Showing papers on "Volterra series published in 1983"

Measuring Volterra kernels

Abstract: Volterra series have been in the engineering literature for some time now, and yet there have been few attempts to measure Volterra kernels. This paper discusses techniques for measuring the Volterra kernels of weakly nonlinear systems. We introduce a new quick method for measuring the second Volterra kernel which is analogous to pseudonoise testing of a linear device. To illustrate the discussion we present an experimental example, an electro-acoustic transducer. Throughout the paper we emphasize the practical aspects of kernel measurement.

On Volterra expansions for time-varying nonlinear systems

Abstract: We extend in an important direction recent results by this writer which show for the first time the existence of a locally convergent Volterra series representation for the input-output relation of a certain large class of nonlinear systems containing an arbitrary finite number of nonlinear elements. (Systems of the type considered arise, for example, in the area of communication system modeling.) The extension covers cases in which the linear part of the system is of a more general type that arises often, and unavoidably leads to considerably more complicated general expressions for the terms in the corresponding series expansions.

An adaptive nonlinear digital filter with lattice orthogonalization

Abstract: A novel approach to nonlinear filtering with minimum mean square error criterion is presented. This method considers the class of nonlinear filters with Volterra series structures under the assumption that filter inputs are Gaussian, and a relatively simple solution results which is directly applicable in many practical situations. Moreover, two simple parameter adaption algorithms for the second order Volterra filter are presented and it is shown that their convergence speeds depend on the squared ratio of maximum to minimum eigenvalues of the input autocovariance matrix. Finally, the lattice orthogonalization of filter input is considered for faster convergence.

Nonlinear stochastic problems

Abstract: I - Introduction.- Overview of the ASI on Nonlinear Stochastic Problems.- II - Spectral Estimation.- Estimation of structured covariance matrices: A generalization of the Burg technique.- Classification of radar clutter using the maximum-entropy method.- A simple approach to high-resolution spectral analysis.- III - Identification.- Estimation of stochastic parameters for ARMA models by fast filtering algorithms.- Convergence study of two real-time parameter estimation schemes for nonlinear system.- Approximation by a sum of complex exponentials utilizing the pencil of function method.- Minimax estimation of ARMA systems.- IV - System Theory.- On the structure of minimal Markovian representations.- A brief tutorial on calculus on manifolds, with emphasis on applications to identification and control.- Role of multiplicative non-white noise in a nonlinear surface catalytic reaction.- V - Adaptive / Stochastic Control.- Geometric aspects of the convergence analysis of identification algorithms.- Multivariable adaptive regulators based on multistep quadratic cost functionals.- Overparametrization, positive realness and multistep minimum variance adaptive regulators.- Adaptive receding horizon controllers for discrete stochastic systems.- An explicit solution to a problem in nonlinear stochastic control involving the Wiener process.- VI - Optimal Control.- On optimal control for a class of partially-observed systems.- Optimal stochastic control of linear systems with state and control dependent noise: efficient computational algorithms.- VII - Nonlinear Filtering.- Joint information and demodulation.- Generalized finite-dimensional filters in discrete time.- Nonlinear filtering equation for Hilbert space valued processes.- Optimal orthogonal expansion for estimation I: signal in white Gaussian noise.- Optimal orthogonal expansion for estimation II: signal in counting observations.- Approximations for nonlinear filtering.- Phase demodulation: a nonlinear filtering approach.- Volterra series and finite dimensional nonlinear filtering.- VIII - Stochastic Processes.- Causal invertibility: an approach to the innovations problem.- Spectral analysis of nonlinear semi-Markov processes.- Differential calculus for Gaussian random measures.- Finite dimensional causal functionals of Brownian motion.- Transience, recurrence and invariant measures for diffusions.- Eigenfunction expansion for the nonlinear time dependent Brownian motion.- Point process differentials with evolving intensities.- Transformation properties of stochastic differential equations.- Some Applications of stochastic calculus on the nuclear spaces to the nonlinear problems.- IX - Applications.- Reconstruction and compression of two dimensional fields from sampled data by pseudo-potential functions.- Optimum perturbation signal that makes nonlinear systems approach to linear systems.- Stochastic filtering problems in multiradar tracking.- Population extinction probabilities and methods of estimation for population stochastic Differential Equation Models.- Dynamic ship positioning control systems design including nonlinear thrusters and dynamics.- Joint optimization of transmitter and receiver for cyclostationary random signal processes.- Fundamental properties and performance of nonlinear estimators for bearings-only target tracking.- List of Participants.- Index of Subjects.- Index of Authors.

