Finite Volterra Series Research Articles

Practical realization of discrete-time Volterra series for high-order nonlinearities

Full realization of all versions of Volterra series like pure, truncated, and doubly finite Volterra series, and especially, the realization of their high orders is an intractable problem. Hence, practical implementation of Volterra series for high-order nonlinearities is not feasible with reasonable computational cost. For this reason, mathematicians, neuroscientists, and especially, biomedical and electrical engineers are forced to use only the low-order Volterra series. In this paper, we provide a full realization of off-repetitive discrete-time Volterra series (ORDVS) by departure from a traditional approach in favor of choosing a hierarchical structure. The proposed method is named fast full tantamount of off-repetitive discrete-time Volterra series (FFT-ORDVS). We have proven that the proposed off-repetitive discrete-time Volterra series approximates the basic discrete-time Volterra series very well and with much less computational complexity. In a conventional method, if $${M} +1$$ is considered as the memory length of the ORDVS, around $$2^{M}$$ math operations are needed for the full realization of it. In most cases, M is a large number and consequently, $$2^{M}$$ is too large. To solve this problem, we have proposed a simple polynomial time solution and using the proposed method, the same task is done only by 6M math operations. It means that we have found a shortcut to change an intractable problem ($${O}(2^{M}))$$ to a simple P problem (O(M)). This achievement enables researchers to use high-order kernels and consequently covers high-order nonlinearities with the lowest possible computational load. We have proven our claims mathematically and validated the performance of the proposed method using two numerical examples and a real problem.

Volterra-series-based equivalent nonlinear system method for subharmonic vibration systems

ABSTRACTSubharmonics generation in the nonlinear system, directly using the traditional finite Volterra series, cannot generally represent the aimed system. In this paper, a new approach is presented, which is an extension of single finite Volterra series for representation and analysis of the subharmonic vibration system based on equivalent nonlinear system. The equivalent nonlinear system, which is constructed by pre-compensating the subharmonic vibration system with the super-harmonic nonlinear model, yields the input–output relation between the virtual source and the response of the aimed nonlinear system. Orthogonal least square method is employed to identify the truncated order of Volterra series and predominant Volterra kernels of the equivalent nonlinear system. The MGFRFs (modified generalised frequency response functions) of the equivalent nonlinear system is obtained from the data of the virtual source and response, and verified by comparing the response estimated by the MGFRFs with its true value. Therefore, the aimed subharmonic vibration system can be analysed by taking advantage of a truncated Volterra series based on the equivalent nonlinear system. Numerical simulations were carried out, whose results have shown that the proposed method is valid and feasible, and suitable to apply on representation and analysis of subharmonic vibration systems.

Discussion of the application of finite Volterra series for the modeling of the oscillation behavior of ultrasound contrast agents

Ultrasound contrast agents consist of microbubbles with diameters in the micrometer range. Excited by ultrasound, these bubbles exhibit highly nonlinear oscillation. While well developed physical models for microbubble oscillation exist, the efficiency of pulse sequences for sensitive microbubble detection is discussed based on simple mathematical models of general nonlinearity. Typically, Taylor series are used to model microbubble nonlinearity for the development of detection schemes. Recently, pulse sequences were proposed which exploit nonlinear memory of microbubbles, a property that cannot be modeled by a Taylor series but can be explained using a Volterra series. Therefore, this paper discusses and evaluates the usage of Volterra series for the modeling of the scattering behavior of contrast agent microbubbles. A numerically stable linear estimation algorithm is implemented to determine a third order Volterra model for a free gas bubble with a resting radius r 0 = 1 μ m . For insonification pressures up to 100 kPa, the identified model allowed for a mean-square error of less than −16 dB with respect to the reference signal. Analysis of the response to narrowband signals showed that the achievable mean-square error is further reduced for the bandwidth available to typical ultrasound transducers used for clinical diagnostics.

Channel characterization methods using dipulse extraction

We present dibit extraction with a nonlinear representation of the readback signal in terms of a finite Volterra series. We show that the extracted dibit response is the linear combination of the cyclically shifted Volterra kernels. Then we explore several implications of this to dibit extraction and system characterization such as the indistinguishability of physical mechanisms from output observation, bandwidth expansion, and requirements on the length of the pseudorandom binary sequences that arise when viewing the system in the wider context of the finite Volterra series.

Uniform Approximation of Discrete-Time Periodically Varying Systems

It is shown that the elements of a large class of discrete-time periodically varying nonlinear input-output maps can be uniformly approximated arbitrarily well, over infinite time intervals, using a certain structure that can be implemented in many ways using, for example, radial basis functions, polynomial functions, piecewise linear functions, sigmoids, or combinations of these functions. 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 finite discrete-time Volterra series. Results are given for the case in which inputs and outputs are defined on the integers. The case in which inputs and outputs are defined on the nonnegative integers is also addressed.

