Nonlinear Channel Equalization Research Articles

Kernel recursive generalized mixed norm algorithm

This work studies the problem of kernel adaptive filtering (KAF) for nonlinear signal processing under non-Gaussian noise environments. A new KAF algorithm, called kernel recursive generalized mixed norm (KRGMN), is derived by minimizing the generalized mixed norm (GMN) cost instead of the well-known mean square error (MSE). A single error norm such as lp error norm can be used as a cost function in KAF to deal with non-Gaussian noises but it may exhibit slow convergence speed and poor misadjustments in some situations. To improve the convergence performance, the GMN cost is formed as a convex mixture of lp and lq norms to increase the convergence rate and substantially reduce the steady-state errors. The proposed KRGMN algorithm can solve efficiently the problems such as nonlinear channel equalization and system identification in non-Gaussian noises. Simulation results confirm the desirable performance of the new algorithm.

Robust nonlinear channel equalization using WNN trained by symbiotic organism search algorithm

In the present world of ‘Big Data,’ the communication channels are always remaining busy and overloaded to transfer quintillion bytes of information. To design an effective equalizer to prevent the inter-symbol interference in such scenario is a challenging task. In this paper, we develop equalizers based on a nonlinear neural structure (wavelet neural network (WNN)) and train it's weighted by a recently developed meta-heuristic (symbiotic organisms search algorithm). The performance of the proposed equalizer is compared with WNN trained by cat swarm optimization (CSO) and clonal selection algorithm (CLONAL), particle swarm optimization (PSO) and least mean square algorithm (LMS). The performance is also compared with other equalizers with structure based on functional link artificial neural network (trigonometric FLANN), radial basis function network (RBF) and finite impulse response filter (FIR). The superior performance is demonstrated on equalization of two non-linear three taps channels and a linear twenty-three taps telephonic channel. It is observed that the performance of the gradient algorithm based equalizers fails in the presence of burst error. The robustness in the performance of the proposed equalizers to handle the burst error conditions is also demonstrated.

Split FTDNN-DFE Equalizer for Complex Non-linear Wireless Channels with NARMA Approximation

AbstractTime-dependent channel variations in wireless systems have been modeled using autoregressive (AR), moving average, and AR moving average (ARMA) approaches, which in certain conditions fail to provide the most suitable description of the scenarios. It lowers the quality of service, increases the dependence on the traditional tapped delay line model, and requires more reference symbols as fading in the channel increases. Non-linear attributes in channels are best described by non-linear AR (NAR) and non-linear ARMA (NARMA) models, which capture the channel state information (CSI) better. Learning-based systems like artificial neural networks (ANN) are better equipped to deal with NAR and NARMA approximations of channels, recover CSI, and provide proper channel estimation due to their natural association with non-linear computing. Though decision feedback equalizers (DFE) are the traditional and preferred approaches to reduce the ill effects of channel-induced variations, these are unable to mitigate noise-triggered responses that ANNs can. Here, DFEs and special forms of ANNs are combined to recover CSI from NAR and NARMA approximations of time-varying channels. We especially configure a class of fully focused time delay neural network designed in split form in unitary and composite modes with a DFE to track real and imaginary components of signals. Experimental results establish the effectiveness of such hybrid tools in tracking time-varying non-linear channels covering International Telecommunication Union pedestrian to vehicular conditions.

Multilayer Perceptron Based Equalizer with an Improved Back Propagation Algorithm for Nonlinear Channels

Neural network based equalizers can easily compensate channel impairments; such additive noise and inter symbol interference (ISI). The authors present a new approach to improve the training efficiency of the multilayer perceptron (MLP) based equalizer. Their improvement consists on modifying the back propagation (BP) algorithm, by adapting the activation function in addition to the other parameters of the MLP structure. The authors report on experiment results evaluating the performance of the proposed approach namely the back propagation with adaptive activation function (BPAAF) next to the BP algorithm. To further prove its effectiveness, the proposed approach is also compared beside a so known nonlinear equalizer explicitly the multilayer perceptron with decision feedback equalizer MLPDFE. The authors consider various performance measures specifically: signal resorted quality, lower steady state MSE reached and minimum bit error rate (BER) achieved, where nonlinear channel equalization problems are employed.