Hopf bifurcation via Volterra series

Abstract: The Hopf bifurcation theorem gives a method of predicting oscillations which appear in a nonlinear system when a parameter is varied. There are many different ways of proving the theorem and of using its results, but the way which is probably the most useful, to control and system theorists, uses Nyquist loci in much the same way as the describing function method does. The main advantages of this method are dimensionality reduction, which eases the calculation, and the ability to cope with higher-order approximations than are used in the original Hopf theorem. This paper shows how such an approach to the Hopf bifurcation follows naturally and easily from Volterra series methods. Such use of Volterra series in nonlinear oscillations appears to be new. In many problems, the calculations involved are simplified when the Volterra series approach is taken, so the approach has practical merits as well as theoretical ones.

Computer-aided distortion analysis of switched capacitor filters in the frequency domain

Abstract: A Volterra series-based distortion analysis technique for switched capacitor circuits is presented. The algorithm has been implemented in the DIANA-SC program and is completely compatible with other DIANA simulation modes. The efficiency of the method is based on the use of direct z-domain and compaction methods, while only one extra circuit analysis is needed for each higher order distortion fraction of interest. Nonlinear operational amplifiers and capacitors can be handled using polynomial models. Both harmonic and intermodulation distortion analysis modes are available. The method is demonstrated using a practical design example.

Relationship between Volterra series and generalized power series

Abstract: The relationship between the Volterra nonlinear transfer functions of a system and the elements of its generalized power series is established. A formula is derived which enables the Volterra nonlinear transfer functions to be obtained from the power series expansion of the nonlinear system.

Linearization about constant operating points: An input-output viewpoint

Abstract: For a class of stationary nonlinear systems that can be described by Volterra series, input-output definitions of the familiar notions of a constant operating point and the corresponding linearized system are discussed. Some recent results are reviewed with emphasis on the motivation for the input-output formulation. Then, to illustrate the utility of the approach, the problem of finding the linearization of an interconnection of nonlinear systems in terms of the subsystem linearizations is considered.

On the concept of derivatives and Taylor expansions for nonlinear input-output systems

Abstract: Whenever the time interval is not fixed, i.e., when the dynamics is taken into account, Volterra series are not functional Taylor expansions as too often falsely asserted. It is shown here that with respect to a new "causal" derivative, which is related to the shuffle product, noncommutative generating power series are genuine Taylor expansions. An implicit function theorem is proved.

Volterra series analysis of intermodulation distortion in second-order active filters

Abstract: The nonlinear performance of second-order active filters using single and two operational amplifiers at high frequencies and/or large-signal levels is investigated. A general procedure using the Volterra series is presented to allow comparison of different filter types regarding intermodulation distortion, and to provide means for improvement. Experimental and theoretical results compare favorably, demonstrating the accuracy of the analysis technique. It is shown that intermodulation distortion for single amplifier filters is much larger than for two-amplifier circuits.

