Nonlinear Channel Equalization Problem Research Articles

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.

Numerical Study of Parallel Optoelectronic Reservoir Computing to Enhance Nonlinear Channel Equalization

Nonlinear impairment is one of the critical limits to enhancing the performance of high-speed communication systems. Traditional digital signal processing (DSP)-based nonlinear channel equalization schemes are influenced by limited bandwidth, high power consumption, and high processing latency. Optoelectronic reservoir computing (RC) is considered a promising optical signal processing (OSP) technique with merits such as large bandwidth, high power efficiency, and low training complexity. In this paper, optoelectronic RC was employed to solve the nonlinear channel equalization problem. A parallel optoelectronic RC scheme with a dual-polarization Mach–Zehnder modulator (DPol-MZM) is proposed and demonstrated numerically. The nonlinear channel equalization performance was greatly enhanced compared with the traditional optoelectronic RC and the Volterra-based nonlinear DSP schemes. In addition, the system efficiency was improved with a single DPol-MZM.

An Efficient JAYA Algorithm with Lévy Flight for Non-linear Channel Equalization

Neural network (NN) based equalizers are known to outperform the linear equalizers based on finite impulse response (FIR) adaptive filter for highly dispersive and non-linear channels. To overcome the limitations of the back-propagation (BP) algorithm, metaheuristic algorithms are emerging as promising alternatives for training the NN based equalizers. JAYA is a simple and efficient metaheuristic algorithm. Hence, its application to channel equalization problem is worth investigating. Despite its simplicity and efficiency, the JAYA algorithm has problems such as being trapped in local minima due to insufficient diversity of population and weak exploration capability. To alleviate these issues, in this paper the concept of Lévy flight (LF) and greedy selection scheme has been incorporated into the basic JAYA algorithm. The LF concept enhances the population diversity and thus avoids the state of stagnation. The greedy selection scheme is employed to improve the exploitation ability without loss of population diversity. Furthermore, in order to maintain the balance between the exploration and exploitation capabilities of the algorithm, an adaptive Lévy index is proposed based on a linear control parameter strategy. An extensive simulation-based sensitivity analysis of proposed method called JAYA algorithm with Lévy flight (JAYALF) with respect to key parameters is carried out to select the optimized values for these parameters. In order to validate the local optima avoidance ability, exploitation and convergence rate of the proposed JAYALF algorithm, it is tested on seventeen well-known unimodal and multimodal benchmark functions and to verify the effectiveness of the JAYALF for non-linear channel equalization problem, three wireless communication channels with two different nonlinearities have been considered for simulation. In addition, the non-parametric pairwise Wilcoxon rank-sum test has been employed to test the statistical validity of the results obtained from JAYALF. The results of experiments and statistical test demonstrate that the proposed algorithm significantly outperforms JAYA, variants of JAYA, state-of-the-art algorithms and BP algorithm in terms of solution quality, convergence speed, and robustness. Furthermore, the results of experimental analyses conducted indicate that proposed JAYALF algorithm has a better exploration ability and converges quickly without getting trapped into local optima.

The Generalized Complex Kernel Least-Mean-Square Algorithm

We propose a novel adaptive kernel based regression method for complex-valued signals: the generalized complex-valued kernel least-mean-square (gCKLMS). We borrow from the new results on widely linear reproducing kernel Hilbert space (WL-RKHS) for nonlinear regression and complex-valued signals, recently proposed by the authors. This paper shows that in the adaptive version of the kernel regression for complex-valued signals we need to include another kernel term, the so-called pseudo-kernel. This new solution is endowed with better representation capabilities in complex-valued fields, since it can efficiently decouple the learning of the real and the imaginary part. Also, we review previous realizations of the complex KLMS algorithm and its augmented version to prove that they can be rewritten as particular cases of the gCKLMS. Furthermore, important conclusions on the kernels design are drawn that help to greatly improve the convergence of the algorithms. In the experiments, we revisit the nonlinear channel equalization problem to highlight the better convergence of the gCKLMS compared to previous solutions. Also, the flexibility of the proposed generalized approach is tested in a second experiment with non-independent real and imaginary parts. The results illustrate the significant performance improvements of the gCKLMS approach when the complex-valued signals have different properties for the real and imaginary parts.

