Volterra Equalizer Research Articles

Reducing computation complexity by using elastic net regularization based pruned Volterra equalization in a 80 Gbps PAM-4 signal for inter-data center interconnects.

Volterra equalization (VE) presents substantial performance enhancement for high-speed optical signals but suffers from high computation complexity which limits its physical implementations. To address these limitations, we propose and experimentally demonstrate an elastic net regularization-based pruned Volterra equalization (ENPVE) to reduce the computation complexity while still maintain system performance. Our proposed scheme prunes redundant weight coefficients with a three-phase configuration. Firstly, we pre-train the VE with an adaptive EN-regularizer to identify significant weights. Next, we prune the insignificant weights away. Finally, we retrain the equalizer by fine-tuning the remaining weight coefficients. Our proposed ENPVE achieves superior performance with reduced computation complexity. Compared with conventional VE and L1 regularization-based Volterra equalizer (L1VE), our approach show a complexity reduction of 97.4% and 20.2%, respectively, for an O-band 80-Gbps PAM4 signal at a received optical power of -4 dBm after 40 km SMF transmission.

Time skew enabled vestigial sideband modulation for dispersion-tolerant direct-detection transmission.

Fiber dispersion and square-law detection-induced power fading is a fundamental obstacle in intensity modulation with direct detection links. In this Letter, we propose a hardware-efficient vestigial sideband (VSB) transmitter to suppress such an impairment. By introducing an appropriate time skew between the differential arms of the Mach-Zehnder modulator, a VSB signal can be generated based on a single digital-to-analog convertor without optical filtering. At the receiver, a Volterra equalizer is utilized to mitigate the inter-symbol interference. In the proof-of-concept experiment, 32 Gbaud 4/6/8-ary pulse amplitude modulation signals can be successfully transmitted over an 80 km standard single-mode fiber with bit-error rates below the KP4, 7% hard-decision and 20% soft-decision forward error correction thresholds, respectively. The proposed scheme provides a promising and low-cost solution for high-speed metro and data center interconnect applications.

High-speed PAM4 transmission with a GeSi electro-absorption modulator and dual-path neural-network-based equalization.

Equalization based on artificial neural networks (NN) has proved to be an effective way for nonlinearity mitigation in various kinds of optical communication systems. In this Letter, we propose a novel methodology of dual-path neural network (DP-NN)-based equalization. By combining a linear equalizer with an input-pruned NN equalizer, DP-NN can effectively reduce the computation cost compared to a conventional NN equalizer. We confirm its feasibility through 4-ary pulse amplitude modulation (PAM4) transmission at a gross(net) bitrate of 160 Gb/s (133.3 Gb/s), based on a GeSi electro-absorption modulator operating at C-band. After a 2 km transmission, the bit error rate is below the 20% hard-decision forward-error-correction threshold of 1.5×10-2 with the DP-NN equalization, which outperforms the Volterra equalization and is comparable to conventional NN-based equalization.

Blind nonlinearity mitigation of 10G DMLs using sparse Volterra equalizer in IM/DD PAM-4 transmission systems

Abstract The nonlinear characteristics of the electro-optic components are the major limiting factor to realize a multi-level signal transmission. Cascaded multi-modulus algorithm (CMMA) is a popular blind equalization scheme that can compensate for only linear distortions without using any training sequence. In this paper, a modified CMMA is proposed to blindly adapt the tap weights of a receiver-side nonlinear Volterra equalizer (NLVE) to mitigate both linear and nonlinear distortions. Furthermore, a nonlinear sparse Volterra equalizer (NLSVE) is suggested in this paper that can achieve similar performance with a substantial reduction of complexity, in comparison with an NLVE. Experimental results for a 25 Gb/s PAM-4 transmission in the 1310-nm band over a 20-km single-mode fiber (SMF-28e) using a commercial 10 GHz-class directly modulated laser (DML) for an intensity modulation direct-detection (IM/DD) optical access network verify that the proposed NLSVE can achieve a gain of ~ 2-dB in comparison with a traditional CMMA.

