented in terms of system performance, cost, and DSP complexity. In the 20 km downlink transmission scenario, the SVLE scheme outperforms the KK scheme by 4.9 dB in terms of power budget. The total number of multiplications of the SVLE scheme is 37, while that of the KK scheme is 4358. Therefore, the DSP complexity of the SVLE scheme is much lower than that of the KK scheme. The results of the comparison demonstrate that, in downlink transmission scenarios, the SVLE scheme is more suitable than the KK scheme as it exhibits a higher power budget, lower DSP complexity, and lower cost. Consequently, the proposed SVLE scheme could be a highly promising solution for future ultra-high-speed PON downlink transmission.

End-to-end optimization of optical communication systems based on directly modulated lasers

The use of directly modulated lasers (DMLs) is attractive in low-power, cost-constrained short-reach optical links. However, their limited modulation bandwidth can induce waveform distortion, undermining their data throughput. Traditional distortion mitigation techniques have relied mainly on the separate training of transmitter-side pre-distortion and receiver-side equalization. This approach overlooks the potential gains obtained by simultaneous optimization of the transmitter (constellation and pulse shaping) and receiver (equalization and symbol demapping). Moreover, in the context of DML operation, the choice of laser-driving configuration parameters such as the bias current and peak-to-peak modulation current has a significant impact on system performance. We propose, to our knowledge, a novel end-to-end optimization approach for DML systems, incorporating the learning of bias and peak-to-peak modulation current to the optimization of constellation points, pulse shaping, and equalization. The simulation of the DML dynamics is based on the use of the laser rate equations at symbol rates between 15 and 25 Gbaud. The resulting output sequences from the rate equations are used to build a differentiable data-driven model, simplifying the calculation of gradients needed for end-to-end optimization. The proposed end-to-end approach is compared to three additional benchmark approaches: the uncompensated system without equalization, a receiver-side finite impulse response equalization approach, and an end-to-end approach with learnable pulse shape and nonlinear Volterra equalization but fixed bias and peak-to-peak modulation current. The numerical simulations on the four approaches show that the joint optimization of bias, peak-to-peak current, constellation points, pulse shaping, and equalization outperforms all other approaches throughout the tested symbol rates.

Decoding scheme based on CNN for differential free space optical communication system

A high-speed and robust decoding algorithm based on Convolutional Neural Networks (CNN) is proposed for differential free space optical communication system. The system adopts bipolar complementary pulse width modulation (BCPWM) that can enhance the amplitude of the signal and avoid interference from common mode signals in a single channel system. In addition, BCPWM has amplitude and envelope characteristics. The deep learning-based detector proposed by the signal does not require channel state information. It directly feeds the received signal into a deep neural network, and CNN can effectively extract amplitude and envelope features from the signal. The experimental comparison was conducted with bipolar non-return to zero (BNRZ) signals with only amplitude features to verify the demodulation ability of CNN for signals with different features. With a bit error rate of 10−5, the required signal-to-noise ratio for BNRZ is 13 dB, and the required signal-to-noise ratio for BCPWM is 8 dB. The demodulation performance of CNN, maximum likelihood (ML), and Volterra equalizer detector were compared. For BCPWM signals, compared to ML and Volterra equalizer detectors, the proposed CNN-based detector reduces the required signal-to-noise ratio by 2 dB and 8 dB at a bit error rate of 10−5, respectively.

DACNN-aided nonlinear equalizer for a probabilistic shaping coherent optical communication system.

The probabilistic shaping (PS) technique is a key technology for fiber optic communication systems to further approach the Shannon limit. To solve the problem that nonlinear equalizers are ineffective for probabilistic shaping optical communication systems with non-uniform distribution, a distribution alignment convolutional neural network (DACNN)-aided nonlinear equalizer is proposed. The approach calibrates the equalizer using the probabilistic shaping prior distribution, which reduces the training complexity and improves the performance of the equalizer simultaneously. Experimental results show nonlinear equalization of 120Gb/s PS 64QAM signals in a 375km transmission scenario. The proposed DACNN equalizer improves the receiver sensitivity by 2.6dB and 1.1dB over the Volterra equalizer and convolutional neural network (CNN) equalizer, respectively. Meanwhile, DACNN converges with fewer training epochs than CNN, which provides great potential for mitigating the nonlinear distortion of PS signals in fiber optic communication systems.