Complex Gaussian Processes for Regression.

In this paper, we propose a novel Bayesian solution for nonlinear regression in complex fields. Previous solutions for kernels methods usually assume a complexification approach, where the real-valued kernel is replaced by a complex-valued one. This approach is limited. Based on the results in complex-valued linear theory and Gaussian random processes, we show that a pseudo-kernel must be included. This is the starting point to develop the new complex-valued formulation for Gaussian process for regression (CGPR). We face the design of the covariance and pseudo-covariance based on a convolution approach and for several scenarios. Just in the particular case where the outputs are proper, the pseudo-kernel cancels. Also, the hyperparameters of the covariance can be learned maximizing the marginal likelihood using Wirtinger's calculus and patterned complex-valued matrix derivatives. In the experiments included, we show how CGPR successfully solves systems where the real and imaginary parts are correlated. Besides, we successfully solve the nonlinear channel equalization problem by developing a recursive solution with basis removal. We report remarkable improvements compared to previous solutions: a 2-4-dB reduction of the mean squared error with just a quarter of the training samples used by previous approaches.

Widely Linear Complex-Valued Kernel Methods for Regression

Usually, complex-valued RKHS are presented as an straightforward application of the real-valued case. In this paper we prove that this procedure yields a limited solution for regression. We show that another kernel, here denoted as pseudo kernel, is needed to learn any function in complex-valued fields. Accordingly, we derive a novel RKHS to include it, the widely RKHS (WRKHS). When the pseudo-kernel cancels, WRKHS reduces to complex-valued RKHS of previous approaches. We address the kernel and pseudo-kernel design, paying attention to the kernel and the pseudo-kernel being complex-valued. In the experiments included we report remarkable improvements in simple scenarios where real a imaginary parts have different similitude relations for given inputs or cases where real and imaginary parts are correlated. In the context of these novel results we revisit the problem of non-linear channel equalization, to show that the WRKHS helps to design more efficient solutions.

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 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.

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.

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.

Orthogonal least squares based complex-valued functional link network

Functional link networks are single-layered neural networks that impose nonlinearity in the input layer using nonlinear functions of the original input variables. In this paper, we present a fully complex-valued functional link network (CFLN) with multivariate polynomials as the nonlinear functions. Unlike multilayer neural networks, the CFLN is free from local minima problem, and it offers very fast learning of parameters because of its linear structure. Polynomial based CFLN does not require an activation function which is a major concern in the complex-valued neural networks. However, it is important to select a smaller subset of polynomial terms (monomials) for faster and better performance since the number of all possible monomials may be quite large. Here, we use the orthogonal least squares (OLS) method in a constructive fashion (starting from lower degree to higher) for the selection of a parsimonious subset of monomials. It is argued here that computing CFLN in purely complex domain is advantageous than in double-dimensional real domain, in terms of number of connection parameters, faster design, and possibly generalization performance. Simulation results on a function approximation, wind prediction with real-world data, and a nonlinear channel equalization problem exhibit that the OLS based CFLN yields very simple structure having favorable performance.

Adaptive reduced multivariate polynomial equalizers for blu-ray disc channels

The multivariate polynomial model provides an effective way to describe complex nonlinear input-output relationships since it is tractable for optimization, sensitivity analysis, and prediction of confidence intervals. However, for high-dimensional and high-order problems, multivariate polynomial model becomes impractical due to its huge number of product terms. Therefore, multivariate polynomial model cannot be applied for nonlinear channel equalization problems. This is especially true for the case of a full interaction model. In this paper, we propose a reduced multivariate polynomial model to circumvent the dimensionality problem with some compromise in its approximation capability. Then the proposed reduced model is applied to Blu-ray Disc (BD) for channel equalization. Simulation experiments show that the adaptive reduced multivariate polynomial equalizers with moderate complexity can effectively compensate for intersymbol interference in high density optical recording systems such as BD.

