Showing papers on "Volterra series published in 1992"

Quadratic filters for signal processing

Abstract: Polynomial (or Volterra) filters are introduced, and the quadratic filters are presented as the simplest example of such filters. The principle aspects and properties of quadratic filters are derived in the framework of the discrete Volterra expansion. Fixed as well as adaptive filters are considered in one-dimensional and multidimensional environments. Such issues as design and efficient realizations are thoroughly addressed, and standard and advanced adaptation algorithms are presented. Several examples of signal processing applications requiring quadratic filters are discussed. >

Pseudorandom signals to estimate apparent transfer and coherence functions of nonlinear systems: applications to respiratory mechanics

Abstract: For pseudorandom (PRN) input stimuli, general expressions are derived for the apparent transfer (Z) and coherence ( gamma /sup 2/) functions of nonlinear systems that can be represented by a Volterra series. To avoid the problems that are shown here to be associated with harmonic distortions and to minimize the influence of crosstalk, a family of pseudorandom signals which are especially suited for the estimation of Z and gamma /sup 2/ in mechanical measurement of physiological systems at low frequencies is proposed. The components in the signals cannot be reproduced as linear combinations of two or more frequency components of the input. In a second-order system, this completely eliminates the bias, while in higher order but not strongly nonlinear systems, the interactions among the components are reduced to such a level that the response can be considered as if it were measured with independent sine waves of an equivalent amplitude. It is also shown that the values of gamma /sup 2/ are not appropriate for assessing linearity of the system. The theory is supported by simulation results and experimental examples. >

A nonlinear integral model of electron devices for HB circuit analysis

Abstract: A technology-independent large-signal model of electron devices, the nonlinear integral model (NIM), is proposed. It is rigorously derived from the Volterra series under basic assumptions valid for most types of electron devices and is suitable for harmonic-balance circuit analysis. Unlike other Volterra-based approaches, the validity of the NIM is not limited to weakly nonlinear operation. In particular, the proposed model allows the large-signal dynamic response of an electron device to be directly computed on the basis of data obtained either by conventional measurements or by physics-based numerical simulations. In this perspective, it provides a valuable tool for linking accurate device simulations based on carrier transport physics and harmonic-balance circuit analysis algorithms. Simulations and experimental results, which confirm the validity of the NIM, are also presented. >

Mechanisms determining third order intermodulation distortion in AlGaAs/GaAs heterojunction bipolar transistors

Abstract: The third-order intermodulation distortion (IMD3) mechanisms of heterojunction bipolar transistors (HBTs) are analyzed using Volterra series theory. A T-equivalent circuit is used for the large-signal model of the HBT. The third-order nonlinear currents generated by the device nonlinearities are evaluated for this purpose and current cancellation is discussed. It is found that, even though the C/sub je/ and g/sub je/ related currents do not show pronounced cancellation, the total base-emitter current and the total base-collector current cancel partially. Second harmonic loading is addressed in view of IMD3 optimization while at the same time maintaining high gain through conjugate matching at the fundamental frequency. IMD3 is very sensitive to the nonlinear currents generated by g/sub je/ and alpha . Optimum IMD3 occurs at high second-harmonic reflection coefficients corresponding to open load conditions. Minimum and maximum IMD3 occurs for second-harmonic load reflection coefficient phases close to analogous extremes of the dominant nonlinear current of the device. >

Mathematical Tools and Software for Analysis and Design of Nonlinear Control Systems

Abstract: The theory of nonlinear control systems has developed rapidly

Uniform approximation with doubly finite Volterra series

Abstract: 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. >

Identification of linear systems in the presence of nonlinear distortions

Abstract: A method is presented to measure and identify a linear system in the presence of nonlinear distortions. The method is based on a two-step approach. In the first step the influence of nonlinear systems up to degree 4 is eliminated. In the second step the remaining linear system is identified using a weighted least squares method. The kernel of the proposed technique is the excitation of the system with a pure sinusoid. Special attention is paid to the elimination of higher harmonics in the excitation signal which are due to the nonlinear load of the generator.

