Nonlinear Channel Equalization Research Articles

Numerical Approaches in Nonlinear Fourier Transform‐Based Signal Processing for Telecommunications

ABSTRACTWe discuss applications of the inverse scattering transform, also known as the nonlinear Fourier transform (NFT) in telecommunications, both for nonlinear optical fiber communication channel equalization and time‐domain signal processing techniques. Our main focus is on the challenges and recent progress in the development of efficient numerical algorithms and approaches to NFT implementation.

Experimental realization of a performance-enhanced reservoir computer based on a photonic-filter feedback laser

Reservoir computing (RC), especially time-delayed RC, as a lightweight, high-speed machine learning paradigm, shows excellent performance in time-series prediction and recognition tasks. Within this framework, time delays play a vital role in dynamic systems, i.e., significantly affecting the transient behavior and the dimensionality of reservoirs. In this work, we explore a multidelay system as the core computational element of RC, which is constructed using a semiconductor laser with photonic-filter feedback. We demonstrate experimentally that the photonic-filter feedback scheme can improve the mapping of scalar inputs into higher-dimensional dynamics, and thus enhance the prediction and classification ability in time series and nonlinear channel equalization tasks. In particular, the rich neural dynamics in turn boosts its memory capacity, which offers great potential for short-term prediction of time series. The numerical results show good qualitative agreement with the experiment. We show that improved RC performance can be achieved by utilizing a small coupling coefficient and eschewing feedback at integer multiples, which can induce detrimental resonance. This work provides an alternative photonic platform to achieve high-performance neural networks based on high-dimensional dynamic 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.

Experimental Investigation of Nonlinear Vector Autoregressor-Based Channel Equalization for IM/DD Transmission With Silicon Microring Modulators

Experimental Investigation of Nonlinear Vector Autoregressor-Based Channel Equalization for IM/DD Transmission With Silicon Microring Modulators

Enhanced performances of photonic reservoir computing using a semiconductor laser with random distributed optical feedback

We propose and experimentally demonstrate a photonic time-delay reservoir computing (TDRC) system with random distributed optical feedback under optical injection. To evaluate the performance, we calculate the memory ability and perform two benchmark tasks, i.e., chaotic time series prediction and nonlinear channel equalization task. Our numerical results show that the proposed TDRC has a superior performance compared with the case with conventional single optical feedback. This is attributed to the fact that the random distributed optical feedback offers multiple external cavity modes, which enhance the nonlinearity of the reservoir laser. Additionally, the experimental result also shows that our proposed TDRC scheme outperforms the computer with single optical feedback in the chaotic time series prediction task. To the best of our knowledge, our work offers a novel path to improve the performance of TDRC by introducing random distributed optical feedback.

Deep Time-Delay Reservoir Computing With Cascading Injection-Locked Lasers

Reservoir computing is a simplified recurrent neural network, which requires training only at the output layer and hence significantly reduces the training cost. Time-delay reservoir computing (TDRC) introduces a large number of virtual neurons based on a single physical neuron and a feedback loop, which is friendly for hardware implementations. This work proposes a scheme for implementing the deep TDRC architecture based on cascading injection-locked semiconductor lasers. In each layer, the reservoir consists of a quantum dot laser and an optical feedback loop. The output of each reservoir layer is fed into the subsequent one through the optical injection-locking technique. This all-optical approach (for reservoir layers) has the merit of high scalability without any depth limitation. Theoretical analysis shows that the deep TDRC improves the performance on multiple benchmark tasks, including the memory capacity, the prediction of chaos, the nonlinear channel equalization, and the recognition of spoken digits.

Adaptive CL-BFGS Algorithms for Complex-Valued Neural Networks.

Complex-valued limited-memory BFGS (CL-BFGS) algorithm is efficient for the training of complex-valued neural networks (CVNNs). As an important parameter, the memory size represents the number of saved vector pairs and would essentially affect the performance of the algorithm. However, the determination of a suitable memory size for the CL-BFGS algorithm remains challenging. To deal with this issue, an adaptive method is proposed in which the memory size is allowed to vary during the iteration process. Basically, at each iteration, with the help of multistep quasi-Newton method, an appropriate memory size is chosen from a variable set {1,2,..., M} by approximating complex Hessian matrix as close as possible. To reduce the computational complexity and ensure desired performance, the upper bound M is adjustable according to the moving average of memory sizes found in previous iterations. The proposed adaptive CL-BFGS (ACL-BFGS) algorithm can be efficiently applied for the training of CVNNs. Moreover, it is suggested to take multiple memory sizes to construct the search direction, which further improves the performance of the ACL-BFGS algorithm. Experimental results on some benchmark problems including the pattern classification, complex function approximation, and nonlinear channel equalization problems are given to illustrate the advantages of the developed algorithms over some previous ones.