A hierarchical alternative updated adaptive Volterra filter with pipelined architecture

The pipelined adaptive Volterra filters (PAVFs) with a two-layer structure constitute a class of good low-complexity filters. They can efficiently reduce the computational complexity of the conventional adaptive Volterra filter. Their major drawbacks are low convergence rate and high steady-state error caused by the coupling effect between the two layers. In order to remove the coupling effect and improve the performance of PAVFs, we present a novel hierarchical pipelined adaptive Volterra filter (HPAVF)-based alternative update mechanism. The HPAVFs with hierarchical decoupled normalized least mean square (HDNLMS) algorithms are derived to adaptively update weights of its nonlinear and linear subsections. The computational complexity of HPAVF is also analyzed. Simulations of nonlinear system adaptive identification, nonlinear channel equalization, and speech prediction show that the proposed HPAVF with different independent weight vectors in nonlinear subsection has superior performance to conventional Volterra filters, diagonally truncated Volterra filters, and PAVFs in terms of initial convergence, steady-state error, and computational complexity.

Kernel fractional affine projection algorithm

This paper extends the kernel affine projection algorithm to a rich, flexible and cohesive taxonomy of fractional signal processing approach. The formulation of the algorithm is established on the inclusion of Riemann–Liouville fractional derivative to gradient-based stochastic Newton recursive method to minimize the cost function of the kernel affine projection algorithm. This approach extends the idea of fractional signal processing in reproducing kernel Hilbert space. The proposed algorithm is applied to the prediction of chaotic Lorenz time series and nonlinear channel equalization. Also the performance is validated in comparison with the least mean square algorithm, kernel least mean square algorithm, affine projection algorithm and kernel affine projection algorithm.

Adaptive Filter Design Using Type-2 Fuzzy Cerebellar Model Articulation Controller.

This paper aims to propose an efficient network and applies it as an adaptive filter for the signal processing problems. An adaptive filter is proposed using a novel interval type-2 fuzzy cerebellar model articulation controller (T2FCMAC). The T2FCMAC realizes an interval type-2 fuzzy logic system based on the structure of the CMAC. Due to the better ability of handling uncertainties, type-2 fuzzy sets can solve some complicated problems with outstanding effectiveness than type-1 fuzzy sets. In addition, the Lyapunov function is utilized to derive the conditions of the adaptive learning rates, so that the convergence of the filtering error can be guaranteed. In order to demonstrate the performance of the proposed adaptive T2FCMAC filter, it is tested in signal processing applications, including a nonlinear channel equalization system, a time-varying channel equalization system, and an adaptive noise cancellation system. The advantages of the proposed filter over the other adaptive filters are verified through simulations.

Simultaneous Computation of Two Independent Tasks Using Reservoir Computing Based on a Single Photonic Nonlinear Node With Optical Feedback.

In this brief, we numerically demonstrate a photonic delay-based reservoir computing system, which processes, in parallel, two independent computational tasks even when the two tasks have unrelated input streams. Our approach is based on a single-longitudinal mode semiconductor ring laser (SRL) with optical feedback. The SRL emits in two directional optical modes. Each directional mode processes one individual task to mitigate possible crosstalk. We illustrate the feasibility of our scheme by analyzing the performance on two benchmark tasks: 1) chaotic time series prediction and 2) nonlinear channel equalization. We identify some feedback configurations for which the results for simultaneous prediction/classification indicate a good performance, but with slight degradation (as compared with the performance obtained for single task processing) due to nonlinear and linear interactions between the two directional modes of the laser. In these configurations, the system performs well on both tasks for a broad range of the parameters.

Robust Blind Learning Algorithm for Nonlinear Equalization Using Input Decision Information.

In this paper, we propose a new blind learning algorithm, namely, the Benveniste-Goursat input-output decision (BG-IOD), to enhance the convergence performance of neural network-based equalizers for nonlinear channel equalization. In contrast to conventional blind learning algorithms, where only the output of the equalizer is employed for updating system parameters, the BG-IOD exploits a new type of extra information, the input decision information obtained from the input of the equalizer, to mitigate the influence of the nonlinear equalizer structure on parameters learning, thereby leading to improved convergence performance. We prove that, with the input decision information, a desirable convergence capability that the output symbol error rate (SER) is always less than the input SER if the input SER is below a threshold, can be achieved. Then, the BG soft-switching technique is employed to combine the merits of both input and output decision information, where the former is used to guarantee SER convergence and the latter is to improve SER performance. Simulation results show that the proposed algorithm outperforms conventional blind learning algorithms, such as stochastic quadratic distance and dual mode constant modulus algorithm, in terms of both convergence performance and SER performance, for nonlinear equalization.