Uniform Approximation of Periodically-Varying Systems

It is shown that the elements of a large class of periodically varying nonlinear input-output maps can be uniformly approximated arbitrarily well, over infinite time intervals, using a certain structure that can be implemented in many ways using, for example, radial basis functions, polynomial functions, piecewise linear functions, sigmoids, or combinations of these functions. 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 finite Volterra series. Results are given for the case in which inputs and outputs are defined on IR. The case in which inputs and outputs are defined on the half-line IR/sub +/ is also addressed. In both cases inputs need not be functions that are continuous.

Uniform approximation of time‐invariant non‐linear systems

AbstractAn Erratum has been published for this article in International Journal of Circuit Theory and Applications 2004; 32(6):633.It is shown that the elements of a large class of time‐invariant non‐linear input–output maps can be uniformly approximated arbitrarily well, over infinite time intervals, using a certain structure that can be implemented in many ways using, for example, radial basis functions, polynomial functions, piecewise linear functions, sigmoids, or combinations of these functions. 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 finite Volterra series. Results are given for the case in which inputs and outputs are defined on ℝ. The case in which inputs and outputs are defined on the half‐line ℝ+ is also addressed, and in both cases inputs need not be functions that are continuous. Copyright © 2004 John Wiley & Sons, Ltd.

Evaluation of Signal-to-Noise and Distortion Ratio Degradation in Nonlinear Systems

This paper presents a new figure-of-merit to evaluate signal-to-noise and distortion (SINAD) ratio degradation in nonlinear systems, herein referred to as the noise and distortion figure (NDF). In order to obtain a mathematical formula for this NDF, the best linear approximation calculation is presented for memoryless and dynamic nonlinear systems, which can be modeled by a finite Volterra series. To the best of the authors' knowledge, this is the first time such an attempt of calculating the NDF for a nonlinear and dynamic system is made. NDF results are discussed in both types of systems by means of numerical simulations of systems up to the third order.

/spl Ropf//sub +/ fading memory and extensions of input-output maps

Much is known about time-invariant nonlinear systems with inputs and outputs defined on /spl Ropf//sup +/ that possess approximately finite memory. For example, under mild additional conditions, they can be approximated arbitrarily well by the maps of certain interesting simple structures. An important fact that gives meaning to results concerning such systems is that the approximately finite memory condition is known to be often met. Here we consider the known proposition that if a causal time-invariant continuous-time input-output map H has fading memory on a certain set of bounded functions defined on all of /spl Ropf/, then H can be approximated arbitrarily well by a finite Volterra series operator. We show that in a certain sense, involving the existence of extensions of system maps, this result too has wide applicability.

Z+fading memory and extensions of input–output maps

AbstractAn Erratum for this article has been published in theInternational Journal of Circuit Theory and Applications 30(4) 2002, 179.Much is known about time‐invariant non‐linear systems with inputs and outputs defined onZ+that possess approximately‐finite memory. For example, under mild additional conditions, they can be approximated arbitrarily well by the maps of certain interesting simple structures. An important fact that gives meaning to results concerning such systems is that the approximately‐finite‐memory condition is known to be often met. Here we consider the known proposition that if a causal time‐invariant discrete‐time input–output mapHhas fading memory on a set of bounded functions defined on all of the integersZ, thenHcan be approximated arbitrarily well by a finite Volterra series operator. We show that in a certain sense, involving the existence of extensions of system maps, this result too has wide applicability. Copyright © 2001 John Wiley & Sons, Ltd.

Separation conditions and approximation of continuous-time approximately finite memory systems

We consider causal time-invariant nonlinear input-output maps that take a set of locally pth-power integrable functions into a set of real-valued functions, and we give criteria under which these maps can be uniformly approximated arbitrarily well using a certain structure consisting of a not-necessarily-linear dynamic part, followed by a nonlinear memoryless section that may contain sigmoids or radial basis functions, etc. In our results, certain separation conditions, of the kind associated with the Stone-Weierstrass theorem, play a prominent role. Here they emerge as criteria for approximation, not just sufficient conditions under which an approximation exists. As an application of the results and for p=2 we show that system maps of the type addressed can be uniformly approximated arbitrarily well by certain doubly finite Volterra-series approximants if and only if these maps have approximately finite memory and satisfy certain continuity conditions.