Performance-enhanced time-delayed photonic reservoir computing system using a reflective semiconductor optical amplifier.

We propose a time-delayed photonic reservoir computing (RC) architecture utilizing a reflective semiconductor optical amplifier (RSOA) as an active mirror. The performance of the proposed RC structure is investigated by two benchmark tasks, namely the Santa Fe time-series prediction task and the nonlinear channel equalization task. The simulation results show that both the prediction and equalization performance of the proposed system are significantly improved with the contribution of RSOA, with respect to the traditional RC system using a mirror. By increasing the drive current of the RSOA, the greater nonlinearity of the RSOA gain saturation is achieved, as such the prediction and equalization performance are enhanced. It is also shown that the proposed RC architecture shows a wider consistency interval and superior robustness than the traditional RC structure for most of the measured parameters such as coupling strength, injection strength, and frequency detuning. This work provides a performance-enhanced time-delayed RC structure by making use of the nonlinear transformation of the RSOA feedback.

Non-linear Channel Equalization using Modified Grasshopper Optimization Algorithm

In this paper, a modified grasshopper optimization algorithm is proposed for equalization of non-linear wireless communication channels. Even though grasshopper optimisation algorithm (GOA) is an efficient algorithm, it gets trapped into local optima after some iterations due to loss of swarm diversity. Furthermore, GOA does not involve any provision to retain the elite grasshoppers found so far at each index which weakens the exploitation capability and convergence rate of GOA. These limitations of GOA are alleviated in this paper by incorporating three key modifications into GOA. A threshold parameter is introduced to detect the inefficient search region. Lévy Flight is integrated with the basic GOA to improve the diversity of grasshopper swarm and the greedy selection operator is used to preserve the elite grasshoppers found so far at every index. The superiority of a modified grasshopper optimization algorithm (MGOA) is illustrated over the existing metaheuristic algorithms. The key parameters of MGOA are selected by performing the sensitivity analysis. The simulation results on four non-linear channels demonstrate the equalization capability of the proposed MGOA in terms of MSE and BER performance. A statistical validity of results provided by MGOA is confirmed through Wilcoxon rank-sum test.

Neural network architectures for optical channel nonlinear compensation in digital subcarrier multiplexing systems.

In this work, we propose to use various artificial neural network (ANN) structures for modeling and compensation of intra- and inter-subcarrier fiber nonlinear interference in digital subcarrier multiplexing (DSCM) optical transmission systems. We perform nonlinear channel equalization by employing different ANN cores including convolutional neural networks (CNN) and long short-term memory (LSTM) layers. First, we develop a fiber nonlinearity compensation for DSCM systems based on a fully-connected network across all subcarriers. In subsequent steps, and borrowing from the perturbation analysis of fiber nonlinearity, we gradually upgrade proposed designs towards modular structures with better performance-complexity advantages. Our study shows that putting proper macro structures in design of ANN nonlinear equalizers in DSCM systems can be crucial in development of practical solutions for future generations of coherent optical transceivers.

Photonic reservoir computing with a silica microsphere cavity.

We experimentally demonstrate a photonic reservoir computing (RC) system using a passive silica microsphere cavity. The microsphere cavity exhibits a consistent nonlinear response to the non-return-to-zero signal and the multiple-level signal due to strong interference between numerous whispering gallery modes in the "over-coupling" state. Benefiting from the fact that the long photon lifetime inside the microsphere cavity provides a memory of past inputs, this photonic reservoir does not require a delayed feedback loop. We evaluate the generalization property of the RC system and obtain a correlation coefficient of 0.923. In addition, we obtain a NMSE of 0.06 for the Santa-Fe chaotic time series prediction task and a SER of 0.02 at a SNR of 12 dB for the nonlinear channel equalization task. Moreover, a microsphere cavity with a higher quality factor can provide a larger memory capacity. The application of the silica microsphere cavity as a small-volume passive device in a reservoir furnishes a new avenue for achieving a low-consumption and integrated RC system.