Non-linear multi-degree-of-freedom system random vibration by equivalent statistical quadratization

Abstract: The statistical linearization method is often inadequate for estimating spectral properties of random responses of non-linear systems. This is sometimes due to the fact that the power spectra of responses of linear systems span only the frequency range of the excitation spectrum, whereas significant responses outside this range are possible for non-linear systems. Recently, the concept of the statistical “quadratization” method was introduced to address this shortcoming of the linearization methods. The effectiveness of statistical quadratization was demonstrated on several single-degree-of-freedom systems. In this paper the method is generalized to multi-degree-of-freedom systems. The statistical quadratization solution procedure involves replacing the non-linear system by an “equivalent” system with polynomial non-linearities up to quadratic order. The non-linear equivalent system has a form whose solutions can be approximated by using the Volterra series method. The non-Gaussian joint response probability distribution is approximated by a third-order Gram-Charlicr expansion. The method is formulated for systems with general non-linearities and with non-linearities of a special form. To demonstrate the usefulness of the method, solutions are obtained for a specific system. The corresponding results compare well with Monte Carlo simulation data. Further, it is shown that the quadratization method is notably superior to the linearization method for the considered system.

Identification of a nonlinear system modeled by a sparse Volterra series

Abstract: An algorithm based on recursive approximation and estimation is proposed for the identification of nonlinear systems which can be modeled by a sparse Volterra series. The algorithm detects the terms of the Volterra series on which the output depends and estimates the associated Volterra kernels using a least squares criterion. The performance of the algorithm is primarily dependent on the number of nonzero Volterra kernels and not on their distribution in the whole series. The input sequence can be either i.i.d. or correlated. The algorithm can also be directly applied to the delay estimation of a sparse finite impulse response (FIR) filter. >

Identification of Volterra systems with a polynomial neural network

Abstract: The Volterra series finds wide application in the general representation of nonlinear systems. A method of identifying linear and second-order time-invariant nonlinear systems is proposed using a variation of the group method of data handling (GMDH) algorithm, a polynomial network, employing a combination of quadratic polynomial and linear layers. The principal advantage of this method is that the degree of nonlinearity and the memory of the system do not have to be known a priori and are determined recursively. The GMDH method allows a Volterra series to be modeled solely from a set of input-output data. System identification using GMDH consists of applying a set of input-output data to train the network by computing the necessary coefficient sets and to select the optimum combination of these coefficient sets to obtain the model parameters. >

Neural network modeling of nonlinear systems based on volterra series extension of a linear model

Abstract: A Volterra series approach has been applied to the identification of nonlinear systems which are described by a neural network model. A procedure is outlined by which a mathematical model can be developed from experimental data obtained from the network structure. Applications of the results to control of robotic systems are discussed. >

Nonlinear Incoherent Spectroscopy: Noisy

Abstract: The basic treatment of spectroscopy with incoherent excitation is summarized. The response in coherently pulsed experiments is compared to the nonlinear correlation functions of stochastic NMR. The processing of the stochastic response is reviewed as well as the analysis of nonlinear cross-correlation functions. The design of a nonlinear interferometer (cross-correlator) for NMR spectroscopy is outlined. Experimental results are presented. The incoherent optical echo is treated in the Volterra functional series. The stochastic analogs of pump-and-probe experiments are reviewed as examples of nonlinear interference. Finally, it is shown how COSY-like 2D spectra can be generated by linear interference of the nonlinear response with part of the excitation

Transformation of the absolute descriptions of linear and non-linear processes to their incremental forms

Abstract: The discrete-time process models usually contain functions of the input signal. For control purpose it is practical if the model equation contains the increments of the input signal instead of the input signal itself. Transformation equations from the absolute (or non-incremental) to the incremental description form are given for both linear and non-linear, non-parametric and parametric models. The non-linear models include the Hammerstein and Volterra series, the generalized Hammerstein, the parametric Volterra, the bilinear and the linear-in-input signal non-linear-in-output signal models. Several examples of second-order, linear and quadratic models are also presented.

Pipelined Volterra filter

Abstract: A pipelined realisation for a kth order Volterra filter is introduced. Although it needs input data broadcasting, the realisation uses a minimum number of simple identical processing elements (PEs), produces a fixed output delay equal to the order of the filter, and is suitable for implementing adaptive Volterra filters.

