Showing papers on "Volterra series published in 1994"

Accurate simulation of GaAs MESFET's intermodulation distortion using a new drain-source current model

Abstract: An accurate characterization of the nonlinear distortion caused by the Ids(Vgs,Vds) current in a MESFET, does not allow the common approach of splitting this nonlinear equivalent circuit element in two voltage dependent nonlinear current sources, Gm(Vgs) and Gds(Vds). By an improved laboratory characterization procedure, it was possible to extract the cross terms of the Ids(Vgs,Vds) Taylor series expansion. Measurements and Volterra series simulations, made at 2 GHz, have shown that they can give an important contribution to the prediction and understanding of MESFET's intermodulation load-pull behavior. >

Random and pseudorandom inputs for Volterra filter identification

Abstract: This paper studies input signals for the identification of nonlinear discrete-time systems modeled via a truncated Volterra series representation. A Kronecker product representation of the truncated Volterra series is used to study the persistence of excitation (PE) conditions for this model. It is shown that i.i.d. sequences and deterministic pseudorandom multilevel sequences (PRMS's) are PE for a truncated Volterra series with nonlinearities of polynomial degree N if and only if the sequences take on N+1 or more distinct levels. It is well known that polynomial regression models, such as the Volterra series, suffer from severe ill-conditioning if the degree of the polynomial is large. The condition number of the data matrix corresponding to the truncated Volterra series, for both PRMS and i.i.d. inputs, is characterized in terms of the system memory length and order of nonlinearity. Hence, the trade-off between model complexity and ill-conditioning is described mathematically. A computationally efficient least squares identification algorithm based on PRMS or i.i.d. inputs is developed that avoids directly computing the inverse of the correlation-matrix. In many applications, short data records are used in which case it is demonstrated that Volterra filter identification is much more accurate using PRMS inputs rather than Gaussian white noise inputs. >

Application of higher order spectral analysis to cubically nonlinear system identification

Abstract: In this study, digital higher-order spectral analysis and frequency-domain Volterra system models are utilized to yield a practical methodology for the identification of weakly nonlinear time-invariant systems up to third order. The primary focus is on consideration of random excitation of nonlinear systems and, thus, the approach makes extensive use of higher-order spectral analysis to determine the frequency-domain Volterra kernels, which correspond to linear, quadratic, and cubic transfer functions. Although the Volterra model is nonlinear in terms of its input, it is linear in terms of its unknown transfer functions. Thus, a least squares approach is used to determine the optimal (in a least squares sense) set of linear, quadratic, and cubic transfer functions. Of particular practical note, is the fact that the approach of this paper is valid for non-Gaussian, as well as Gaussian, random excitation. It may also be utilized for multitone inputs. The complexity of the problem addressed in this paper arises from two principal causes: (1) the necessity to work in a 3D frequency space to describe cubically nonlinear systems, and (2) the necessity to characterize the non-Gaussian random excitation by computing higher-order spectral moments up to sixth order. A detailed description of the approach used to determine the nonlinear transfer functions, including considerations necessary for digital implementation, is presented. >

Calculation of the Volterra kernels of non-linear dynamic systems using an artificial neural network

Abstract: The Volterra series is a well-known method of describing non-linear dynamic systems. A major limitation of this technique is the difficulty involved in the calculation of the kernels. More recently, artificial neural networks have been used to produce black box models of non-linear dynamic systems. In this paper we show how a certain class of artificial neural networks are equivalent to Volterra series and give the equation for the nth order Volterra kernel in terms of the internal parameters of the network. The technique is then illustrated using a specific non-linear system. The kernels obtained by the method described in the paper are compared with those obtained by a Toeplitz matrix inversion technique.

Analysis of Non-Gaussian Surge Response of Tension Leg Platforms Under Wind Loads

Abstract: The nonlinearity in the wind loading expression for a complaint offshore structure, e.g., a tension leg platform (TLP), results in response statistics that deviate from the Gaussian distribution. This paper focuses on the statistical analysis of the response of these structures to random wind loads. The analysis presented here involves a nonlinear system with memory. As an improvement over the commonly used linearization approach, an equivalent statistical quadratization method is presented. The higher-order response cumulants are based on Volterra series. A direct integration scheme and Kac-Siegert technique is utilized to evaluate the response cumulants. Based on the first four cumulants, the response probability density function, crossing rates, and peak value distribution are derived. The results provide a good comparison with simulation. A nonlinear wind gust loading factor based on the derived extreme value distribution of nonlinear wind effects is formulated.