Performance assessment of dual-polarized 5G waveforms and beyond in directly modulated DFB-laser using Volterra equalizer

We investigate the performance of 25-Gbps dual-polarized orthogonal frequency division multiplexing (OFDM)-based modulation in a directly modulated distributed feedback (DFB)-laser over 25 km of single-mode fiber. A Volterra equalizer is used to compensate for the nonlinear effects of the optical fiber. The results show that FBMC-OQAM modulation outperforms OFDM, universal filtered multicarrier (UFMC), and generalized frequency division multiplexing (GFDM) waveforms. Indeed, a target bit error rate of similar to 3.8 x 10(-3) [forward error correction (FEC) limit] for FBMC, UFMC, OFDM, and GFDM can be achieved at -30.5, -26, -16, and -14.9 dBm, respectively. The effect of the DFB laser is also investigated for UFMC, OFDM, and GFDM, and they undergo a Q penalty of 2.44, 2.77, and 4.14 dB, respectively, at their FEC limit points. For FBMC-OQAM, the signal is perfectly recovered when excluding the DFB laser at -30.5 dBm. (C) 2020 Society of Photo-Optical Instrumentation Engineers (SPIE)

Computationally efficient 104 Gb/s PWL-Volterra equalized 2D-TCM-PAM8 in dispersion unmanaged DML-DD system.

Two-dimensional eight-level pulse amplitude modulation with trellis-coded modulation (2D-TCM-PAM8) is proposed to overcome the bandwidth limitation for high-speed signal transmission due to its high spectral efficiency. However, the high coding gain of the TCM can only be achieved in bandlimited additive white Gaussian noise (AWGN) channels and cannot be achieved in nonlinear channels without any equalizers. In the directly modulated laser and direct detection (DML-DD) transmission system, the transceiver nonlinearities and the interaction between DML chirp and fiber dispersion will introduce nonlinear distortion. To compensate for the nonlinear distortion, we propose a computationally efficient piecewise linear (PWL)-Volterra equalizer. In this equalizer, we first use the PWL to correct the skewed eye diagram and then employ a simple 2nd order Volterra to compensate for the residual nonlinear distortions. By using the PWL-Volterra equalizer prior to the Viterbi decoder, the high coding gain of TCM can be achieved. In the experiment, a 104 Gb/s 8-state 2D-TCM-PAM8 signal generated in a ∼ 20 GHz DML is successfully transmitted over 10 km standard single-mode fiber (SSMF) in C band, with the bit error ratio (BER) below the HD-FEC limit of 3.8 × 10-3. Compared to only using the conventional 2nd order Volterra equalizer with a similar BER performance, the PWL-Volterra equalizer shows 29% computational complexity reduction.

Piecewise Linear Equalizer for DML Based PAM-4 Signal Transmission Over a Dispersion Uncompensated Link

Directly modulated laser (DML) and direct detection (DD) based pulse-amplitude modulation (PAM) for short reach optical communications has attracted lots of research interests due to its low cost and simple configuration. However, the directly modulated PAM signal usually suffers from distortions after the transmission over a dispersion uncompensated link. Due to the adiabatic chirp of the DML working in a high output power region, different intensity levels of the PAM signal correspond to different frequency offsets, thus resulting in amplitude-dependent skew after the transmission over a dispersion uncompensated link. It will degrade signal quality and limit transmission distance. In this article, we, for the first time, propose a computational efficient piecewise linear (PWL) equalizer to correct the amplitude-dependent skew due to the interaction between the DML chirp and chromatic dispersion in the fiber. By using the PWL equalizer, the amplitude-dependent skew is corrected, and we successfully transmit 56 Gb/s PAM-4 signal over 40 km and 84 Gb/s PAM-4 signal over 20 km dispersion uncompensated link in the C band, with a bit error ratio (BER) below the HD-FEC threshold (3.8 × 10− 3). To demonstrate the low computational complexity of the PWL, we compare its complexity with 2nd order Volterra equalizer for a similar BER performance. The results show that the computational complexity can be reduced by 61.4% and 40.4% for the 56 Gb/s and 84 Gb/s PAM-4 signal transmission, respectively.