Modified fuzzy c-means and Bayesian equalizer for nonlinear blind channel

In this study, we present a modified fuzzy c-means (MFCM) clustering algorithm in the problem of nonlinear blind channel equalization. The proposed MFCM searches for the optimal channel output states of a nonlinear channel based on received symbols. In contrast to the commonly exploited Euclidean distance, in this method we consider the usage of the Bayesian likelihood fitness function. In the search procedure, all possible sets of desired channel states are constructed by considering the combinations of estimated channel output states and the set of desired states characterized by the maximal value of the Bayesian fitness is selected. By using these desired channel states, the Bayesian equalizer is implemented to reconstruct transmitted symbols. In the simulation studies, binary signals are generated at random with Gaussian noise. The performance of the proposed method is compared with that of a hybrid genetic algorithm (GA) augmented by the mechanism of simulated annealing (SA), GASA for brief. It is demonstrated that a relatively high accuracy and a fast search speed have been achieved.

Communication channel equalization using complex-valued minimal radial basis function neural networks

A complex radial basis function neural network is proposed for equalization of quadrature amplitude modulation (QAM) signals in communication channels. The network utilizes a sequential learning algorithm referred to as complex minimal resource allocation network (CMRAN) and is an extension of the MRAN algorithm originally developed for online learning in real-valued radial basis function (RBF) networks. CMRAN has the ability to grow and prune the (complex) RBF network's hidden neurons to ensure a parsimonious network structure. The performance of the CMRAN equalizer for nonlinear channel equalization problems has been evaluated by comparing it with the functional link artificial neural network (FLANN) equalizer of J.C. Patra et al. (1999) and the Gaussian stochastic gradient (SG) RBF equalizer of I. Cha and S. Kassam (1995). The results clearly show that CMRANs performance is superior in terms of symbol error rates and network complexity.

Fuzzy techniques for adaptive nonlinear equalization

This paper presents a review of the application of fuzzy filters to nonlinear channel equalization problem. The paper presents the development of fuzzy filters for channel equalization applications first. The close relationship between the fuzzy equalizers and the equalizer based on maximum a posteriori probability principle (MAP) is derived. The MAP equalizers are also called Bayesian equalizers since their decision function is based on Bayes’ decision theory. It is shown that the fuzzy implementation of the MAP equalizer provides major computational advantages. The structure of this fuzzy equalizer also provides scope for co-channel interference (CCI) suppression in the presence of inter symbol interference (ISI) and additive white Gaussian noise (AWGN) with the help of a pre-processor. The advantages associated with the fuzzy equalizers can make them a candidate for mobile communication applications.

Complex-valued neural networks with adaptive spline activation function for digital-radio-links nonlinear equalization

In this paper, a new complex-valued neural network based on adaptive activation functions is proposed. By varying the control points of a pair of Catmull-Rom cubic splines, which are used as an adaptable activation function, this new kind of neural network can be implemented as a very simple structure that is able to improve the generalization capabilities using few training samples. Due to its low architectural complexity (low overhead with respect to a simple FIR filter), this network can be used to cope with several nonlinear DSP problems at a high symbol rate. In particular, this work addresses the problem of nonlinear channel equalization. In fact, although several authors have already recognized the usefulness of a neural network as a channel equalizer, one problem has not yet been addressed: the high complexity and the very long data sequence needed to train the network. Several experimental results using a realistic channel model are reported that prove the effectiveness of the proposed network on equalizing a digital satellite radio link in the presence of noise, nonlinearities, and intersymbol interference (ISI).

Fuzzy adaptive filters, with application to nonlinear channel equalization

Two fuzzy adaptive filters are developed: one uses a recursive-least-squares (RLS) adaptation algorithm, and the other uses a least-mean-square (LMS) adaptation algorithm. The RLS fuzzy adaptive filter is constructed through the following four steps: (1) define fuzzy sets in the filter input space Rn whose membership functions cover U; (2) construct a set of fuzzy IF-THEN rules which either come from human experts or are determined during the adaptation procedure by matching input-output data pairs; (3) construct a filter based on the set of rules; and (4) update the free parameters of the filter using the RLS algorithm. The design procedure for the LMS fuzzy adaptive filter is similar. The most important advantage of the fuzzy adaptive filters is that linguistic information (in the form of fuzzy IF-THEN rules) and numerical information (in the form of input-output pairs) can be combined in the filters in a uniform fashion. The filters are applied to nonlinear communication channel equalization problems. >