Adaptive polynomial filtering algorithms

Abstract: This dissertation presents several algorithms for adaptive nonlinear filters equipped with polynomial system models. In particular, truncated Volterra and bilinear system models are considered. These models are capable of representing a wide class of nonlinear systems. A fast, recursive least-squares (RLS) second-order Volterra filter and a fast extended RLS adaptive bilinear filter are introduced. These algorithms exploit the ideas employed for developing fast RLS adaptive multichannel filters and have computational complexity that is an order of magnitude lower than the most efficient previously available RLS algorithms. A theoretical performance analysis of the steady-state behavior of the second-order Volterra filter operating in both stationary and nonstationary environments is presented. The analysis shows that, when the input is zero-mean, Gaussian distributed and the adaptive filter is operating in a stationary environment, the steady-state excess mean-squared error is independent of the statistics of the input signal. The idea of developing fast RLS higher-order Volterra filters is also discussed. Despite the recursive system model employed, the extended RLS adaptive bilinear filter is shown to be stable whenever the desired response signal is bounded. Simulation results that exhibit good parsimony in the use of coefficients of the bilinear filters over the Volterra filters are presented. Mainly motivated by the computational simplicity, gradient-type output-error adaptive bilinear filters are also developed. These algorithms are among a large class of adaptive bilinear filtering algorithms that exhibit stability problems. Several input-dependent stability conditions for the output of a bilinear system to be bounded when the input is bounded are presented. A set of simple sufficient stability conditions of a very general bilinear system model is introduced. The derivation shows that the bilinear system model considered has a unique Volterra series expansion, and then utilizes the convergence concepts associated with the Volterra series expansion. An efficient implementation of the stability test is then introduced. A set of necessary stability conditions for a general bilinear system model and a set of sufficient and necessary stability conditions for a specific bilinear system model are also derived. These conditions are attractive because they are very easy to check. The stability conditions presented in this dissertation are essential for developing a large class of adaptive bilinear filters.

Nonlinear system identification using multilayer perceptrons with locally recurrent synaptic structure

Abstract: It is proved that a multilayer perceptron (MLP) with infinite impulse response (IIR) synapses can represent a class of nonlinear block-oriented systems. This includes the well-known Wiener, Hammerstein, and cascade or sandwich systems. Previous methods used to model these systems such as the Volterra series representation are known to be extremely inefficient, and so the IIR MLP represents an effective method of modeling block-oriented nonlinear systems. This was demonstrated by simulations on two models within the class. The significance of the IIR MLP is that it demonstrates that a useful range of systems can be modeled by a network architecture based on the MLP and adaptive linear filters. >

New techniques for nonlinear system identification : a rapprochement between neural networks and linear systems

Abstract: System identification is an area of research that has important implications for many problems Techniques for system identification have been developed over a long period of time, with significant contributions made by Gauss (1890), Mener (1942), Kahnan (1956), and many others since It is known however, that many physical processes are intrinsically nonlinear, and cannot be adequately approximated by linear models Various methods have been devised to overcome this, including a generalization of the linear transfer function model to the nonlinear case, resulting in the Volterra series Describing Functions, and linearization techniques using the extended Kalman filter have also been popular There are disadvantages with these models however: the Volterra series has the problem of requiring a large number of terms, and in many cases a nonlinear model is required rather than a linearization of the system about some operating point A number of other techniques have been proposed which include the State-Dependent Model, Bilinear model Threshold Autoregressive model, and Exponential Autoregressive model These models in one way or another attempt to generalize the linear transfer function description to the nonlinear case More recently, neural networks have undergone rapid development and become recognized as powerful nonlinear approximation methods Specifically, the proof that there exists a network, within the class of two-layer multilayer perceptrons, that is capable of approximating any nonlinear function to an arbitrary degree of accuracy has been a major milestone The usefulness of this model has raised the question of whether it is possible to extend its static modelling capabilities to time-dependent systems Methods which have been proposed to do this, include the use of sampled raw input data fed to several input units, transforms of the input data, thereby producing a "static" representation of the input data, (that is, at least for the duration of the window, as in a Short-Time Fourier Transform (STFT)) Other techniques have included the use of feedback connections in the network, and time-delays in various pathways In spite of the desire to produce time-depedent models, there has been little use made of the existing principles of adaptive filter theory, and system identification This thesis develops a new set of nonlinear system identification techiuques, biased on a rational approach to combining the powerful nonlinear approximation capabilities of multilayer perceptrons, with the known methods of adaptive filtering and system identification It is shown that this approach is successful, through the introduction of new nonlinear models with useful properties which extend the capabilities of existing linear adaptive filters, and neural networks models for time-dependent dataKeywords: Neural Networks, Multilayer Perceptrons, Nonlinear System Identification, Adaptive Filters

Analysis and optimization of third order intermodulation distortion mechanisms in AlGaAs/GaAs heterojunction bipolar transistors