Compensation of nonlinearities in high-density magnetic recording channels

Abstract: Volterra-series models of magnetic-saturation recording channels are used to derive readback structures that compensate for channel nonlinearities. These structures are based on a canceler of linear and nonlinear channel distortions, and can achieve significant improvement in terms of mean-square error and error probability. Proper operation of the canceler requires reliable preliminary decisions to be taken on the information symbols. These decisions are obtained by passing the received signal through a linear equalizer, then processing the equalized signal through a symbol-by-symbol detector or a Viterbi detector. By using the data obtained in [4] for magneto-resistive heads, it was found that symbol-by-symbol preliminary detection performs adequately. A Volterra model was also obtained experimentally for the recording channel generated by magneto-inductive heads that exhibit higher-order nonlinear effects. In order to recover data from this highly distorted channel the preliminary detection scheme needs a 4-state Viterbi detector. >

A stability condition for certain bilinear systems

Abstract: The authors derive a simple sufficient condition for the output of a discrete-time, time-invariant bilinear system to be bounded whenever the input signal to the system is bounded by a finite constant. >

Bilinear recurrent neural network

Abstract: A recurrent neural network and its training algorithm are proposed in this paper. Since the proposed algorithm is based on the bilinear polynomial, it can model many nonlinear systems with much more parsimony than the higher order neural networks based on Volterra series. The proposed bilinear recurrent neural network (BLRNN) is compared with multilayer perceptron neural networks (MLPNN) for time series prediction problems. The results show that the BLRNN is robust and outperforms the MLPNN in terms of prediction accuracy. >

Adaptive nonlinear recursive state-space filters

Abstract: Adaptive nonlinear filters previously reported often employ truncated Volterra series and have a finite impulse response (FIR). This paper introduces a nonlinear state-space structure for adaptive nonlinear filters. The adaptive filters are recursive and thus generally have an infinite impulse response (IIR). They are expected to be useful for many applications and are especially attractive for those with long memories where adaptive nonlinear FIR filters are too expensive to use. Efficient methods, which significantly reduce computation for gradients, have been developed to facilitate further their application in real-time signal processing. Numerical simulations have been performed to demonstrate the properties of the proposed algorithms. >

Volterra and Algebraic Approaches to the Zero Dynamics

Abstract: The zero dynamics is a property of affi.ne nonlinear systems w hich has been extensively studied during the last decade. Its properties are important in several contexts such as exact linearization, stabilization and sliding mode control. We will first give a result which considers how zeros of the sequence of transfer functions that emerge from the Laplace transform of the regular kernels in a Volterra series are connected to the zero dynamics. It is shown that if a certain factorization can be performed then zeros in the right half plane gives an unstable zero dynamics. This can be viewed as a generalization of the linear case.Further, a result is given which shows how differential algebra, in particular the Ritt algorithm, can be used to calculate zero dynamics. For a large dass of affi.ne SISO state space descriptions the Ritt algorithm, with a certain ranking, is shown to give the zero dynamics. This indicates that the concept of zero dynamics can be generalized to more complex state space descriptions.

Volterra models of ocean structures : extreme and fatigue reliability

Abstract: A new method is developed to estimate the reliability of nonlinear structural systems, modeled with second‐order Volterra series, against both extreme and fatigue failures. It includes non‐Gaussian response effects for a given input spectrum as well as long‐term variations in the parameters of this input spectrum. Moment influence coefficients are introduced to give response moments for arbitrarily scaled input spectra. Rates of extremes and fatigue damage are then estimated analytically from a non‐Gaussian (Hermite) random process model. The method is applied to study the pitch and heave “springing” motions of a tension leg platform (TLP) under random wave loads. State‐of‐the‐art nonlinear diffraction results are used to model the second‐order loads on the TLP. These forces are then applied to a coupled six‐degree‐of‐freedom (6DOF) linear structural model of the TLP. Statistics of tether tensions, platform motions, and accelerations are estimated. These results are weighted over probabilistic models of t...

