Nonlinear system identification using Gaussian inputs

The paper is concerned with the identification of nonlinear systems represented by Volterra expansions and driven by stationary, zero mean Gaussian inputs, with arbitrary spectra that are not necessarily white. Procedures for the computation of the Volterra kernels both in the time as well as in the frequency domain are developed based on cross-cumulant information. The derived kernels are optimal in the mean squared error sense for noncausal systems. Order recursive procedures based on minimum mean squared error reduction are derived. More general input output representations that result when the Volterra kernels are expanded in a given orthogonal base are also considered     

Nonlinear system identification

This paper provides an overview of nonlinear system identification methodologies. The theory and application of nonlinear system identification is vast, and this overview is not intended to be comprehensive. Rather, the attempt here is to illustrate some of the salient features and key aspects of nonlinear system identification, especially those most relevant to the practitioner. In particular, this overview focuses on important issues in nonlinear system idenfication that differ from those encountered in linear system identification, including tests for nonlinearity, model selection, and input signal considerations.     

Nonlinear vector prediction using feed-forward neural networks

The performance of a classical linear vector predictor is limited by its ability to exploit only the linear correlation between the blocks. However, a nonlinear predictor exploits the higher order correlations among the neighboring blocks, and can predict edge blocks with increased accuracy. We have investigated several neural network architectures that can be used to implement a nonlinear vector predictor, including the multilayer perceptron (MLP), the functional link (FL) network, and the radial basis function (RBF) network. Our experimental results show that a neural network predictor can predict the blocks containing edges with a higher accuracy than a linear predictor.     

Nonlinear channel modeling and identification using baseband Volterra-Parafac models

Baseband Volterra models are very useful for representing nonlinear communication channels. These models present the specificity to include only odd-order nonlinear terms, with kernels characterized by a double symmetry. The main drawback is their parametric complexity. In this paper, we develop a new class of Volterra models, called baseband Volterra-Parafac models, with a reduced parametric complexity, by using a doubly symmetric Parafac decomposition of high order Volterra kernels viewed as tensors. Three adaptive algorithms are then proposed for estimating the parameters of these models. Some Monte Carlo simulation results are presented to compare the performance of the proposed estimation algorithms, in the case of third-order baseband Volterra systems excited by PSK and QAM inputs.     

Nonlinear system identification and overparameterization effects in multisensory evoked potential studies

Traditional signal processing techniques have not been suitable in establishing contributions from different sensory paths in multisensory evoked potentials. In this paper, a nonlinear modeling technique is proposed to demonstrate the possible mechanisms of interaction between sensory paths. The nonlinear autoregressive with exogenous inputs (NARX) model is explored to establish a relationship between electrical activities of the brain obtained by unimodal and by bimodal stimulation. The intersensory phenomenon concept is extended using nonlinear system theory and applied to show the possible interactions between the visual and auditory sensory paths. In addition, the paper addresses the compensation phenomenon caused by overparameterization in the NARX algorithm when it is applied to event-related potentials. It is hoped that the nonlinear modeling approach will generate hypotheses about the intersensory interaction phenomenon, improving and advancing its theoretical formulation.     

Online system identification using matrix pseudoinverse

Recursive algorithms for online identification of discrete-time systems have been described. These provide minimum-norm estimates of the parameter vector when insufficient data are available, and least-squares estimates with sufficient data. Matrix inversion is not required; nor is the a priori knowledge of noise statistics or probability densities of the parameters.     

FIR system identification using third- and fourth-order cumulants

A new set of equations relating the coefficients of a finite-impulse-response (FIR) system and the third- and forth-order cumulants of the system output are derived. Based on these equations, two new methods to estimate FIR parameters are presented. Simulation results show that these methods perform better than other recently published linear methods in the additive coloured Gaussian noise case. This improvement is due to the fact that they do not make use of any correlation information and that they employ several slices of third- and forth-order cumulants     

Nonlinear time series modelling and prediction using Gaussian RBFnetwork with evolutionary structure optimisation

An evolutionary structure optimisation method for the Gaussian radial basis function network is presented for modelling and predicting nonlinear time series. The generalisation performance is significantly improved with a much smaller network, compared with that of the previous clustering and least square learning method     

Nonlinear dynamic system modeling using recurrent wavelet network

A recurrent wavelet network for the dynamic system nonparametric modeling is proposed in this paper. It is noted that the suitable recurrent units are introduced so that the dynamics of the wavelet network has been greatly improved. The recurrent backpropagation identification algorithm is also given. The simulation results show that regress system model with large-dimension can be better constructed and the useful guidelines for initialization of the network parameter are also provided with recurrent wavelet network identification.     

A system identification algorithm using orthogonal functions

An adaptive filter (ADF) structure is proposed for applications in which large-order ADFs are required. It is based on modeling the impulse response of the system to be identified as a linear combination of a set of discrete Legendre orthogonal functions. The proposed adaptive filter structure has desirable stability features and a unimodal mean-square error surface as well as a modular structure that permits an easy increase of the filter order without changing the previous stages. Computer simulations in which the proposed structure is used to identify actual acoustic echo path impulse responses show that the Legendre ADF has better convergence performance than the transversal ADF when identifying systems with long impulse response     

Blind identification of autoregressive system using chaos

The problem of identifying an autoregressive (AR) model using chaos is investigated here. Based on the Cramer-Rao bound (CRB) analysis, it is proved here that when chaos is used to drive an AR system, identification using only the output signal can be as good as that based on using both input and output signals, that is, blind identification is equivalent to nonblind identification. A deterministic maximum likelihood (ML) is, therefore, developed to blindly identify an AR system driven by chaos. Combined with the global search technique genetic algorithm (GA), the proposed GA-ML method is found to achieve the optimal identification performance imposed by the CRB. The theoretical mean square error (MSE) performance of the proposed GA-ML method is derived, and the result is validated using computer simulations. Compared to conventional methods based on white Gaussian driving signal, the chaos approach is shown to have superior performance. The improvement is proved to be the result of the positive and finite Lyapunov exponent of the chaotic signal. The proposed chaos identification method is applied to blind equalization of a spread spectrum (SS) communication system where chaos is used to modulate the information signal. Computer simulations show that the proposed chaos approach has a satisfactory equalization performance even under strong channel effects.     

