Discrete Volterra Series Research Articles

Nonlinear growth dynamics of date palms responding to environmental parameters

Scientific analysis of plant growth helps in improving the efficiency of cultivation practices through optimization of their environmental conditions. The ultimate aim of this research was to derive an optimal policy for better growth of date palms by considering its dynamical response to environmental parameters such as solar radiation, soil moisture, and temperature. Field experiments were conducted at an irrigation scheme located in the Jordan Rift Valley. A drip irrigation system is installed to water ten trees of date palms either with fresh or saline water depending on the soil matric potential. The circumference of the trunk of a tree was measured using a dendrometer at 30 mins interval and recorded in a data logger. Environmental parameters including the soil matric potential, solar radiation, and soil temperature were also logged every 30 mins. This study focused on determining a nonlinear model representing the growth dynamics of the date palm tree responding to those environmental parameters. The linear regression was applied to estimate the kernel coefficients of discrete Volterra series modeling the time series. The non-linearity of the model is expected to explain diurnal shrinkage and swelling of the tree trunk under different environmental conditions.

Fault Diagnosis of an Analog Circuit Based on Hierarchical DVS

Analog circuit fault diagnosis technology is widely used in the diagnosis of various electronic devices. The basic strategy is to extract circuit fault characteristics and then to use a clustering algorithm for diagnosis. The discrete Volterra series (DVS) is a common feature extraction method; however, it is difficult to calculate its parameters. To solve the problem of feature extraction in fault diagnosis, we propose an improved hierarchical Levenberg–Marquardt (LM)–DVS algorithm (IDVS). First, the DVS is simplified on the basis of the hierarchical symmetry of the memory parameters, the LM strategy is used to optimize the coefficients, and a Bayesian information criterion based on the symmetry of entropy is introduced for order selection. Finally, we propose a fault diagnosis method by combining the improved DVS algorithm and a condensed nearest neighbor algorithm (CNN) (i.e., the IDVS–CNN method). A simulation experiment was conducted to verify the feature extraction and fault diagnosis ability of the IDVS–CNN. The results show that the proposed method outperforms conventional methods in terms of the macro and micro F1 scores (0.903 and 0.894, respectively), which is conducive to the efficient application of fault diagnosis. In conclusion, the improved method in this study is helpful to simplify the calculation of the DVS parameters of circuit faults in analog electronic systems, and provides new insights for the prospective application of circuit fault diagnosis, system modeling, and pattern recognition.

Recovery of the Input Signals of Eddy-Current Displacement Transducers Under Thermal-Shock Actions

We propose a method for the recovery of input signals of eddy-current displacement transducers under thermal-shock action. The recovery of the input signals was performed with correction of the output data of a primary measuring transducer. The procedure of correction is based on the methods of identification of nonlinear dynamic systems described by the discrete Volterra series.

Obtaining Spectrum Matching Time Series Using a Reweighted Volterra Series Algorithm (RVSA)

In this paper, we introduce a novel algorithm for morphing any accelerogram into a spectrum matching one. First, the seed time series is re-expressed as a discrete Volterra series. The first-order Volterra kernel is estimated by a multilevel wavelet decomposition using the stationary wavelet transform. Second, the higher-order Volterra kernels are estimated using a complete multinomial mixing of the first-order kernel functions. Finally, the weighting of every term in this Volterra series is optimally adapted using a Levenberg–Marquardt algorithm such that the modified time series matches any target response spectrum. Comparisons are made using the SeismoMatch algorithm, and this reweighted Volterra series algorithm is demonstrated to be considerably more robust,matching the target spectrum more faithfully. This is achieved while qualitatively maintaining the original signal’s non-stationary statistics, such as general envelope, time location of large pulses, and variation of frequency content with time.

Multilayer Perceptrons: Approximation Order and Necessary Number of Hidden Units

This paper considers the approximation of sufficiently smooth multivariable functions with a multilayer perceptron (MLP). For a given approximation order, explicit formulas for the necessary number of hidden units and its distributions to the hidden layers of the MLP are derived. These formulas depend only on the number of input variables and on the desired approximation order. The concept of approximation order encompasses Kolmogorov-Gabor polynomials or discrete Volterra series, which are widely used in static and dynamic models of nonlinear systems. The results are obtained by considering structural properties of the Taylor polynomials of the function in question and of the MLP function.