Low-complexity and low-latency equalization technique - probabilistic noise cancellation.

Both inside data centers (DCs) and in short optical links between data centers (DC campuses), intensity-modulation and direct-detection (IMDD) systems using four-level pulse amplitude modulation (PAM4) will dominate this decade due to low transceiver price and power consumption. The next DC transceiver generation based on 100 Gbaud PAM4 will require advanced digital signal processing (DSP) algorithms and more powerful forward error correction (FEC) codes. Because of bandwidth limitations, the conventional DC DSP based on a few-tap linear feed-forward equalizer (FFE) is likely to be upgraded to more complex but still low-complexity Volterra equalizers followed by a noise whitening filter and either a maximum likelihood sequence estimation (MLSE) or a maximum a posteriori probability (MAP) algorithm. However, stringent power consumption and latency requirements may limit the use of complex algorithms such as decision feedback equalizer (DFE) or MLSE/MAP in DC networks (DCN). In this paper, we introduce a low-complexity, low-latency algorithm based on a feedforward structure, yielding a performance between DFE and MLSE. We call the novel equalization algorithm probabilistic noise cancellation (PNC), since it weights noise patterns based on their probabilities in the presence of bandwidth limitations. The probabilistic weighting is efficiently exploited in correcting correlated errors caused by noise coloring in the FFE.

OAM mode-division multiplexing IM/DD transmission at 4.32 Tbit/s with a low-complexity adaptive-network-based fuzzy inference system nonlinear equalizer.

Stochastic nonlinear impairment is the primary factor that limits the transmission performance of high-speed orbital angular momentum (OAM) mode-division multiplexing (MDM) optical fiber communication systems. This Letter presents a low-complexity adaptive-network-based fuzzy inference system (LANFIS) nonlinear equalizer for OAM-MDM intensity-modulation direct-detection (IM/DD) transmission with three OAM modes and 15 wavelength division multiplex (WDM) channels. The LANFIS equalizer could adjust the probability distribution functions (PDFs) of the distorted pulse amplitude modulation (PAM) symbols to fit the statistical characteristics of the WDM-OAM-MDM transmission channel. Therefore, although the transmission symbols in the WDM-OAM-MDM system are subjected to a stochastic nonlinear impairment, the proposed LANFIS equalizer can effectively compensate the distorted signals. The proposed equalizer outperforms the Volterra equalizer with improvements in receiver sensitivity of 2, 1.5, and 1.3 dB for three OAM modes at a wavelength of 1550.12 nm, respectively. It also outperforms a CNN equalizer, with improvements in receiver sensitivity of 1, 0.5, and 0.3 dB, respectively. Moreover, complexity reductions of 67%, 74%, and 99.9% are achieved for the LANFIS equalizer compared with the Volterra, CNN, and ANFIS equalizers, respectively. The proposed equalizer has high performance and low complexity, making it a promising candidate for a high-speed WDM-OAM-MDM system.

Multi-Tb/s Transmission Based on DMT-Modulated VCSELs in Metro Networks Assisted by Volterra Equalization

Multi-Tb/s Transmission Based on DMT-Modulated VCSELs in Metro Networks Assisted by Volterra Equalization

Performance and Computational Efficiency of LMS Adaptive Volterra Equalizers for Nonlinear TWTA Distortion in Satellite Communications

Performance and Computational Efficiency of LMS Adaptive Volterra Equalizers for Nonlinear TWTA Distortion in Satellite Communications

A Simplified Volterra Equalizer Based on System Characteristics for Direct Modulation Laser (DML)-Based Intensity Modulation and Direct Detection (IM/DD) Transmission Systems