Nonlinear time series modelling and prediction using Gaussian RBFnetworks with enhanced clustering and RLS learning

An improved clustering and recursive least squares (RLS) learning algorithm for Gaussian radial basis function (RBF) networks is described for modelling and predicting nonlinear time series. Significant performance gain can be achieved with a much smaller network compared with the usual clustering and RLS method     

LMS coupled adaptive prediction and system identification: astatistical model and transient mean analysis

The LMS algorithm has been successfully used in many system identification problems. However, when the input data covariance matrix is ill-conditioned, the algorithm converges slowly. To overcome the slow convergence, an adaptive structure is studied, which incorporates an LMS adaptive predictor (prewhitener) prior to the LMS algorithm for system identification (canceler). Since the prewhitener is also adaptive, the input to the LMS canceler is nonstationary, even when the input is stationary. Because of the coupling and the nonstationarity of LMS canceler input, analysis of the performance of the two adaptations is extremely difficult. A simple theoretical model of the coupled adaptations is presented and analyzed. First and second moment analysis indicates that the adaptive predictor significantly speeds up the LMS canceler as compared to a system without prewhitening and enlarges the stability domain of the canceler (larger allowable μ). Monte-Carlo simulations are presented which are in good agreement with the predictions of the mathematical model     

Accurate system identification using inputs with imperfectautocorrelation properties

The application of periodic pseudorandom sequences to system identification via input-output crosscorrelation is well established. Accurate estimates of system impulse response can be obtained if the pseudorandom test signal has a perfect (impulsive) periodic autocorrelation function. The authors present theoretical results which show that it is also possible to use sequences with imperfect autocorrelation functions as test inputs while maintaining identification accuracy. Simulation results are provided which verify the theoretical basis of this method     

Nonlinear distortion prediction of active filters

The represention of circuit variables in frequency-domain complex matrix and nonlinearities by power series gives rise to a nonlinear distortion prediction technique, the intermodulation-balance method. This operates entirely in the frequency domain. The technique is verified by a single-amplifier filter with a two-tone input signal     

Nonlinear system identification for cascaded block model: anapplication to electrode polarization impedance

A new algorithm has been developed to identify a system divided into cascaded blocks of dynamic linear (L), static nonlinear (N), and dynamic linear (L) subsystems based strictly on the input-output relationship. The nonlinear element is assumed to be equicontinuous, or must be satisfied by the Weierstrass criterion. Therefore, it could either be continuous type as represented by polynomial approximation or abrupt type as represented by piecewise-linear segments. The process uses a series of multilevel input to decouple the two linear subsystems from the nonlinear subsystem and then applies the predictor-corrector algorithm to minimize a cost function to obtain the parameter of the subsystem. The method does not restrict the type of input signal and no prior knowledge of the subsystems is necessary. Numerical example for a prescribed system is given and the results show almost identical values by any one of the three types of input, namely: step, sinusoidal, or white noise. Three computer programs have been developed for the identification of the system with odd, even, and piecewise abrupt types of nonlinearity. The method is applied to model the interfacial phenomenon of noble metal electrode (Pt) at the nonlinear range and the algorithm is verified by comparison with the result developed previously.     

Nonlinear adaptive prediction of nonstationary signals

We describe a computationally efficient scheme for the nonlinear adaptive prediction of nonstationary signals whose generation is governed by a nonlinear dynamical mechanism. The complete predictor consists of two subsections. One performs a nonlinear mapping from the input space to an intermediate space with the aim of linearizing the input signal, and the other performs a linear mapping from the new space to the output space. The nonlinear subsection consists of a pipelined recurrent neural network (PRNN), and the linear section consists of a conventional tapped-delay-line (TDL) filter. The nonlinear adaptive predictor described is of general application. The dynamic behavior of the predictor is demonstrated for the case of a speech signal; for this application, it is shown that the nonlinear adaptive predictor outperforms the traditional linear adaptive scheme in a significant way     

A SYSTEM IDENTIFICATION METHOD USING VECTOR SPACE BASE FUNCTION

This paper presents a new system identification approach using vector space base functions, and proposes two network structures based on Gamma sequence and Laguerre sequence. After analyzing and comparing these structures in detail, some simulation results to demonstrate the conclusions are given.     

Adaptive system identification using robust LMS/F algorithm

Adaptive system identification (ASI) problems have attracted both academic and industrial attentions for a long time. As one of the classical approaches for ASI, performance of least mean square (LMS) is unstable in low signal‐to‐noise ratio (SNR) region. On the contrary, least mean fourth (LMF) algorithm is difficult to implement in practical system because of its high computational complexity in high SNR region, and hence it is usually neglected by researchers. In this paper, we propose an effective approach to identify unknown system adaptively by using combined LMS and LMF algorithms in different SNR regions. Experiment‐based parameter selection is established to optimize the performance as well as to keep the low computational complexity. Copyright © 2013 John Wiley & Sons, Ltd.