ESTIMATION OF PARSIMONIOUS DISCRETE VOLTERRA SERIES

The discrete Volterra series represents an important model for the representation, analysis and synthesis of nonlinear dynamical systems. A major problem with this approach to system identification is that the number of terms to be estimated grows combinatorially with memory length and polynomial degree often leading to redundancy, over-fitting and numerical difficulties. Many attempts have been made to recover only the salient terms in the expansion but none is, to our knowledge, optimal. In this paper we apply a simple, iterative approach derived from a majorization-minorization technique that delivers parsimony as a result of optimization and allows a trade-off between model accuracy and sparsity. While this is viable for problems of moderate size, a new algorithm is derived that requires no matrix inversion and is suited to larger scale systems. The method is demonstrated, successfully, on synthetic data.

A Method for Parallel Adaptive Volterra Filter Using RLS Algorithm

In recent years, the adaptive Volterra filter (AVF) based on discrete Volterra series has been used to identify a nonlinear system, remove nonlinear distortion etc. Since AVF is an extension of the conventional linear digital filters, the algorithms used in linear filters are easily used in AVF, and AVF is sufficiently approximate to the complicated systems due to its general function expansion. However, AVF has a problem that the calculation work becomes extremely large for long memory length, or for filtering high-order nonlinear characteristic. This paper presents a type of parallel adaptive Volterra filter (PAVF) to decrease the calculation complexity. Finally, to illustrate the effectiveness of the proposed method, the results of computer simulation with recursive least squares (RLS) algorithm are presented.

IDENTIFICATION OF INFINITE DEGREE VOLTERRA SERIES IN THE TIME AND FREQUENCY DOMAINS

Volterra series (VS) are widely used in non-linear dynamical system identification. Much physical information about a system can be extracted from the corresponding VS model. Most non-linear frequency domain representations have been based on VS models through the application of the Fourier transform. But the fact that the number of the parameters to be identified in a VS model increases exponentially with the size of the VS model restricts its application. The involvement of kernel methods has been shown significantly to reduce the burden of the computation, with the potential to increase the practical usability of VS and the methods that are based on it. This paper presents the identification of infinite degree, finite memory length, time-invariant, discrete VS from the general reproducing kernel Hilbert space point of view and introduces its extension to the estimation of generalized frequency response functions.

Error propagation and recovery in decision-feedback equalizers for nonlinear channels

Nonlinear intersymbol interference is often present in communication and digital storage channels. Decision-feedback equalizers (DFEs) can decrease this nonlinear effect by including appropriate nonlinear feedback filters. Although various applications of these types of equalizers have been published in the literature, the analysis of their stability and error recovery has not appeared. We consider a DFE with a nonlinear feedback filter based on a discrete Volterra series. We extend error propagation, error probability, stability, and error recovery time results for Nth order nonlinear channels.

Notes on uniform approximation of shift-varying systems

It is shown that the elementsG of a large class of input-output maps can be uniformly approximated arbitrarily well using a certain structure if and only ifG is continuous. For the case considered the system inputs and outputs are defined on a discrete set {0, 1,...,a1}×...{0, 1,...,am}, in which a1,...,am are positive integers. Our approximating structure involves certain functions that can be chosen in different ways. For the special case in which these functions are taken to be certain polynomial functions, the input-output map of our structure is a generalized discrete Volterra series. Our results provide an analytical basis for the use of such series.

Identification of discrete Volterra series using maximum length sequences

An efficient method is described for the determination of the Volterra kernels of a discrete nonlinear system. It makes use of the Wiener general model for a nonlinear system to achieve a change of basis. The orthonormal basis required by the model is constructed from a modified binary maximum sequence (MLS). A multilevel test sequence is generated by time reversing the MLS used to form the model and suitably summing delayed forms of the sequence. This allows a sparse matrix solution of the Wiener model coefficients to be performed. The Volterra kernels are then obtained from the Wiener model by a change of basis.