Separation conditions and approximation of discrete-time and discrete-space systems

We first consider causal time-invariant nonlinear input-output maps that take a set of bounded functions into a set of real-valued functions, and we give criteria under which these maps can be uniformly approximated arbitrarily well using a certain structure consisting of a not necessarily linear dynamic part followed by a nonlinear memoryless section that may contain sigmoids or radial basis functions, etc. In our results certain separation conditions of the kind associated with the Stone-Weierstrass theorem play a prominent role. Here they emerge as criteria for approximation, and not just as sufficient conditions under which an approximation exists. As an application of the results, we show that system maps of the type addressed can be uniformly approximated arbitrarily well by doubly finite Volterra-series approximantsif and only if these maps have approximately finite memory and satisfy certain continuity conditions. Corresponding results are then given for (not necessary causal) multivariable input-output maps. Such multivariable maps are of interest in connection with image processing.

Design method of exact model matching control for finite Volterra series systems

This paper is concerned with exact model matching control (EMM) for finite Volterra series systems. First, we show a structure of a causal controller which can achieve the exact model matching to a reference model and clarify the relationship between the proposed method and the EMM for linear systems. Second, to analyse the stability of the proposed control system, we present an input dependent small gain theorem for the system with an external input, then extend it for the system with two external inputs. With the help of this theorem, we clarify the condition under which the control system is stable for the reference input magnitude within a certain range, and is also robust for small disturbances. Finally, the effectiveness of the proposed method is illustated through numerical simulations.

IIR Volterra filtering with application to bilinear systems

A new nonlinear filtering technique by means of infinite impulse response (IIR) Volterra functionals is developed. It yields the projection onto the closed class of finite Volterra series with separable kernels of arbitrary degree k. Such an optimal estimator is finitely realizable as a bilinear system with parameters that are computable off line. Moreover, if the original system model is itself bilinear, this computation is finitely recursive through higher moments of degree 2 k. Two simple illustrating examples are developed: (i) estimation of the covariance of the internal white noise driving a linear system and (ii) filtering of a non-Gaussian linear system (driven by a Poisson process). The robustness with respect to the observation noise distribution is finally examined.

I.I.R. Volterra Filtering with Application to Bilinear Systems

A new nonlinear filtering technique by means of infinite impulse response (IIR) Volterra functionals is developed. It yields the projection onto the closed class of finite Volterra series with separable kernels of arbitrary degree k. Such an optimal estimator is finitely realizable as a bilinear system with parameters that are computable off line. Moreover, if the original system model is itself bilinear, this computation is finitely recursive through higher moments of degree 2 k. Two simple illustrating examples are developed: (i) estimation of the covariance of the internal white noise driving a linear system and (ii) filtering of a non-Gaussian linear system (driven by a Poisson process). The robustness with respect to the observation noise distribution is finally examined.

有限ボルテラ級数システムに対する厳密なモデルマッチング

有限ボルテラ級数システムに対する厳密なモデルマッチング

Adaptive Model Matching for Finite Volterra Series System

In this paper, the model matching problem is investigated and its application to the model reference adaptive control (MRAC) for a class of finite Volterra series system is also considered. At first, a class of finite Volterra series system which has a bilinear realization is treated, and a scheme for matching the input-output relation to a given linear reference model is presented by using a bilinear observer. Secondly, a model reference adaptive control scheme is proposed. Furthermore, the exact model matching problem for an extended class of the finite Volterra series system is also discussed.

Nonlinear channel models for digital magnetic recording

Methods for identifying the digital magnetic recording channel in which data dependent nonlinear effects are present are described. The models used include the finite state machine (FSM), Volterra series, and channel state dependent transition shifting. Procedures to compute the model parameters are described, and the models are compared with a view to their accuracy and use for studying receiver performance. Results are included from prototype channels with various recording densities with significant nonlinear distortion. >

Uniform approximation and the circle criterion

The input-output maps G of causal time-invariant systems are considered, and a complete characterization of those maps that can be uniformly approximated by the maps of certain simple structures is given. The criterion is that G must satisfy certain continuity and approximately-finite-memory conditions. It is proved that the conditions are satisfied by the input-output maps of control systems of a familiar type containing a sector nonlinearity for which the circle condition for stability is met. In particular, this shows that such feedback systems, with inputs drawn from a certain large set of bounded functions, possess arbitrarily good finite Volterra-series (or radial-basis-function) approximations.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Uniform approximation with doubly finite Volterra series

The assumption that a system possesses a certain discrete-time Volterra series representation frequently forms the basis for studies in the areas of signal processing and communication theory. A further assumption often made, always without discussion, is that the representation can be suitably approximated by a corresponding 'double finite' series. It is shown that, for a very large class of nonlinear discrete-time systems, such doubly finite approximations exist in the sense of uniform approximation on a ball of bounded inputs. Several additional results are also given. These concern, for example, asymptotic properties of the expansions. The results provide a more firm foundation for applications involving Volterra series.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>