A new training scheme for neural networks and application in non-linear channel equalization

This paper deals with the problem of equalization of channels in a digital communication system. In the literature, artificial neural network (ANN) has been increasingly used for the said problem. However, traditional methods of ANN training fall short of desired performance in the problem of equalization. In this paper, we propose a recently proposed training method for ANN for the problem. This training uses directed search optimization (DSO) as a trainer to neural networks. Then, we apply the same to the problem of nonlinear channel equalization and in that way, this paper introduces a novel strategy for equalization of nonlinear channels. Proposed method of channel equalization performs better than contemporary equalization methods used in the literature, as evident from extensive simulation results presented in this paper.

Quaternion Reproducing Kernel Hilbert Spaces: Existence and Uniqueness Conditions

The existence and uniqueness conditions of quaternion reproducing kernel Hilbert spaces (QRKHS) are established in order to provide a mathematical foundation for the development of quaternion-valued kernel learning algorithms. This is achieved through a rigorous account of left quaternion Hilbert spaces, which makes it possible to generalise standard RKHS to quaternion RKHS. Quaternion versions of the Riesz representation and Moore-Aronszajn theorems are next introduced, thus underpinning kernel estimation algorithms operating on quaternion-valued feature spaces. The difference between the proposed quaternion kernel concept and the existing real and vector approaches is also established in terms of both theoretical advantages and computational complexity. The enhanced estimation ability of the so-introduced quaternion-valued kernels over their real- and vector-valued counterparts is validated through kernel ridge regression applications. Simulations on real world 3D inertial body sensor data and nonlinear channel equalisation using novel quaternion cubic and Gaussian kernels support the approach.

An Algebraic Translation of Cayley-Dickson Linear Systems and Its Applications to Online Learning

The m-dimensional Cayley-Dickson number system Am is a standard extension of real (m=1), complex (m=2), quaternion (m=22), octonion (m=23) and sedenion (m=24) etc. In this paper, we present a systematic algebraic translation of the Cayley-Dickson hypercomplex valued linear systems into a real vector valued linear model. This translation is designed by using jointly two new isomorphisms between real vector spaces and enables us to straightforwardly apply the well established schemes in real domain to problems for the hypercomplex linear model. We also clarify useful algebraic properties of the proposed translation. As an example of many potential algorithms through the proposed algebraic translation, we present Am-adaptive projected subgradient method ( Am-APSM) for Am valued adaptive system identification, and show that many hypercomplex adaptive filtering algorithms can be viewed as special cases of this algorithm. Moreover, we also apply the Am-APSM to nonlinear adaptive filtering by using the kernel trick. Numerical examples show that the effectiveness of the Am-APSM in many Cayley-Dickson valued linear system identification and nonlinear channel equalization problems.

Artificial Neural Network trained by Particle Swarm Optimization for non-linear channel equalization

In this paper, we apply Artificial Neural Network (ANN) trained with Particle Swarm Optimization (PSO) for the problem of channel equalization. Existing applications of PSO to Artificial Neural Networks (ANN) training have only been used to find optimal weights of the network. Novelty in this paper is that it also takes care of appropriate network topology and transfer functions of the neuron. The PSO algorithm optimizes all the variables, and hence network weights and network parameters. Hence, this paper makes use of PSO to optimize the number of layers, input and hidden neurons, the type of transfer functions etc. This paper focuses on optimizing the weights, transfer function, and topology of an ANN constructed for channel equalization. Extensive simulations presented in this paper shows that, as compared to other ANN based equalizers as well as Neuro-fuzzy equalizers, the proposed equalizer performs better in all noise conditions.

Square Root Extended Kernel Recursive Least Squares Algorithm for Nonlinear Channel Equalization

This study presents a square root version of extended kernel recursive least square algorithm. Basically main idea is to overcome the divergence phenomena arise in the computation of weights of the extended kernel recursive least squares algorithm. Numerically stable givens orthogonal transformations are used to obtain the next iteration of the algorithm. The usefulness of the proposed algorithm is illustrated by discussing its application on the nonlinear multipath fading channel equalization based on Rayleigh distribution. Experiments are performed on slow fading Rayleigh channel with scattered signals.