The nonlinear Volterra equalizer has been proved to be able to solve the problem of nonlinear distortion, but it has high computational complexity and is difficult to implement. In this paper, a simplified second-order Volterra nonlinear equalizer designed for intensity modulation/direct detection systems based on direct modulated laser is proposed and demonstrated, taking into account the characteristics of the system. It has been proved that the received signal of direct modulation laser/direct detection system can be expressed in Volterra series form, but its form is too complex, and the device parameters should also be considered. We re-derived it and obtained a more concise form. At the same time, we proposed a method to simplify the second-order Volterra nonlinear equalizer without relying on device parameters. The performance of the proposed Volterra nonlinear equalizer is evaluated experimentally on a 56 Gb/s 4-ary pulse amplitude modulation link implemented by using a 1.55 µm direct modulation laser. The results show that, compared with the traditional Volterra nonlinear equalizer, the receiver sensitivity of the equalizer is only reduced by 0.2 dB at most, but the complexity can be reduced by 50%; compared with diagonally pruned Volterra nonlinear equalizers, the complexity of the equalizer is the same, but the reception sensitivity can be improved by 0.5 dB.

Kramers–Kronig Transmission with a Crosstalk-Dependent Step Multiple-Input Multiple-Output Volterra Equalizer in a Seven-Core Fiber

In this paper, we experimentally demonstrate a net bit rate of 261.7 Gbit/s in a seven-core transmission system with a Kramers–Kronig (KK) receiver. The 10 GBaud 16-level quadrature amplitude modulation (QAM) signal is transmitted over a 2.5 km seven-core fiber, and the relationship between carrier-to-signal power ratio, signal power, frequency spacing, and optical power is analyzed. Moreover, a multiple-input multiple-output (MIMO) Volterra equalization algorithm with crosstalk-dependent steps is proposed to compensate for inter-core crosstalk and impairments induced by other devices. Compared to the single-input single-output (SISO) Volterra equalizer, the CSPR can be reduced by 1.3 dB, and the received power gain can reach up to 0.7 dB.

400 Gbit/s 4 mode transmission for IM/DD OAM mode division multiplexing optical fiber communication with a few-shot learning-based AffinityNet nonlinear equalizer.

Nonlinear impairment in a high-speed orbital angular momentum (OAM) mode-division multiplexing (MDM) optical fiber communication system presents high complexity and strong stochasticity due to the massive optoelectronic devices. In this paper, we propose an Affinity Network (AffinityNet) nonlinear equalizer for an OAM-MDM intensity-modulation direct-detection (IM/DD) transmission with four OAM modes. The labeled training and testing signals from the OAM-MDM system can be regarded as "small sample" and "large target", respectively. AffinityNet can be used to build an accurate nonlinear model using "small sample" based on few-shot learning and can predict the stochastic characteristic nonlinearity of OAM-MDM with a high level of generalization. As a result, the AffinityNet nonlinear equalizer can effectively compensate the stochastic nonlinearity in the OAM-MDM system, despite the large difference between the training and testing signals due to the stochastic nonlinear impairment. An experiment was conducted on a 400 Gbit/s transmission with four OAM modes using a pulse amplitude modulation-8 (PAM-8) signal over a 2 km ring-core fiber (RCF). Our experimental results show that the proposed nonlinear equalizer outperformed the conventional Volterra equalizer with improvements in receiver sensitivity of 1.7, 1.8, 3, and 3.3 dB for the four OAM modes at the 15% forward error correction (FEC) threshold, respectively. In addition, the proposed equalizer outperformed a convolutional neural network (CNN) equalizer with improvements in receiver sensitivity of 0.8, 0.5, 0.9, and 1.4 dB for the four OAM modes at the 15% FEC threshold. In the experiment, a complexity reduction of 37% and 83% of the AffinityNet equalizer is taken compared to the conventional Volterra equalizer and CNN equalizer, respectively. The proposed equalizer is a promising candidate for a high-speed OAM-MDM optical fiber communication system.