Abstract: The third-order intermodulation distortion (IMD3) mechanisms of HBTs (heterojunction bipolar transistors) are analyzed using Volterra series theory. The third-order nonlinear currents generated by the device nonlinearities are evaluated for this purpose. Second-harmonic loading is addressed in view of IMD3 optimization while, at the same time, maintaining high gain through conjugate matching at the fundamental frequency. It is shown that IMD3 depends on a complex process involving interactions between various nonlinear elements and is highly sensitive to C/sub bc/ generated nonlinear current. The interaction of the latter with the other HBT elements significantly affects the IMD3. Optimum IMD3 occurs at high second-harmonic reflection coefficients corresponding to open load conditions. An IMD3 improvement of up to 27 dBm can be obtained by proper loading. >

Estimation of nonlinear distortion using digital higher-order spectra and Volterra series

Abstract: The concept of linear and nonlinear coherence functions is utilized to evaluate the second- and third-order distortion of a time-invariant nonlinear system, which can be represented by a Volterra series up to the third order, without assuming a Gaussian input. The proposed approach considers the most significant nonlinear distortion products (second-order harmonic, third-order harmonic, second-order intermodulation, and some forms of third-order intermodulation distortion), which are typical in audio components such as loudspeakers, to obtain relatively accurate estimates with reasonable data requirements and computational complexity. This method has been applied to a loudspeaker at low frequencies to model and qualify the linear response, second-order distortion, and third-order distortion. >

Identification of second-order Volterra filters driven by non-Gaussian stationary processes

Abstract: Some recent results relating to system identification are described and illustrated in this contribution. The system considered is nonlinear and time-invariant, being represented by a Volterra series up to second order. Closed-form expressions for the transfer functions of first and second order are derived for a class of non-Gaussian stationary input processes. It is shown that the obtained parameters are optimum in the mean square sense. Once the system is identified, we derive a closed-form expression for the quadratic coherence that is a measure of the goodness of fit of the quadratic model. It is shown that this expression simplifies to well known results when the system is linear or its input is Gaussian. Furthermore, we develop estimates for the transfer functions and the quadratic coherence from spectral and bispectral estimates, based on averaged periodograms and biperiodograms of data stretches of the observed input and output of the system. This method is tested and validated by using simulated input-output data of a known quadratically nonlinear system, with known input signal statistic, Finally, we discuss the problem of testing a specified value of the quadratic coherence.

Comment on “Identification of a constrained nonlinear hydrological system described by Volterra Functional Series” by J. Xia

Abstract: In his recent paper Xia [1991] presents an interesting penalty function technique for the identification of kernel functions for watershed systems which are represented by Volterra series. It is unfortunate that the published results of research that has been carried out in hydrology on rainfall­ runoff modeling by means of Volterra series in the last 13 years were not taken into account. Hence, there are some errors and misconceptions in the paper. It is silently assumed in the paper by Xia that the water­ shed system can be represented by a lumped nonlinear operator P, which maps a space of lumped rainfalls into a space of corresponding flood waves at a measuring station on a river. If this unknown operator P is differentiable as many times as may be required in the Frechet sense, and the system is deterministic, time invariant, initially relaxed and nonanticipating, then its behavior can be approximated within the radius of convergence to any predetermined accuracy by a truncated Volterra series [Volterra, 1959]:

Modeling the linear and nonlinear behavior of gallium arsenide p-i-n diode control circuits

Abstract: A model describing the linear and nonlinear behavior of GaAs p-i-n diodes that includes the effects of the highly anisotropic i-region carrier mobilities is presented. A Volterra series approach is used in the analysis. The nth-order transfer functions of the forward biased p-i-n diode are evaluated to the third order. These transfer functions are used in computing the first-order impedance of the device and the second- and third-order distortion intercept points. Experimental measurements show good agreement with the theoretical calculations, verifying the approach used in the model. >

Estimation errors of volterra kernels measured by use of nonwhite input

Abstract: It is well known that employment of nonwhite input in system identification often causes large estimation errors. While the mechanism of inducing these errors has been well understood for linear system identification, it has not been made clear yet for nonlinear systems. This paper analyzes this mechanism for nonlinear system identification with Volterra functional series models. A vector space approach gives a clear physical interpretation of the error generation from a viewpoint of the excitation intensity of the input against the identified system. Based on this analysis, new algorithms of reducing estimation errors are proposed.