Complex-Valued Filtering Based on the Minimization of Complex-Error Entropy

In this paper, we consider the training of complex-valued filter based on the information theoretic method. We first generalize the error entropy criterion to complex domain to present the complex error entropy criterion (CEEC). Due to the difficulty in estimating the entropy of complex-valued error directly, the entropy bound minimization (EBM) method is used to compute the upper bounds of the entropy of the complex-valued error, and the tightest bound selected by the EBM algorithm is used as the estimator of the complex-error entropy. Then, based on the minimization of complex-error entropy (MCEE) and the complex gradient descent approach, complex-valued learning algorithms for both the (linear) transverse filter and the (nonlinear) neural network are derived. The algorithms are applied to complex-valued linear filtering and complex-valued nonlinear channel equalization to demonstrate their effectiveness and advantages.

Channel equalization using quasi type-2 fuzzy strategies

This paper presents a proposal for channel equalization of non-linear time varying communication channels that uses strategies based on quasi type-2 fuzzy sets. Both triangular and trapezoidal quasi type-2 fuzzy equalizers are designed by means of an incremental design approach. A comparative experimental study among several fuzzy equalization architectures is carried out, showing that the equalizer based on trapezoidal quasi type-2 fuzzy sets exhibits the best performance for low and medium levels of uncertainty. The results of this work also suggest that high order fuzzy sets make the equalization process more robust to the variability of the communications channel.

Identification and Equalization of Nonlinear Channel Based on ANFIS

The demand of high-speed communication leads that the application for resource of channel exceeds the range of linear model. The channel should be described as a nonlinear model. So, the paper uses the adaptive neuron-fuzzy inference system (ANFIS) to identify and equalize the nonlinear channel of the high-speed communication system. Meanwhile, the subtract cluster is applied to identify the construction of the ANFIS and the hybrid learning algorithm based on the least square and back-propagation is used to train network. The simulation results show that the convergence rate and identification accuracy of the ANFIS are better than BP network and the efficiency of the ANFIS based on subtract cluster partition algorithm is higher than that of the ANFIS based on the grid partition algorithm.

KIMEL: A kernel incremental metalearning algorithm

The Kernel method is a powerful tool for extending an algorithm from linear to nonlinear case. Metalearning algorithm learns the base learning algorithm, thus to improve performance of the learning system. Usually, metalearning algorithms exhibit faster convergence rate and lower Mean-Square Error (MSE) than the corresponding base learning algorithms. In this paper, we present a kernelized metalearning algorithm, named KIMEL, which is a metalearning algorithm in the Reproducing Kernel Hilbert Space (RKHS). The convergence analyses of the KIMEL algorithm are performed in detail. To demonstrate the effectiveness and advantage of the proposed algorithm, we firstly apply the algorithm to a simple example of nonlinear channel equalization. Then we focus on a more practical application in blind Image Quality Assessment (IQA). Experimental results show that the KIMEL algorithm has superior performance.

Adaptive learning in complex reproducing kernel Hilbert spaces employing Wirtinger's subgradients.

This paper presents a wide framework for non-linear online supervised learning tasks in the context of complex valued signal processing. The (complex) input data are mapped into a complex reproducing kernel Hilbert space (RKHS), where the learning phase is taking place. Both pure complex kernels and real kernels (via the complexification trick) can be employed. Moreover, any convex, continuous and not necessarily differentiable function can be used to measure the loss between the output of the specific system and the desired response. The only requirement is the subgradient of the adopted loss function to be available in an analytic form. In order to derive analytically the subgradients, the principles of the (recently developed) Wirtinger's calculus in complex RKHS are exploited. Furthermore, both linear and widely linear (in RKHS) estimation filters are considered. To cope with the problem of increasing memory requirements, which is present in almost all online schemes in RKHS, the sparsification scheme, based on projection onto closed balls, has been adopted. We demonstrate the effectiveness of the proposed framework in a non-linear channel identification task, a non-linear channel equalization problem and a quadrature phase shift keying equalization scheme, using both circular and non circular synthetic signal sources.

Optoelectronic Reservoir Computing

Reservoir computing is a recently introduced, highly efficient bio-inspired approach for processing time dependent data. The basic scheme of reservoir computing consists of a non linear recurrent dynamical system coupled to a single input layer and a single output layer. Within these constraints many implementations are possible. Here we report an optoelectronic implementation of reservoir computing based on a recently proposed architecture consisting of a single non linear node and a delay line. Our implementation is sufficiently fast for real time information processing. We illustrate its performance on tasks of practical importance such as nonlinear channel equalization and speech recognition, and obtain results comparable to state of the art digital implementations.