Mode identification of bilinear systems

Abstract: A method is presented for the identification of the eigenvalues of systems characterized by bilinear differential equation models. The identification procedure is based on the existence of a zeroth harmonic, or DC shift, in the output of non-linear systems perturbed by a periodic input. The magnitude of the DC shift is a function of system properties and of the input waveform and frequency. After the appropriate equations based on a truncated Volterra series representation of a general bilinear system are presented, the regression problem associated with the identification procedure is discussed in detail. It is shown that in many cases good approximations of the system eigenvalues are obtained. Furthermore, a minimal order for any proposed system model is obtained. The method is robust in the sense that it employs system inputs which are relatively easy to generate. Also, the method requires only time averages of the system output, thereby minimizing the effects of measurement noise.

Theory of nonlinear systems

Abstract: This paper gives a general review of the Theory of Nonlinear Systems. In 1960, the author presented a paper “Theory of Nonlinear Control” at the First IFAC Congress at Moscow. Professor Norbert Wiener, who attended this Congress, drew attention to his work on the synthesis and analysis of nonlinear systems in terms of Hermitian polynomials in the Laguerre coefficients of the past of the input. Wiener's original idea was to use white noise as a probe on any nonlinear system. Applying this input to a Laguerre network gives u1, u2,…, us, and then to a Hermite polynomial generator gives V(α)'s. Applying the same input to the actual nonlinear system gives output c(t). Putting c(t) and V(α)'s through a product averaging device, we get c(t)V(α) = A α 2л s 2 , where the upper bar denotes time average and Aα's can be considered as characteristic coefficients of the nonlinear system. A desired output z(itt) may replace c(itt) to get a new set of Aα's. The Volterra functional method suggested by Wiener in 1942 has been greatlydeveloped from 1955 to the present. The method involves a multi-dimensional convolution integral with multi- dimensional kernels. The associated multi-dimensional transforms are given by Y.H. Ku and A.A. Wolf (J. Franklin Inst., Vol. 281, pp. 9–26, 1966). Wiener extended the Volterra functionals by forming an orthogonal set of functionals known as G-functionals, using Gaussian white noise as input. Volterra kernels and Wiener kernels can be correlated and form the characteristic functions of nonlinear systems. From an extension of the linear system to the nonlinear system, the input-output crosscorrelation φxy can be shown to be equal to the convolution of system impulse response h1 with the autocorrelation φxx. Using the white noise as input, where its power density spectrum is a constant, say, A, the crosscorrelation is given by φxy(σ) = Ah1(σ), while the autocorrelation is φxx(τ) = Au(τ). This extension forms the basis of an optimum method for nonlinear system identification. Measurement of kernels can be made through proper circuitry. Parallel to the Volterra series and the Wiener series, another series based on Taylor-Cauchy transforms developed since 1959 are given for comparison. The Taylor-Cauchy transform method can be applied in the analysis of simultaneous nonlinear systems. It is noted that the Volterra functional method and the Taylor-Cauchy transform method give identical final results. A selected Bibliography is appended not only to include other aspects of nonlinear system theory but also to show the wide application of nonlinear system characterization and identification to problems in biology, ecology, physiology, cybernetics, control theory, socio- economic systems, etc.

Harmonic distortion in a one-dimensional p-n-p transistor

Abstract: A method to obtain nonlinear distortion of small a.c. signals in a one-dimensional bipolar transistor is described. The fundamental physical semiconductor equations are the basis from which the small-signal relations are derived. A finite difference scheme is employed to achieve space-discretization; temporal dependence of input-output relations is given in terms of Volterra functional series. The internal distribution of quasi-Fermilevels, potential and excess carrier densities is discussed and some computational results are produced. The influence of frequency, bias point, and external circuit elements on distortion properties is illustrated. Throughout, a common-emitter p-n-p transistor structure is assumed. The theory of Volterra series analysis is briefly summarized.