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.

Secure 100 Gb/s IMDD Transmission Over 100 km SSMF Enabled by Quantum Noise Stream Cipher and Sparse RLS-Volterra Equalizer

In this paper, we have experimentally demonstrated a secure 100 Gb/s intensity-modulation and direct-detection transmission over 100 km standard single-mode fiber (SSMF) based on quantum noise stream cipher (QNSC) for the first time. The original 4-level pulse-amplitude modulation (PAM4) data is mapped to M-level data sequence randomly, and quantum noise such as the generated amplified spontaneous emission noise and the shot noise in the channel can mask these adjacent signal levels. The legitimate receiver needs to distinguish not M-level encrypted data but 4-level data with the shared key. In our scheme, a calculated detection failure probability of 98.72% is achieved. Apparently, more quantum noise contributes to higher security but the worse signal to noise ratio. To balance algorithm complexity, transmission performance and security performance, a sparse $\ell _{1} $ regularization based on recursive least square (RLS) algorithm is proposed and firstly used in Volterra equalizer. To eliminate the power fading induced by fiber dispersion, a dual-drive Mach-Zehnder modulator is used to achieve single-sideband modulation, and no dispersion compensation is required. By these means, 100 Gb/s PAM4-QNSC signal transmission over 100 km SSMF with the bit error rate (BER) below the 7% overhead hard-decision forward error correction threshold of $3.8 \times 10^{-3}$ is achieved, and > 59% complexity reduction of Volterra equalizer is realized. Moreover, the measured BER of 150 Gb/s PAM8-QNSC signal transmission over 50 km SSMF could go below the 20% overhead soft-decision forward error correction threshold of $2.0\times 10^{-2}$ . The results validate that the proposed scheme is effective to realize low-cost, high-speed (> 100 Gb/s), and secure optical fiber transmission in the data center.

Optimized self-interference cancellation based on optical dual-parallel MZM for co-frequency and co-time full duplex wireless communication under nonlinear distortion and emulated multipath effect.

Co-frequency and co-time full duplex (CCFD) technique has the potential to further improve the capacity of the fifth-generation (5G) system. However, in a CCFD-based transceiver, the self-interference-cancellation (SIC) module is essential because the transmitted signal is also captured by the receiving antenna. In this paper, an optimized SIC technique is proposed based on a dual-parallel Mach-Zehnder modulator (DP-MZM). In addition to the multipath effect between the transmitting and receiving antennas, the nonlinear distortion induced by the used optical and electrical components is considered and analyzed for the first time. In this scheme, one of the children MZMs is biased at 90°. The bias points of the other child MZM and the parent MZM are swept for SIC optimization. Compared to the traditional SIC scheme using DP-MZM, the power of the received signal is increased by 6 dB and the robustness to the bias point drift is improved. In our experiment, over 35 dB cancellation ratio is achieved for 200 MHz filtered orthogonal frequency division multiplexing (F-OFDM) interference signal with carrier frequency from 2.5 GHz to 6 GHz. To mitigate the influence of multipath effect and nonlinear distortion, the recursive least squared (RLS) linear equalizer and the RLS Volterra equalizer are designed and compared. Finally, 42 dB cancellation ratio is achieved for 200 MHz F-OFDM signal with carrier frequency of 4.5 GHz in the presence of nonlinear distortion and emulated multipath effect.