Hardware-Efficient Duobinary Neural Network Equalizers for 800 Gb/s IM/DD PAM4 Transmission Over 10 km SSMF

In this article, we discuss challenges and options for scaling IM/DD transceivers towards 800Gbps. Our focus is CWDM4 PAM4 transmission and our target distance is 10km in O-band, which is a most urgent use case for next generation optical short reach systems like data centre interconnects and networks. <p xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">At this reach and rate, chromatic dispersion (CD) becomes the main challenge. Its mitigation is essential and primarily done with digital signal processing. State of the art techniques, however, make transceivers quickly too complex. <p xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">We show upon measurement results how neural network equalization can meet Volterra equalization performance with 30% less hardware multiplier complexity. When also applying magnitude weight pruning, an additional 43% reduction is possible without performance loss across all CWDM4 lanes. If needed, an added MLSE stage can further push performance in both cases. <p xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">In any of these configurations, a key enabler against strong CD penalties is duobinary training, which is applicable to all feed-forward equalization architectures.

Echo State Network Based Nonlinear Equalization for 4.6 km 135 GHz D-Band Wireless Transmission

Reservoir computing (RC) is a novel computational framework derived from recurrent neural networks (RNN). It can reduce the training complexity of RNN and is suitable for time-series learning tasks. The echo state network (ESN) is one of the most popular forms of RC. In this paper, an ESN-based equalizer is applied to perform signal equalization in a wireless D-band communication system to compensate for the nonlinear distortion. Based on the photonics-based technology and multiple amplifiers, a long-range wireless transmission system is successfully established at D-band. We experimentally demonstrated that our proposed wireless transmission link can realize up to 4.6-km wireless delivery of over 8 Gbit/s quadrature phase shift keying (QPSK) millimeter-wave signal at 135 GHz with a bit-error-rate (BER) less than the hard decision forward error correction (HD-FEC) threshold of 3.8 × 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−3</sup> . Compared with the traditional CMA equalizer and the Volterra equalizer, the experimental results show that the ESN-based equalizer can achieve a better balance between the BER performance and the computational complexity.

C-band 100-GBaud/λ PAM-4 transmission enabled by hardware-efficient nonlinear equalizer with weight-sharing pruning.

A Volterra nonlinear equalizer (VNLE) can effectively mitigate both linear and nonlinear impairments arising in intensity-modulation direct-detection (IM-DD) transmission systems. However, the high computational complexity of the VNLE hinders its applications. Here, we propose a hardware-efficient Volterra equalizer with weight-sharing pruning (VE-WSP). Such an equalizer first uses the k-means++ clustering algorithm for the weight sharing within the same cluster, and then ranks the cluster centroid weight for pruning, leading to a significant computational complexity reduction without sacrificing any equalization performance. We experimentally verified that the use of VE-WSP can enable the C-band 100-GBaud/λ PAM-4 transmission over a 1-km standard single-mode fiber (SSMF), to reach the 20% soft-decision forward error correction (SD-FEC) threshold. Meanwhile, the proposed VE-WSP can reduce the computational complexity by 51% and 21%, in terms of multiplication and addition, respectively, in comparison with the VNLE.

Sparse Volterra Equalizer to Mitigate the Nonlinear Distortion in High Power Budget IM-DD OFDM PON

To improve the power budget for intensity-modulation direct-detection (IM-DD) orthogonal frequency division multiplexing passive optical network (OFDM PON), a high power budget downstream scheme by using the Volterra equalizer in digital signal processing (DSP) processing is proposed in this paper. When the launch power is increased, the subcarrier-to-subcarrier beating interference (SSBI) and other nonlinear distortion are well mitigated by the Volterra equalizer. Moreover, to reduce the complexity of receiver DSP unit, the sparse Volterra equalizer is obtained through optimizing full Volterra equalizer employing orthogonal search method. Simulation and experiment results are presented to evaluate the transmission performance and analyze the distribution of taps of Volterra equalizer under different respects. The results show that the number of tap coefficient of sparse Volterra equalizer is reduced by 60% compared with that of conventional second-order full Volterra equalizer.

