Second-order Volterra System Research Articles

Optimal nonlinear system identification using fractional delay second-order volterra system

The aim of this work is to design a fractional delay second order Volterra filter that takes a discrete time sequence as input and its output is as close as possible to the output of a given nonlinear unknown system which may have higher degree nonlinearities in the least square sense. The basic reason for such a design is that rather than including higher than second degree nonlinearities in the designed system, we use the fractional delay degrees of freedom to approximate the given system. The advantage is in terms of obtaining a better approximation of the given nonlinear system than is possible by using only integer delays ( since we are giving more degrees of freedom via the fractional delays ) and simultaneously it does not require to incorporate higher degree nonlinearities than two. This work hinges around the fact that if the input signal is a decimated version of another signal by a factor of M, then fractional delays can be regarded as delays by integers less than M. Using the well known formula for calculating the discrete time Fourier transform ( TFT ) of a decimated signal, we then arrive at an expression for the DTFT of the output of a fractional delay system in terms of the unknown first and second order Volterra system coefficients and the fractional delays. The final energy function to be minimized is the norm square of the difference between the DTFT of the given output and the DTFT of the output of the fractional delay system. Minimization over the filter coefficients is a linear problem and thus the final problem is to minimize a highly nonlinear function of the fractional delays which is accomplished using search techniques like the gradient-search and nature inspired optimization algorithms. The effectiveness of the proposed method is demonstrated using two nonlinear benchmark systems tested with five different input signals. The accuracy of the stated models using the globally convergent metaheuristic, cuckoo-search algorithm ( CSA ) are observed to be superior when compared with other techniques such as real-coded genetic algorithm ( RGA ), particle swarm optimization ( PSO ) and gradient-search ( GS ) methods. Finally, statistical analysis affirms the potential of the proposed designs for its successful implementation.

Optimal nonlinear system identification using fractional delay second-order volterra system

The aim of this work is to design a fractional delay second order Volterra filter that takes a discrete time sequence as input and its output is as close as possible to the output of a given nonlinear unknown system which may have higher degree nonlinearities in the least square sense. The basic reason for such a design is that rather than including higher than second degree nonlinearities in the designed system, we use the fractional delay degrees of freedom to approximate the given system. The advantage is in terms of obtaining a better approximation of the given nonlinear system than is possible by using only integer delays U+0028 since we are giving more degrees of freedom via the fractional delays U+0029 and simultaneously it does not require to incorporate higher degree nonlinearities than two. This work hinges around the fact that if the input signal is a decimated version of another signal by a factor of M, then fractional delays can be regarded as delays by integers less than M. Using the well known formula for calculating the discrete time Fourier transform U+0028 DTFT U+0029 of a decimated signal, we then arrive at an expression for the DTFT of the output of a fractional delay system in terms of the unknown first and second order Volterra system coefficients and the fractional delays. The final energy function to be minimized is the norm square of the difference between the DTFT of the given output and the DTFT of the output of the fractional delay system. Minimization over the filter coefficients is a linear problem and thus the final problem is to minimize a highly nonlinear function of the fractional delays which is accomplished using search techniques like the gradient-search and nature inspired optimization algorithms. The effectiveness of the proposed method is demonstrated using two nonlinear benchmark systems tested with five different input signals. The accuracy of the stated models using the globally convergent metaheuristic, cuckoo-search algorithm U+0028 CSA U+0029 are observed to be superior when compared with other techniques such as real-coded genetic algorithm U+0028 RGA U+0029, particle swarm optimization U+0028 PSO U+0029 and gradient-search U+0028 GS U+0029 methods. Finally, statistical analysis affirms the potential of the proposed designs for its successful implementation.

Accelerated Gradient Descent Estimation for Rational Models by Using Volterra Series: Structure Identification and Parameter Estimation

This brief studies a rational model identification problem using a novel gradient descent algorithm. The rational model is approximated by a second-order Volterra system, whose parameters and structure are estimated through an accelerated gradient descent algorithm. This algorithm improves the convergence rates by computing an optimal step-size and introducing a preconditioning method. In addition, it can avoid the eigenvalue calculation if the preconditioning matrix is equal to the inverse of the information matrix. A numerical experiment is given to illustrate the performances of the proposed algorithm.