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

Abstract: 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. >

Analysis of MESFET injection-locked oscillators in fundamental mode of operation

Abstract: In this paper, nonlinear analyses of MESFET oscillator in free-running and injection-locked states are studied based on the Volterra series method and stability analysis. The approach shown is capable of quantitatively analyzing the injection locking performance of oscillator using three-terminal devices in the fundamental mode of operation. Both the transmission-type and reflection-type injection locked oscillators (ILO) are simulated and experimentally verified. >

Distortion Analysis of Weakly Nonlinear Filters Using Volterra Series

Abstract: An analysis of weakly nonlinear band pass filters using Volte rra series is presented. The Volterra transfer functions for a typical Gm-C biquad are derived analytically and used to quantify and unify the distortion that arises with multiple input sinusoids, specifically gain compression, desensitization, and intermodulation disto rt on. A feedback structure that can reduce distortion is analyzed algebraically, and pract ic l examples of the structure are simulated and their distortion terms extracted using a nove l technique. The latter is applied to an actualGm-C biquad in the laboratory and measured performance agrees wi th that predicted by the Volterra analysis.

Spectral modeling of switched-mode power converters

Abstract: A new modeling approach for the spectral analysis of pulsewidth modulated (PWM) converters with independent inputs is developed. The key of this approach is to extend the Volterra functional series to nonlinear systems with multiple independent inputs. After formulating the state-space equations describing the dynamical behavior of PWM converters, the Volterra transfer function characterizing the output frequency response can be obtained, which is then symmetrised to form the spectral model. Since the model is developed in a closed form, it is suitable for computer analysis. The modeling approach has been applied to various PWM converters, and the results are verified. The spectral models of different power converters can readily be obtained by using this general approach. >

MMD-an efficient approximation to the 2nd order Volterra filter

Abstract: Nonlinear filtering based on the Volterra series expansion is very popular. A serious problem thereby is the increased filter complexity. This paper presents an efficient approximation to the 2nd order Volterra filter. The proposed filter structure is called multi memory decomposition (MMD) and is composed of 3 linear FIR filters with one multiplier. Therefore, the number of required filter operations is comparable to that of linear filters, i.e. O(N). Two algorithms for the determination of the optimal FIR coefficients of the MMD model are presented. The first one approximates the effective MMD kernel to a quadratic reference kernel. The second algorithm determines the MMD coefficients adaptively from input and output measurements. Simulations as well as real time applications show the good performance of the MMD approximation. >

An extension of the m-sequence technique for the analysis of multi-input nonlinear systems

Abstract: White-noise analysis and related methods of nonlinear systems identification describe a physical system’s response to its input in terms of “kernels” of progressively higher orders. A popular analytic scheme in the laboratory uses a class of pseudorandom binary sequences, m-sequences, as a test signal. The m-sequence method has several advantages for investigating linear and nonlinear systems: ease of implementation, rapid calculation of system kernels, and a solid theoretical framework. One difficulty with this method for nonlinear analysis comes from the algebraic structure of m-sequences: linear and nonlinear terms can be confounded, especially in the analysis of systems with many inputs. We have developed a modification of the m-sequence method which allows control of these anomalies. This method is based on input signals consisting of a superposition of m-sequences whose lengths are relatively prime. The fast computational methods which facilitate kernel calculation for a single m-sequence input are readily extended to this new setting. We describe the theoretical foundation of this method and present an application to the study of ganglion cells of the macaque retina.

Control-relevant identification of Volterra series models

Abstract: This paper considers the control-relevant identification of nonlinear models for nonlinear systems. The discussion covers: Volterra series models, uncertainty modeling for the truncation error and kernel mismatch, robustness analysis of a nonlinear internal model control system, and development of a control-relevant identification criterion. The criterion is expressed by the uncertainty coefficient and the local gain of an inverse Volterra series in such a way that the identified nominal model enables the design of a high performance controller for the plant. A numerical example is presented to illustrate the proposed technique.