Low-Complexity Second-Order Volterra Equalizer for DML-Based IM/DD Transmission System

Volterra nonlinear equalizers (VNLEs) are known to be effective in compensating for nonlinear distortions. However, the major drawback of these equalizers is the huge implementation and computation complexity. In this article, we propose and demonstrate low-complexity quadratic VNLEs specifically designed for directly modulated laser (DML)-based intensity-modulation/direct-detection (DD) systems. We first derive an analytical expression of the quadratic Volterra model for DML/DD systems and then identify the nonlinear distortion terms mainly contributing to the system performance. We show that the interplay between the DML's adiabatic frequency chirp and fiber chromatic dispersion produces the quadratic beating between the signal and the sum of time-sampled signals, which, in turn, makes the coefficients of some beating terms in Volterra series similar. Based on these findings, we reduce the number of coefficients to be determined in the quadratic VNLEs by grouping those beating terms. The performance of the proposed VNLEs is evaluated experimentally on a 56-Gb/s 4-ary pulse amplitude modulation link implemented by using a 1.55-μm DML. We measure the bit-error ratio performance of various equalizers, including feedforward equalizer, polynomial nonlinear equalizer, diagonally-pruned (DP) VNLE, and proposed quadratic VNLE, and compare them in terms of receiver sensitivity and implementation complexity. The results show that the proposed VNLE outperforms the DP-VNLE, although the implementation complexity of the two equalizers is similar. Also presented in this article is an algorithm for optimizing the parameter required in the proposed VNLE.

A SiGe HBT BiCMOS 1-to-4 ADC Frontend Enabling Low Bandwidth Digitization of 100 GBaud PAM4 Data

Market forecasts for intra and inter data center interconnects predict high growth rates in the upcoming years. This is associated with increasing requirements in terms of costs, power consumption, and bit rate per wavelength. Today, PAM4 systems are frequently used at up to 100 Gb/s per wavelength. The next logical step is to double the symbol rate to allow bit rates of 200 Gb/s at the same modulation format and at the same direct detection scheme. Such systems require three-digit gigasample analog-to-digital converters with bandwidths larger than 50 GHz, which are not available today as CMOS components. To extend the bandwidth of CMOS converters, we suggest the use of ADC frontends, which sample and demultiplex the analog input signals simultaneously. At the outputs of such ADC frontends, the signals can then be digitized by parallel converters with lower bandwidth and at a lower conversion rate. In this article, we present the circuit implementation of a 1-to-4 ADC frontend in IHP 130 nm SiGe HBT BiCMOS and demonstrate its integration into a 100 GBaud PAM4 transmission system with only 14 GHz of digitizer bandwidth. For optimum performance, the mitigation of intersymbol and interchannel interferences is mandatory in the transmission system. We describe and compare two different receiver DSP architectures for reduction of these interferences and evaluate their performances in electrical and optical back-to-back experiments. Furthermore, we report on transmission experiments over 500 m, 10 km, and 40 km SMF at 1550 nm wavelength. With linear equalization, we obtained a BER below the HD-FEC threshold of 3.8 × 10−3 and with 3 rd order Volterra equalization, we even achieved a BER below the KP4 threshold of 2.3 × 10 −4 .

Deep Neural Network Equalization for Optical Short Reach Communication

Nonlinear distortion has always been a challenge for optical communication due to the nonlinear transfer characteristics of the fiber itself. The next frontier for optical communication is a second type of nonlinearities, which results from optical and electrical components. They become the dominant nonlinearity for shorter reaches. The highest data rates cannot be achieved without effective compensation. A classical countermeasure is receiver-side equalization of nonlinear impairments and memory effects using Volterra series. However, such Volterra equalizers are architecturally complex and their parametrization can be numerical unstable. This contribution proposes an alternative nonlinear equalizer architecture based on machine learning. Its performance is evaluated experimentally on coherent 88 Gbaud dual polarization 16QAM 600 Gb/s back-to-back measurements. The proposed equalizers outperform Volterra and memory polynomial Volterra equalizers up to 6th orders at a target bit-error rate (BER) of 10 − 2 by 0.5 dB and 0.8 dB in optical signal-to-noise ratio (OSNR), respectively.