Development of a novel RBFNN‐trained nonlinear channel equalizer based on GDEBOA technique

AbstractThe equalization of digital channels is generally known to be a nonlinear classification problem. Applications such as these can benefit from networks that approximate nonlinear mappings. It gets good performance by adjusting only one coefficient and one center closest to the input vector of the radial basis function network (RBFN), which is a simplified version of stochastic gradient techniques. Artificial Neural Networks (ANNs) are suitable for channel equalization because they have the capability to map between variables. Having only one hidden layer in Radial Basis Function Neural Network (RBFNN) makes it the most preferred equalizer to mitigate distortions in the channels. The ability to equalize nonlinear channels is strength of radial RBFNN, which is a simplified version of stochastic gradient techniques. The conventional “hit and trial” approach poses the greatest difficulty when designing RBFNN Equalizers. As a solution to these limitations, this work suggests a training strategy based on Gaussian distribution estimation strategy (GDE) in hybrid butterfly optimization algorithm (GDEBOA) for RBFNN channel equalizer. The proposed RBFNN equalizer is being trained using a population‐based optimization algorithm. An algorithm, called GDEBOA, based on a GDE approach is developed in this paper. The proposed training scheme significantly outperforms existing metaheuristic algorithms in terms of mean square error (MSE) and bit error rate (BER). Furthermore, based on burst error scenarios and bit error probability (BEP), the proposed method has demonstrated greater robustness than other methods when faced with such scenarios. The effectiveness of the suggested scheme over a wide range of signal‐to‐noise ratios has been validated by several simulation studies. Additionally, statistical importance of the suggested technique is analyzed. It is visualized that GDEBOA performs better than existing algorithms for training RBFNN equalizer.

Adaptive orthogonal gradient descent algorithm for fully complex-valued neural networks

For optimization algorithms of fully complex-valued neural networks, complex-valued stepsize is helpful to make the training escape from saddle points. In this paper, an adaptive orthogonal gradient descent algorithm with complex-valued stepsize is proposed for the efficient training of fully complex-valued neural networks. The basic idea is that, at each iteration, the search direction is constructed as a combination of two orthogonal gradient directions by using the algebraic representation of complex-valued stepsize. It is then shown that the determination of suitable complex-valued stepsize is facilitated by a decoupling method such that the computational complexity involved in the training process is greatly reduced. The experiments are finally conducted on pattern classification, nonlinear channel equalization and signal prediction to confirm the advantages of the proposed algorithm.

An Efficient Filtering Algorithm against Impulse Noise in Communication Systems

<p>The kernel adaptive filter (KAF), which processes data in the reproducing kernel Hilbert space (RKHS), can improve the performance of conventional adaptive filters in nonlinear systems. However, the presence of impulse noise can seriously degrade the performance of KAF. In this paper, we propose a kernel modified-sign least-mean-square algorithm (KMSLMS) to mitigate the impact of impulse noise in communication systems. Moreover, we apply the nearest-instance-centroid estimation (NICE) algorithm to reduce the computational complexity of our KMSLMS algorithm, called the NICE-KMSLMS algorithm. Finally, computer simulations were used to evaluate the effectiveness of our proposed method. Compared with the conventional kernel least-mean-square algorithm (KLMS), our proposed method can improve the testing mean-squared error (MSE) by 2.32 dB and 7.39 dB for the nonlinear channel equalization and Mackey-Glass chaotic time series prediction problems, respectively. Furthermore, the testing MSE degradation caused by combining the NICE algorithm with our KMSLMS algorithm is negligible but can save about 55% computational cost in terms of the required mean size.</p> <p> </p>

Hardware optimization for photonic time-delay reservoir computer dynamics

Reservoir computing (RC) is one kind of neuromorphic computing mainly applied to process sequential data such as time-dependent signals. In this paper, the bifurcation diagram of a photonic time-delay RC system is thoroughly studied, and a method of bifurcation dynamics guided hardware hyperparameter optimization is presented. The time-evolution equation expressed by the photonic hardware parameters is established while the intrinsic dynamics of the photonic RC system is quantitively studied. Bifurcation dynamics based hyperparameter optimization offers a simple yet effective approach in hardware setting optimization that aims to reduce the complexity and time in hardware adjustment. Three benchmark tasks, nonlinear channel equalization (NCE), nonlinear auto regressive moving average with 10th order time lag (NARMA10) and Santa Fe laser time-series prediction tasks are implemented on the photonic delay-line RC using bifurcation dynamics guided hardware optimization. The experimental results of these benchmark tasks achieved overall good agreement with the simulated bifurcation dynamics modeling results.