Finite Dimensional Deterministic Nonlinear Filters via Riccati Transformation and Volterra Series

Abstract: Filtering in two classes of nonlinear differential and difference systems is studied. The system structures admit the representation of the optimal recursive filter in the form of a finite dimensional differential or difference system. The filtering problem is posed as a set of fixed interval optimization problems. The deterministic least-squares problem statement results in the filter which is an ordinary nonstochastic differential or difference system driven by the observed signal. No stochastic concepts are used. The system classes considered are described by two linear subsystems with a polynomial link map between them. In one class the link map affects the input of the latter subsystem. In the other class the coefficient matrix of the latter subsystem is a polynomial in the state of the preceding subsystem satisfying a Lie algebraic nilpotency condition. The former class is a special case of the systems described by finite Volterra series for which finite dimensionality of the optimal filter is also ...

The relationships between different measures of nonlinear distortion in single-amplifier active filters

Abstract: The different nonlinear distortion measures in single-amplifier active filters are discussed and the relationships between these measures given. The analysis presented is based on the Volterra-Wiener series approach used in many articles to model nonlinear circuits and systems with mild nonlinearities. Examples of active RC filters illustrate the relationships developed.

Volterra Transfer Functions from Pulse Tests for Mildly Nonlinear Channels.

Abstract: : Multichannel communications systems are often mildly nonlinear, hence they are characterizable by the Volterra series. The methodology described herein represents a numerical implementation of an RADC in-house concept formulation for pulse testing linear and quadratic Volterra systems. This analytic formulation, in terms of the appropriate convolutions, expressed the linear and quadratic responses to square pulse input waveforms. These responses contain, in canonic form, the system poles and residues which are then determined by suitable identification methods and algorithms to provide the Volterra Transfer functions. In this paper we describe a method for determining the Volterra transfer-functions H1 (s1) and H2 (s1, s2) from pulse tests. The method involves two transient tests in the laboratory, followed by analysis by the computer. The latter consists of (a) pole determination using the pencil-of-functions method, and (b) computation of the residues by a least-squares technique. Advantages of the method include the rapidity of the laboratory tests, as contrasted with traditional frequency-scan approaches, and the explicit determination of the transfer functions. Furthermore, the method is readily extendible to H3 (s1, s2, s3) and even to higher order transfer functions, although the computations grow very rapidly for these cases. (Author)

Volterra transfer functions from pulse tests for mildly nonlinear channels

Abstract: : Multichannel communications systems are often mildly nonlinear, hence they are characterizable by the Volterra series. The methodology described herein represents a numerical implementation of an RADC in-house concept formulation for pulse testing linear and quadratic Volterra systems. This analytic formulation, in terms of the appropriate convolutions, expressed the linear and quadratic responses to square pulse input waveforms. These responses contain, in canonic form, the system poles and residues which are then determined by suitable identification methods and algorithms to provide the Volterra Transfer functions. In this paper we describe a method for determining the Volterra transfer-functions H1 (s1) and H2 (s1, s2) from pulse tests. The method involves two transient tests in the laboratory, followed by analysis by the computer. The latter consists of (a) pole determination using the pencil-of-functions method, and (b) computation of the residues by a least-squares technique. Advantages of the method include the rapidity of the laboratory tests, as contrasted with traditional frequency-scan approaches, and the explicit determination of the transfer functions. Furthermore, the method is readily extendible to H3 (s1, s2, s3) and even to higher order transfer functions, although the computations grow very rapidly for these cases. (Author)

Volterra Series and Finite Dimensional Nonlinear Filtering

Abstract: Filtering in a class of nonlinear differential systems described by two subsystems connected with a polynomial link map is studied. The first subsystem is linear. The second one is given by a finite Volterra series driven by polynomials of the state of the preceeding system. The filtering problem is posed in a deterministic framework as a set of fixed interval optimization problems. All the systems studied are described by ordinary non-stochastic differential equations. A stochastic interpretation of the problem statement is discussed. It is shown that the given system structure admits finite dimensional representation for the optimal recursive filter. Explicit nonlinear filters are constructed in some examples. Comparisons with the corresponding stochastic systems and optimal finite dimensional conditional mean filters are performed. An extension to the observations described by an implicit state-linear equation is also discussed.