Implementation of an optical GFDM system based on graded-index fiber

For the first time, in this paper, generalized frequency division multiplexing (GFDM) waveform, through multimode fiber (MMF), is proposed and demonstrated for 5G communications at a data rate of 10 Gb/s. Based on the offset launch technique, the MMF fiber is simulated and evaluated for different offsets. Thanks to the GFDM technique, we reduce the out of band (OOB) radiation by 5 dB compared to orthogonal frequency division multiplexing (OFDM) and we reach even lower values of bit error rate (BER). In addition, we prove that increasing the taps of the Volterra equalizer enables us to obtain acceptable performances of BER, with different values of offsets.

Mitigation of Nonlinear Impairments by Using Support Vector Machine and Nonlinear Volterra Equalizer

A support vector machine (SVM) based detection is applied to different equalization schemes for a data center interconnect link using coherent 64 GBd 64-QAM over 100 km standard single mode fiber (SSMF). Without any prior knowledge or heuristic assumptions, the SVM is able to learn and capture the transmission characteristics from only a short training data set. We show that, with the use of suitable kernel functions, the SVM can create nonlinear decision thresholds and reduce the errors caused by nonlinear phase noise (NLPN), laser phase noise, I/Q imbalances and so forth. In order to apply the SVM to 64-QAM we introduce a binary coding SVM, which provides a binary multiclass classification with reduced complexity. We investigate the performance of this SVM and show how it can improve the bit-error rate (BER) of the entire system. After 100 km the fiber-induced nonlinear penalty is reduced by 2 dB at a BER of 3.7 × 10 − 3 . Furthermore, we apply a nonlinear Volterra equalizer (NLVE), which is based on the nonlinear Volterra theory, as another method for mitigating nonlinear effects. The combination of SVM and NLVE reduces the large computational complexity of the NLVE and allows more accurate compensation of nonlinear transmission impairments.

Performance analysis of UFMC waveform in graded index fiber for 5G communications and beyond

For the first time, we demonstrate in this paper an intensity-modulation direct-detection (IMDD) universal filtered multi-carrier (UFMC) system through several hundred meters of multimode fiber (MMF) that can be adopted for 5G communications and beyond. The proposed system is simulated and evaluated in terms of bit error rate (BER), Q factor, out of band (OOB) emission and Peak-to-Average Power Ratio (PAPR), at a data rate of 10 Gb/s. It is shown that filtering used with UFMC reduces the OOB radiation by 40 dB compared to orthogonal frequency division multiplexing (OFDM). Moreover, we prove that UFMC gives lower values of BER than OFDM, and even if the PAPR ratio of UFMC is higher than OFDM, the gap remains small. The results also show that using a Volterra equalizer with increasing taps enables to obtain acceptable performance through the MMF in terms of BER, even with high number of excited mode groups.

Numerical Investigation of Long-Haul Coherent Optical Generalized Frequency Division Multiplexing Signal

Generalized frequency division multiplexing (GFDM) is a recent physical layer scheme proposed for next-generation cellular systems. Like orthogonal frequency division multiplexing (OFDM), it is based on multicarrier modulation, but the filter shape is no longer rectangular (roll off ≠ 0) and the orthogonality between subcarriers is lost. With higher flexibility, low peak to average power ratio (PAPR) and reduced out-of-band emissions compared with OFDM, GFDM modulation is an attractive candidate for 5G application scenarios. This paper explores the performance of 10 Gb/s GFDM system over the optical link in terms of PAPR, quality factor ( Q factor), and mean squared error using a third-order adaptive Volterra filter with 3 and 5 taps. The impact of linear impairments (chromatic dispersion, polarization mode dispersion), laser line width, and Mach–Zehnder non-linearity have been addressed. The results of our simulations indicate that the transmitted 10 Gb/s GFDM signal through an optical fiber can reach 930 Km and Q = 11.9 dB after 700 Km at –8 dBm of launch power using 5 taps third-order adaptive Volterra equalizer with an improvement of approximately 210 Km and 1.25 dB compared with the same equalizer with 3 taps.