Numeric Variable Forgetting Factor RLS Algorithm for Second-Order Volterra Filtering

Nonlinear adaptive filtering techniques for system identification (based on the Volterra model) are widely used for the identification of nonlinearities in many applications. In this correspondence, the improved tracking capability of a numeric variable forgetting factor recursive least squares (NVFF-RLS) algorithm is presented for first-order and second-order time-varying Volterra systems under a nonstationary environment. The nonlinear system tracking problem is converted into a state estimation problem of the time-variant system. The time-varying Volterra kernels are governed by the first-order Gauss–Markov stochastic difference equation, upon which the state-space representation of this system is built. In comparison to the conventional fixed forgetting factor recursive least squares algorithm, the NVFF-RLS algorithm provides better channel estimation as well as channel tracking performance in terms of the minimum mean square error (MMSE) for first-order and second-order Volterra systems. The NVFF-RLS algorithm is adapted to the time-varying signals by using the updating prediction error criterion, which accounts for the nonstationarity of the signal. The demonstrated simulation results manifest that the proposed method has good adaptability in the time-varying environment, and it also reduces the computational complexity.

Estimating second-order Volterra system parameters from noisy measurements based on an LMS variant or an errors-in-variables method

This paper deals with the identification of a nonlinear SISO system modelled by a second-order Volterra series expansion when both the input and the output are disturbed by additive white Gaussian noises. Two methods are proposed. Firstly, we present an unbiased on-line approach based on the LMS. It includes a bias correction scheme which requires the variance of the input additive noise. Secondly, we suggest solving the identification problem as an errors-in-variables issue, by means of the so-called Frisch scheme. Although its computational cost is high, this approach has the advantage of estimating the Volterra kernels and the variances of both the additive noises and the input signal, even if the signal-to-noise ratios at the input and the output are low.

Separation of Nonlinear Ultrasound Signals Based on Second-Order Volterra System Identification

This paper introduces a system identification based on second Volterra filter (SVF) as a mathematical model to separate harmonic components of a radio frequency (RF) line from a simulated nonlinear pulse–echo system (NPS). Filter coefficients of both the linear (the first-order Volterra) and quadratic (the second-order Volterra) kernels of the SVF can be determined by processing the input and output data in the frequency domain and solving a system of linear equations. Results from the validation of the approach show that while the linear kernel correctly predicts the linear output in the fundamental energy band, the quadratic kernel appropriately captures energy primarily from the second harmonic and low frequency bands. In addition, it is shown that the system identification based on the SVF is capable of separating the second order nonlinearity embedded under the level of noise signal. This is the significant advantage of the SVF approach over other static models such as linear bandpass filters. In addition, the feasibility of regularization in decreasing the number of input–output sequences used in system identification is presented.

Minimum Mean-Square Error Equalization for Second-Order Volterra Systems

In this paper, a novel nonlinear Volterra equalizer is presented. We define a framework for nonlinear second-order Volterra models, which is applicable to different applications in engineering. We use this framework to define the channel and to model the equalizer. We then solve the minimum mean-square error (MMSE) problem explicitly for the tandem connection of the two second-order Volterra systems. Optimal solutions for a simplified, linear version of the MMSE equalizer are also presented. The novel equalizer was tested when applied to a nonlinear ultra-wideband transmitted reference receiver front end. As a comparison, a least mean squares (LMS) equalizer with a training sequence has been used to verify the performance of the newly proposed equalizer. The simulation results show that the LMS equalizer is only able to attain the proposed MMSE equalizer after very long training, which might not be desirable in a communication system.

2E7a Separation of Nonlinear Ultrasound Signals Based on Second-Order Volterra System Identification(Aural English Session IV)

2E7a Separation of Nonlinear Ultrasound Signals Based on Second-Order Volterra System Identification(Aural English Session IV)

Selection of generalized orthonormal bases for second-order Volterra filters

Volterra models are very useful for signal and system representation due to their general nonlinear structure and their property of linearity with respect to their parameters, the kernel coefficients. However, when using Volterra models we are confronted with a complexity problem that results from the very large number of parameters required by such models. Expanding the kernels on a generalized orthonormal basis allows to significantly reduce this parametric complexity. In the present paper, a new constructive procedure is described for selecting such a generalized orthonormal basis in the case of second-order Volterra systems. A pruning method is also proposed for eliminating the least significant terms in the kernel expansions.