Structural identification of quadratic block-oriented models based on estimated Volterra kernels

Block-oriented models having only one multiplier and the Volterra kernels that describe them are summarized. The kernel estimation of the discrete Volterra series is dealt with. It is pointed out that while the Volterra kernels can easily be derived from the block-oriented models, the structure and the parameters of the block-oriented models cannot be computed in a trivial way from the estimated Volterra kernels. Both graphical and analytical procedures are recommended for the structural identification. The method is supported by several simulation examples.

Functional expansions for nonlinear discrete-time systems

Given an input-output map associated with a nonlinear discrete-time state equationx(t + 1) =f(x(t);u(t)) and a nonlinear outputy(t) =h(x(t)), we present a method for obtaining a “discrete Volterra series” representation of the outputy(t) in terms of the controlsu(0), ...,u(t − 1). The proof is based on Taylor-type expansions of the iterated composition of analytic functions. It allows us to make an explicit construction of each kernel, that is, each coefficient of the series expansion ofy(t) in powers of the controls. This is achieved by making use of successive directional derivatives associated with a family of vector fields which are deduced from the discrete state equations. We discuss the use of these vector fields for the analysis and control of nonlinear discrete-time systems.

A computational method for the design of 2-D nonlinear Volterra filters

A class of nonlinear 2-D filters based on the truncated discrete Volterra series is considered, together with a suited matrix notation. An optimization method is developed for the design of the filters according to the required input/output relation. The performance of different optimization algorithms is studied, specifically, the method of steepest descent, Powell's conjugate directions algorithm, and simulated annealing. It is found that the Powell technique is the most suited to the problem. A design example, together with the results obtained after processing a test image by the proposed filters and other standard techniques, is used to compare the performance of the three methods.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A variable-step adaptation algorithm for memory-oriented Volterra filters

A recently proposed adaptation algorithm for digital non-linear adaptive filters based on the truncated discrete Volterra series is modified and improved in order to get a faster and more accurate convergence. Performance examples are presented.

Finite Volterra-series realizations and input-output approximations of non-linear discrete-time systems

Abstract It is shown that a separability property of a given finite family of stationary kernels turns out to be necessary and sufficient for the realization of the associated functional by means of a polynomial affine system (i.e. a system that is polynomial in the input and affine in the state). Moreover, the discrete Volterra kernels associated with the input-output map of a non-linear analytic discrete-lime system, initialized at an equilibrium point, are shown to possess a separability property. On this basis, we state an approximation result for the given input-output map by considering the first kernels of the discrete Volterra series. Two explicit constructions of the approximating polynomial affine systems are proposed.

Adaptive nonlinear digital filters using distributed arithmetic

A nonlinear adaptive filter structure, based upon the theory of the truncated discrete Volterra series, is presented. A memory-oriented implementation exploiting distributed arithmetic is considered, and the conventional LMS adaptation algorithms are suitably modified. Memory-size reduction methods are developed to obtain simpler actual realizations. Computer simulation results are presented.

Edge extraction by a class of second-order nonlinear filters

The application of a class of nonlinear 2-D digital filters, based on the discrete Volterra series, as edge detectors is considered. Independence on edge orientation is achieved by the use of symmetry conditions on the coefficients. Two examples are presented.

Nonlinear digital filter realization by distributed arithmetic

It is assumed that a nonlinear shift-invariant digital filter is represented by a truncated discrete Volterra series. A matrix notation, based on the properties of the Kronecker product of matrices, is derived for the nonlinear operator of order K in the Volterra expansion. The matrix notation shows how the well-known distributed arithmetic, originally proposed for linear digital filters, can be adopted for the realization of nonlinear operators. The characteristics of combinatorial and memory-oriented structures are compared, and the problem of complexity reduction is discussed.

On the realization of nonlinear discrete-time systems

It is shown, on the basis of a compact expression of the kernels of the discrete Volterra series associated to a linear analytic discrete-time system, that the kernels satisfy a suitable property which enables their inductive characterization. We also give a first result on the realization of a given family of functions by a linear analytic discrete-time system with linear invertible drift term.