We investigate the approximation ability of a multilayer perceptron (MLP) network when it is extended to the complex domain. The main challenge for processing complex data with neural networks has been the lack of bounded and analytic complex nonlinear activation functions in the complex domain, as stated by Liouville’s theorem. To avoid the conflict between the boundedness and the analyticity of a nonlinear complex function in the complex domain, a number of ad hoc MLPs that include using two real-valued MLPs, one processing the real part and the other processing the imaginary part, have been traditionally employed. However, since nonanalytic functions do not meet the Cauchy-Riemann conditions, they render themselves into degenerative backpropagation algorithms that compromise the efficiency of nonlinear approximation and learning in the complex vector field. A number of elementary transcendental functions (ETFs) derivable from the entire exponential function e z that are analytic are defined as fully complex activation functions and are shown

In this paper a new adaptive non linear function for blind signal separation is presented. It is based on a spline approximation whose control points are adaptively changed using information maximization techniques. The monotonously increasing characteristic is obtained using suitable B-spline functions imposing simple constraints on its control points. In particular, the problem of adaptively maximizing the entropy of the output is considered in the context of blind separation of independent sources. We derive a simple form of the learning algorithm which allows not only to adapt the separation matrix coefficients but also the shape of the non linear functions. A comparison with the Mixture--OfDensities approach is also presented on some experimental data that demonstrates the effectiveness and efficiency of the proposed method. 1.

Abstract: In this paper, 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 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 Patra et. al. and the Gaussian Stochastic Gradient (SG) RBF equalizer of Cha and Kassam. The results clearly show that CMRAN’s performance is superior in terms of symbol error rates and network complexity.

In this paper a new neural network model for blind demixing of nonlinear mixtures is proposed. We address the use of the Adaptive Spline Neural Network recently introduced for supervised and unsupervised neural networks. These networks are built using neurons with flexible B-spline activation functions and in order to separate signals from mixtures, a gradient-ascending algorithm which maximize the outputs entropy is derived. In particular a suitable architecture composed by two layers of flexible nonlinear functions for the separation of nonlinear mixtures is proposed. Some experimental results that demonstrate the effectiveness of the proposed neural architecture are presented. 1.

component analysis paradigm

Abstract—In this paper, neural networks based on an adaptive nonlinear function suitable for both blind complex time domain signal separation and blind frequency domain signal deconvolution, are presented. This activation function, whose shape is modified during learning, is based on a couple of spline functions, one for the real and one for the imaginary part of the input. The shape control points are adaptively changed using gradient-based techniques. B-splines are used, because they allow to impose only simple constraints on the control parameters in order to ensure a monotonously increasing characteristic. This new adaptive function is then applied to the outputs of a one-layer neural network in order to separate complex signals from mixtures by maximizing the entropy of the function outputs. We derive a simple form of the adaptation algorithm and present some experimental results that demonstrate the effectiveness of the proposed method. Index Terms—Blind deconvolution, blind separation of signals, flexible activation function, frequency domain signal deconvolution, independent component analysis, spline neural networks, unsupervised adaptive learning algorithms. I.

In this paper a novel paradigm for blind source separation in the presence of nonlinear mixtures is presented and described. The proposed approach employs a neural model based on adaptive B-spline functions. Signal separation is achieved through an information maximization criterion. Experimental results and comparison with existing solutions confirm the effectiveness of the proposed architecture.

This paper proposes the Blind Separation of complex signals using a novel neural network architecture based on an adaptive non-linear bi-dimensional activation function; the separation is obtained maximizing the output joint entropy. Avoiding the restriction due to the Louiville’s theorem, the activation function is composed by a couple of bi-dimensional spline functions, one for the real and one for the imaginary part of the signal. The surface of this function is flexible and it is adaptively modified according to the learning process performed by a gradient-based technique. The use of the bi-dimensional spline defines a new class of flexible activation functions which are bounded and locally analytic. This paper aims at demonstrate that this novel bi-dimentional complex activation function outperforms the separation in every environment in which the real and the imaginary part of the complex signal are not decorrelated. This situation is realistic in a large number of cases. Key words: Blind signal separation, independent component analysis, complex neural networks, flexible activation functions, spline neural networks. 1

Abstract – In this paper a novel paradigm for blind source separation in the presence of nonlinear mixtures is presented. In particular the paper address the problem of post nonlinear mixing followed by another instantaneous mixing system. This model is here called postnonlinear-linear (PNL-L). The method is based on the use of the recently introduced flexible activation function whose control points are adaptively changed: a neural model based on adaptive B-spline functions is employed. The signal separation is achieved through an information maximization criterion. Experimental results and comparison with existing solutions confirmed the effectiveness of the proposed architecture.

Traditional approaches to channel equalization are based on the inversion of the global (linear or nonlinear) channel response. However, in digital links the complete channel inversion is neither required nor desirable. Since transmitted symbols belong to a discrete alphabet, symbol demodulation can be effectively recasted as a classification problem in the space of received symbols. In this paper a novel neural network for digital equalization is introduced and described. The proposed approach is based on a decision-feedback architecture trained with a complex-valued discriminative learning strategy for the minimization of the classification error. Main features of the resulting neural equalizer are the high rate of convergence with respect to classical neural equalizers and the low degree of complexity. Its effectiveness has been demonstrated through computer simulations for several typical digital transmission channels.