Threshold-Based Pruned Retraining Volterra Equalization for 100 Gbps/Lane and 100-m Optical Interconnects Based on VCSEL and MMF

The high-speed multimode optical interconnects based on vertical-cavity surface-emitting laser and multimode fiber (MMF) are severely constrained by band-limited devices, modulation nonlinearity, and laser induced noises such as relative intensity noise, mode partition noise, and fiber-induced noises such as chromatic dispersion and mode dispersion. The advanced digital signal processing technologies have been viewed as the inevitable tool to mitigate the physical limitations of electrical and optical components and the noise interactions during the transmission. In order to achieve multimode 100 Gbps/lane over 100-m transmission with the existing devices, a Volterra equalizer has been proven as a very effective equalization solution to deal with nonlinear impairments. However, the practical implementation of this nonlinear equalizer has been constrained by its tremendously high computation complexity. In this paper, we propose the threshold-based pruned retraining Volterra equalization (TRVE) to reduce the computation complexity, while maintaining the transmission performance. Multimode 100-Gbps/lane transmission over 100-m OM3 MMF has been experimentally demonstrated with up to 70.7% computation complexity reduction, while keeping the bit error rate under the 7% hard-decision forward error correction limit. In addition, the TRVE with $\ell _1$ -regularization is also proposed and experimentally verified to optimize the threshold selection.

End-to-end optimized transmission over dispersive intensity-modulated channels using bidirectional recurrent neural networks.

We propose an autoencoding sequence-based transceiver for communication over dispersive channels with intensity modulation and direct detection (IM/DD), designed as a bidirectional deep recurrent neural network (BRNN). The receiver uses a sliding window technique to allow for efficient data stream estimation. We find that this sliding window BRNN (SBRNN), based on end-to-end deep learning of the communication system, achieves a significant bit-error-rate reduction at all examined distances in comparison to previous block-based autoencoders implemented as feed-forward neural networks (FFNNs), leading to an increase of the transmission distance. We also compare the end-to-end SBRNN with a state-of-the-art IM/DD solution based on two level pulse amplitude modulation with an FFNN receiver, simultaneously processing multiple received symbols and approximating nonlinear Volterra equalization. Our results show that the SBRNN outperforms such systems at both 42 and 84 Gb/s, while training fewer parameters. Our novel SBRNN design aims at tailoring the end-to-end deep learning-based systems for communication over nonlinear channels with memory, such as the optical IM/DD fiber channel.

Amplifier-free 4×96 Gb/s PAM8 transmission enabled by modified Volterra equalizer for short-reach applications using directly modulated lasers.

By using four O-band directly modulated lasers (DMLs), for the first time a 384 Gb/s (4 × 96 Gbit/s) 8-level pulse amplitude modulation (PAM8) signal is successfully transmitted over a 15 km standard single mode fiber (SSMF) with no optical amplifier. The nonlinear Volterra equalizer is usually used to cope with the distortions induced by the nonlinearity of DML and the bandwidth-limited components. However, the Volterra equalizer would also enhance the noise at high frequency, which is harmful, especially to PAM8 signal because it is more sensitive to noise. Thus, the Volterra equalizer is modified in our scheme by adding a decision feedback process behind. With the help of the modified Volterra equalizer, the enhanced noise at high frequency is effectively eliminated, and a power gain of 0.5 dB and 3.3 dB for 4 × 64 Gbit/s PAM4 signal transmission over 30 km SSMF and 4 × 96 Gbit/s PAM8 signal transmission over 15 km SSMF at the HD-FEC limit can be obtained, respectively. Moreover, the computation complexity of the modified Volterra equalizer could be reduced by 38% compared with the conventional Volterra equalizer.