Nonlinear Channel Equalization Using Gaussian Processes Regression in IMDD Fiber Link

Gaussian processes regression (GPR)-aided nonlinear channel equalizer (CE) is experimentally demonstrated in a multi-level intensity modulation and direct detection fiber link. In this scheme, the GPR model is used to estimate the transmitted symbols or the corresponding nonlinear distortions after pre-processing. The experimental results show that GPR-aided nonlinear CE has better nonlinear tolerance than conventional linear and nonlinear filter-based CE. It is also shown that the GPR model in the nonlinear channel equalization process can be understood as an optimized single-layer neural network model with infinite width. Finally, we reveal the relationship between the key coefficients in GPR model and parameters in fiber link through both experiment and simulation.

Nonlinear Channel Equalization based on Gaussian Processes for Regression in Fiber Link

Nonlinear Channel Equalization based on Gaussian Processes for Regression in Fiber Link

Parallel and deep reservoir computing using semiconductor lasers with optical feedback.

Photonic reservoir computing has been intensively investigated to solve machine learning tasks effectively. A simple learning procedure of output weights is used for reservoir computing. However, the lack of training of input-node and inter-node connection weights limits the performance of reservoir computing. The use of multiple reservoirs can be a solution to overcome this limitation of reservoir computing. In this study, we investigate parallel and deep configurations of delay-based all-optical reservoir computing using semiconductor lasers with optical feedback by combining multiple reservoirs to improve the performance of reservoir computing. Furthermore, we propose a hybrid configuration to maximize the benefits of parallel and deep reservoirs. We perform the chaotic time-series prediction task, nonlinear channel equalization task, and memory capacity measurement. Then, we compare the performance of single, parallel, deep, and hybrid reservoir configurations. We find that deep reservoirs are suitable for a chaotic time-series prediction task, whereas parallel reservoirs are suitable for a nonlinear channel equalization task. Hybrid reservoirs outperform other configurations for all three tasks. We further optimize the number of reservoirs for each reservoir configuration. Multiple reservoirs show great potential for the improvement of reservoir computing, which in turn can be applied for high-performance edge computing.

Characteristic Study on Parallel Reservoir Computation Based on VCSEL Dynamics With Optoelectronic Feedback

We proposed a new reservoir computation (RC) structure using polarization dynamics of vertical cavity surface emitting lasers (VCSELs) with optoelectronic feedback. Santa-Fe time-series prediction and nonlinear channel equalization were used to evaluate its parallel processing capability. We numerically discussed the effects of feedback strength, injection strength, frequency detuning, injection current and scaling factors on the RC performance of the system. The results show that, the operation range of negative feedback is larger than that of positive feedback; under appropriate feedback and lager injection strengths, good RC performance can be achieved for a wider injection current; with continuous variation of feedback strength from negative to positive values, the optimized location for frequency detuning moves from positive to negative values. We also evaluated the processing rate of proposed RC structure. The related study is helpful for the design and optimization of VCSEL-based RC under optoelectronic feedback.

Experimental implementation of a neural network optical channel equalizer in restricted hardware using pruning and quantization

The deployment of artificial neural networks-based optical channel equalizers on edge-computing devices is critically important for the next generation of optical communication systems. However, this is still a highly challenging problem, mainly due to the computational complexity of the artificial neural networks (NNs) required for the efficient equalization of nonlinear optical channels with large dispersion-induced memory. To implement the NN-based optical channel equalizer in hardware, a substantial complexity reduction is needed, while we have to keep an acceptable performance level of the simplified NN model. In this work, we address the complexity reduction problem by applying pruning and quantization techniques to an NN-based optical channel equalizer. We use an exemplary NN architecture, the multi-layer perceptron (MLP), to mitigate the impairments for 30 GBd 1000 km transmission over a standard single-mode fiber, and demonstrate that it is feasible to reduce the equalizer’s memory by up to 87.12%, and its complexity by up to 78.34%, without noticeable performance degradation. In addition to this, we accurately define the computational complexity of a compressed NN-based equalizer in the digital signal processing (DSP) sense. Further, we examine the impact of using hardware with different CPU and GPU features on the power consumption and latency for the compressed equalizer. We also verify the developed technique experimentally, by implementing the reduced NN equalizer on two standard edge-computing hardware units: Raspberry Pi 4 and Nvidia Jetson Nano, which are used to process the data generated via simulating the signal’s propagation down the optical-fiber system.