Performance and complexity study of a neural network post-equalizer in a 638-nm laser transmission system through over 100-m plastic optical fiber

In recent years, plastic optical fiber (POF) has been considered as a promising cost-effective scheme for short-distance data communications, multimedia communication in cars, and in-house networks. However, due to the intrinsic nature of the relatively large numerical aperture of POF and the high attenuation rate, implementing high data rates over 100-m POF transmission length will be a significant challenge. We propose a scheme of high-speed 100-m POF transmission system based on a visible red laser and a cascaded neural network (NN) post-equalizer. To mitigate the nonlinear distortion induced by the POF, three different NNs, i.e., convolutional NN (CNN), long and short-term memory NN (LSTM), and cascaded NN structure consisting of convolutional layers and LSTM (CNN-LSTM), are employed as the post-equalizer. Experimental results show that using three different post-equalizers can significantly improve the system performance compared with the Volterra equalizer baseline. Among them, CNN-LSTM can outperform the others in terms of the bit error rate (BER) and the system Q-factor in the low nonlinear region. When the system operating in strong nonlinear region, CNN can achieve optimal performance at a lower system overhead of complexity. Finally, we successfully demonstrated a 100-m POF transmission system using 16 quadrature amplitude modulation discrete Fourier transform-spread orthogonal frequency division multiplexing modulation format at 1.8 Gbps with BER below 3.8 × 10 − 3 by utilizing CNN-LSTM.

Probabilistically Shaped PAM-8 Transmission With Simplified Volterra Equalizer

Intensity-modulation direct-detection (IM/DD) probabilistically shaped (PS) 8-level pulse-amplitude modulation (PAM) signal is promising for high-speed intra-data-center networks. However, the challenges of a PS-PAM-8 transmission system are complicated distribution matcher (DM) and nonlinear equalizer. In this work, we propose a simple PS-PAM-8 transmission system, where the probabilistic amplitude shaping (PAS) encoder consists of an energy-level-assigned (ELA)-based DM and a low-density parity-check (LDPC) encoder. In the ELA-based DM, a simple lookup table is designed to realize an approximate bilateral Maxwell-Boltzmann distribution by assigning energy levels to various symbols. Consequently, the DM encoded bits are used as the amplitude bits, and DM label bits and LDPC check bits are designated as sign bits to implement PAS encoding. In addition, a simplified Volterra equalizer (SVE) is introduced to eliminate the inter-symbol interference and improve the receiver sensitivity of PS-PAM-8 signal, where the SVE only includes the 2-order kernel to reduce its complexity. We experimentally demonstrated the proposed PS-PAM-8 scheme and verified its outperformance over uniform PAM-8 signal in the back-to-back and 2-km standard single mode fiber (SSMF) transmission. Compared to the conventional linear equalizer, the SVE contributes approximate 1-dB improvement of the receiver sensitivity in the PS-PAM-8 2-km SSMF transmission, for the error-free performance of post forward error correction bit error ratio.

LED communications with linear complexity compensation of dynamic nonlinear distortion.

An in-depth study of the second order Volterra kernel of light emitting diodes (LEDs) is provided, with a special focus on compensation of dynamic nonlinear distortion generated by the LED when used as a communications signal transmitter. It is shown that from the factorization of the kernel in the frequency domain follows a simplified time-domain model for LED nonlinearity, consisting of three operations: two linear filtrations and squaring. Next, using the pth inverse theory, the corresponding block structures of nonlinear pre- and postdistorters are derived. As it turns out, they exhibit linear computational complexity. It is shown that the model coefficients defining the kernel may be estimated up to the Nyquist frequency. Numerical results indicate that pre- and postdistortion achieve the same performance with respect to receiver signal to noise power ratio (SNR) and have a considerable advantage over linear equalizers. However, in a practical scenario, the predistortion increases the peak to average power ratio of the transmitted signal (unless it is not high already), possibly leading to performance inferior to postdistortion. Finally, it is shown that the simplified equalizer based on the LED model has roughly identical performance to the regular, quadratic Volterra equalizer.