Quadratic system identification using higher order spectra of i.i.d. signals

Abstract: The properties of higher order moment sequences and higher order spectral moments of an i.i.d. (independent, identically distributed) process up to fourth-order are discussed. These properties are utilized to develop algorithms to identify time-invariant nonlinear systems, which can be represented by second-order Volterra series and which are subjected to an i.i.d. input. A relatively simple solution for estimating the linear and quadratic transfer functions, which requires neither the calculation of the higher order spectral moments of the input for various frequencies nor the calculation of the inverse of matrix, is shown to exist, even though the second-order Volterra series is not an orthogonal model for an i.i.d. input (unless the input is a white Gaussian process). >

Parallel modeling and structure of nonlinear Volterra discrete systems

Abstract: A parallel modeling of the nonlinear finite extent Volterra discrete systems, which exploits the inherent symmetries and ensures fast implementation and design with the minimum computational and hardware cost, is presented. The parallel realization model is based on the successive decomposition of the kth order Volterra kernel in terms of lower order kernels, which are ordered in sequential nested subkernels. The resulting parallel realization is a tree structure with inputs the associated quadratic Volterra kernels. Each layer of the tree structure is comprised of nodes that represent the kernels of the same order and can be computed independently and simultaneously. The proposed parallel model is characterized by a great degree of modularity and regularity, since it uses only planar triangular arrays and local communications, and constitutes the basis for efficient fast implementations using VLSI array processors. >

Volterra series based nonlinear simulation of HBTs using analytically extracted models

Abstract: An accurate HBT large signal model has been developed using an analytical equivalent circuit element extraction technique. Nonlinear characteristics are analysed by means of a Volterra series and agree with experiment. Nonlinear current cancellation effects can be analysed. Their presence also enhances convergence and the applicability of Volterra series to large-signal modelling.

A practical nonlinear model for magnetic recording channels

Abstract: In this paper, a practical nonlinear model for high density magnetic recording channels is proposed. Unlike previous Volterra series methods, this model can be constructed based on simple measurements of the isolated transition response and dibit responses. The experimental results show that the proposed nonlinear model accurately predicts the channel output over the entire length of the 1023-bit maximum length pseudo-random binary testing sequence. >

A local small gain theorem and its use for robust stability of uncertain feedback Volterra systems

Abstract: The requirement to evaluate a gain over the whole signal space is one of the restrictions in the well-known small gain theorem. Using the concepts of local gain and strict causality a local form of small gain theorem is proposed, which can be used to analyze input magnitude dependent stability problems of feedback nonlinear systems, such as a Volterra system. An uncertainty model for truncation error and kernel mismatch is derived to address the robustness issue of approximating a nonlinear system by a finite Volterra series in the context of closed-loop control. The local small gain theorem is then used to analyze the feedback properties of the uncertain Volterra system and a sufficient condition for robust stability is obtained.

Identification of Multiple-Input Nonlinear Systems Using Non-White Test Signals

Abstract: The parallel cascade method (Korenberg, 1982, 1991; Palm 1979) provides an elegant means of estimating the Volterra kernels of a nonlinear system. Korenberg (1991) demonstrated methods for producing the individual paths in this parallel array that require relatively few calculations, and are hence practical for use under a variety of conditions. In this paper, we propose alternative methods for determining the paths in a parallel cascade array. Our methods prove to be more robust in the presence of output noise, at the expense of slightly slower convergence under low noise conditions.

Modeling the harmonic distortion of analog-to-digital converter using Volterra series

Abstract: This paper develops a new method to characterize the harmonic distortion of analog-to-digital converter (ADC) using Volterra series. We use Gaussian white noise as the test signal and an orthogonal method to determine the Volterra kernels, when considering quantization noise. This method is not influenced by the quantization noise, which is a serious problem when using the DFT method to measure very weak harmonic distortion. >

Modeling and identification of parallel and feedback nonlinear systems

Abstract: Structural classification and parameter estimation (SCPE) methods have been used for studying single-input single-output (SISO) parallel and feedback nonlinear system models from input-output (I-O) measurements. The uniqueness of the I-O mappings of different models and parameter uniqueness of the I-O mapping of a given structural model are evaluated. The former aids in defining the conditions under which different model structures may be differentiated from one another. The latter defines the conditions under which a given model parameter can be uniquely estimated from I-O measurements. SCPE methods presented in this paper can be further developed to study more complicated multi-input multi-output (MIMO) block-structured models which will provide useful techniques for modeling and identifying highly complex nonlinear systems. >