Two-tone vs. random process inputs for nonlinear distortion estimation

Abstract: Two methods (two-tone input approach and random process input approach) for the estimation of harmonic and intermodulation distortion of nonlinear systems are compared. The random input approach, where a nonlinear system is modeled by a second-order Volterra series, is examined in terms of its statistical properties, and its advantages and limitations over the classical two-tone input approach. Experimental results are shown where these two approaches are applied to evaluate second-order distortion of a loudspeaker and to compare the performance of these approaches in terms of Volterra kernels and distortion factors. >

Equivalent Statistical Quadratization for Multi-Degree-of Freedom Nonlinear Systems

Abstract: The statistical linearization method is often inadequate for estimating spectral properties of random responses of nonlinear systems. This is sometimes due to the fact that the power spectra of responses of linear systems span only the frequency range of the excitation spectrum, whereas significant responses outside this range are possible for nonlinear systems. Recently, the concept of a statistical “quadratization” method was introduced to address this shortcoming of the linearization methods. The effectiveness of statistical quadratization was demonstrated on several single-degree-of-freedom systems. In this paper the method is generalized to multi-degree-of-freedom systems. The statistical quadratization solution procedure involves replacing the nonlinear system with an “equivalent” system with polynomial nonlinearities up to quadratic order. The nonlinear equivalent system has a form whose solutions can be approximated by using the Volterra series method. The non-Gaussian joint response probability distribution is approximated by a third order Gram-Charlier expansion. The method is formulated for systems with general nonlinearities and with nonlinearities of a special form. To demonstrate the method, solutions are obtained for a specific system. The corresponding results compare well with Monte-Carlo simulation data. Further, it is shown that the quadratization method is notably superior to the linearization method for the considered system.

On nonlinear model algorithmic controller design

Abstract: The nonlinear control applications to high angle-of-attack aircraft, as reported here, is of a preliminary nature. However, the analysis does suggest that nonlinear adaptive control can be quite effective to stabilize large rapid maneuvers in angle of attack. Of the comparisons made, the on-linear, nonlinear-time-series and adaptation performed the best and was quite superior to a similar linear MAC.

Nonlinear identification and control of muscle relaxant dynamics.

Abstract: The work reported in this thesis comprised two major parts which are: 1) Off-line nonlinear identification of muscle relaxant dynamics, 2) Simulation-based design of a variety of controllers (ranging from classical PID to nonlinear self-tuners) for the closed-loop control of muscle relaxation. Relaxant drugs namely, Vecuronium and Atracurium are considered throughout. Off-line identification studies, using two special nonlinear identification packages (Nonlinear Identification package and Nonlinear Orthogonal Identification package), were carried out to determine nonlinear difference equation models (NARMAX) that best fit (in the least squares sense) recorded data from trials on humans and dogs for each drug. After validation, these models were assumed to represent, in a nonlinear polynomial form, the muscle relaxant drugs pharmacology. Two different approaches were explored for determining the physiological structure of both relaxant drugs: a) The drug model to comprise a pharmacokinetics part to represent the drug distribution, and pharmacodynamics which are often modelled by using the well known Hill equation. b) A cross-correlation approach based on Volterra series. With the relaxant dynamics structure thus fixed, the work proceeded to the control phase. Simple three-term PID controllers were first designed with their parameters being optimised, off-line, using the Simplex method. The non-adaptive nature of this class of controllers makes their robustness open to question when the system parameters for which they have been optimised change. Hence adaptive controllers in the form of linear and nonlinear generalised minimum variance, self-tuners, generalised predictive and nonlinear k-step ahead predictive controllers were also considered. All these latter control approaches are shown to be satisfactory, in terms of transient and steady state performance.

Advanced techniques in current signature analysis

Abstract: In general, both ac and dc motors can be characterized as weakly nonlinear systems, in which both linear and nonlinear effects occur simultaneously. Fortunately, the nonlinearities are generally well behaved and understood and an be handled via several standard mathematical techniques already well developed in the systems modeling area; examples are piecewise linear approximations and Volterra series representations. Field measurements of numerous motors and motor-driven systems confirm the rather complex nature of motor current spectra and illustrate both linear and nonlinear effects (including line harmonics and modulation components). Although previous current signature analysis (CSA) work at Oak Ridge and other sites has principally focused on the modulation mechanisms and detection methods (AM, PM, and FM), more recent studies have been conducted on linear spectral components (those appearing in the electric current at their actual frequencies and not as modulation sidebands). For example, large axial-flow compressors ({approximately}3300 hp) in the US gaseous diffusion uranium enrichment plants exhibit running-speed ({approximately}20 Hz) and high-frequency vibrational information (>1 kHz) in their motor current spectra. Several signal-processing techniques developed to facilitate analysis of these components, including specialized filtering schemes, are presented. Finally, concepts for the designs of advanced digitally based CSA units are offered, whichmore » should serve to foster the development of much more computationally capable ``smart`` CSA instrumentation in the next several years. 3 refs.« less