A formulation of the convergence property for the second‐order adaptive volterra filter using NLMS algorithm

AbstractIn this paper, a formulation of the convergence property of the NLMS method, an updating algorithm for an adaptive Volterra filter, is discussed. To date, much research has been carried out on the convergence property of the updating algorithm in a linear adaptive filter. However, there has been insufficient study of the updating algorithm for the second‐order adaptive Volterra filter. In this paper, the convergence property of the E‐NLMS method, an updating algorithm in the second‐order adaptive Volterra filter, is formulated and the convergence property of the second‐order adaptive Volterra filter is theoretically determined. In the formulation of the convergence property we use the first‐order recursive filter expression, which is effective for studying various characteristics of the NLMS method in linear adaptive filters. In addition, it is recognized that the nature of the input signal to the diagonal (D) elements of the second‐order Volterra system is different from that of the input to the other (ED) components. By means of computer simulation, it is confirmed that the convergence property given by the derived theoretical equation can sufficiently express the actual convergence property. © 2002 Wiley Periodicals, Inc. Electron Comm Jpn Pt 3, 85(10): 40–50, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjc.1124

Second-order adaptive Volterra system identification based on discrete nonlinear Wiener model

The authors present the nonlinear LMS adaptive filtering algorithm based on the discrete nonlinear Wiener model for second-order Volterra system identification application. The main approach is to perform a complete orthogonalisation procedure on the truncated Volterra series. This allows the use of the LMS adaptive linear filtering algorithm for calculating all the coefficients with efficiency. This orthogonalisation method is based on the nonlinear discrete Wiener model. It contains three sections: a single-input multi-output linear with memory section, a multi-input, multi-output nonlinear no-memory section and a multi-input, single-output amplification and summary section. For a white Gaussian noise input signal, the autocorrelation matrix of the adaptive filter input vector can be diagonalised unlike when using the Volterra model. This dramatically reduces the eigenvalue spread and results in more rapid convergence. Also, the discrete nonlinear Wiener model adaptive system allows us to represent a complicated Volterra system with only few coefficient terms. In general, it can also identify the nonlinear system without over-parameterisation. A theoretical performance analysis of steady-state behaviour is presented. Computer simulations are also included to verify the theory.

Second-order Volterra system identification

Second-order Volterra system identification is discussed. Crosscumulant information is converted into a Fredholm integral equation. Special emphasis is focused on inputs obtained at the output of a linear filter driven by higher order white noise.

An adaptive technique for modeling second-order Volterra systems with sparse kernels

This paper proposes a simple technique for modeling second-order Volterra systems using an adaptive second-order Volterra delay filter (ASOVDF). The developed filter structure essentially extends an adaptive FIR delay filter to include linear and quadratic filter coefficients with an input assumed to be a zero-mean i.i.d. sequence with a symmetric distribution. The implementation of the ASOVDF is based on a stage-by-stage modeling process. At each stage, a dominant delay element is determined, the corresponding adaptive filter coefficient is incorporated to the adaptive filter coefficients from previous stages, and then these filter coefficients are adapted via the recursive least-squares algorithm. The ASOVDF requires few filter coefficients and has better performance and less computational complexity over the conventional adaptive second-order Volterra filter (ASOVF) in modeling second-order Volterra systems with sparse kernels.

Adaptive second-order Volterra delay filter

A new adaptive second-order Volterra delay filter is developed to include the linear and quadratic filter coefficients for system modelling and identification. The filter structure is very effective in modelling second-order Volterra systems with sparse system coefficients.

Lattice algorithms for recursive least squares adaptive second-order Volterra filtering

This paper presents two computationally efficient recursive least-squares (RLS) lattice algorithms for adaptive nonlinear filtering based on a truncated second-order Volterra system model. The lattice formulation transforms the nonlinear filtering problem into an equivalent multichannel, linear filtering problem and then generalizes the lattice solution to the nonlinear filtering problem. One of the algorithms is a direct extension of the conventional RLS lattice adaptive linear filtering algorithm to the nonlinear case. The other algorithm is based on the QR decomposition of the prediction error covariance matrices using orthogonal transformations. Several experiments demonstrating and comparing the properties of the two algorithms in finite and "infinite" precision environments are included in the paper. The results indicate that both the algorithms retain the fast convergence behavior of the RLS Volterra filters and are numerically stable.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>