Showing papers on "Volterra series published in 1996"

Output Frequency Characteristics of Nonlinear Systems

Abstract: Some interesting properties of output frequencies of Volterra-type nonlinear systems are particularly investigated. These results provide a very novel and useful insight into the super-harmonic and inter-modulation phenomena in output frequency response of nonlinear systems, with consideration of the effects incurred by different nonlinear components in the system. The new properties theoretically demonstrate several fundamental output frequency characteristics and unveil clearly the mechanism of the interaction (or coupling effects) between different harmonic behaviors in system output frequency response incurred by different nonlinear components. These results have significance in the analysis and design of nonlinear systems and nonlinear filters in order to achieve a specific output spectrum in a desired frequency band by taking advantage of nonlinearities. They can provide an important guidance to modeling, identification, control and signal processing by using the Volterra series theory in practice.

The identification of nonlinear biological systems: Volterra kernel approaches.

Abstract: Representation, identification, and modeling are investigated for nonlinear biomedical systems. We begin by considering the conditions under which a nonlinear system can be represented or accurately approximated by a Volterra series (or functional expansion). Next, we examine system identification through estimating the kernels in a Volterra functional expansion approximation for the system. A recent kernel estimation technique that has proved to be effective in a number of biomedical applications is investigated as to running time and demonstrated on both clean and noisy data records, then it is used to illustrate identification of cascades of alternating dynamic linear and static nonlinear systems, both single-input single-output and multivariable cascades. During the presentation, we critically examine some interesting biological applications of kernel estimation techniques.

Simulation of a class of non-normal random processes

Abstract: This study addresses the simulation of a class of non-normal processes based on measured samples and sample characteristics of the system input and output. The class of non-normal processes considered here concerns environmental loads, such as wind and wave loads, and associated structural responses. First, static transformation techniques are used to perform simulations of the underlying Gaussian time or autocorrelation sample. An optimization procedure is employed to overcome errors associated with a truncated Hermite polynomial transformation. This method is able to produce simulations which closely match the sample process histogram, power spectral density, and central moments through fourth order. However, it does not retain the specific structure of the phase relationship between frequency components, demonstrated by the inability to match higher order spectra. A Volterra series up to second order with analytical kernels is employed to demonstrate the bispectral matching made possible with memory models. A neural network system identification model is employed for simulation of output when measured system input is available, and also demonstrates the ability to match higher order spectral characteristics.

Nonlinear image operators for the evaluation of local intrinsic dimensionality

Abstract: Local intrinsic dimensionality is shown to be an elementary structural property of multidimensional signals that cannot be evaluated using linear filters We derive a class of polynomial operators for the detection of intrinsically 2-D image features like curved edges and lines, junctions, line ends, etc Although it is a deterministic concept, intrinsic dimensionality is closely related to signal redundancy since it measures how many of the degrees of freedom provided by a signal domain are in fact used by an actual signal Furthermore, there is an intimate connection to multidimensional surface geometry and to the concept of 'Gaussian curvature' Nonlinear operators are inevitably required for the processing of intrinsic dimensionality since linear operators are, by the superposition principle, restricted to OR-combinations of their intrinsically 1-D eigenfunctions The essential new feature provided by polynomial operators is their potential to act on multiplicative relations between frequency components Therefore, such operators can provide the AND-combination of complex exponentials, which is required for the exploitation of intrinsic dimensionality Using frequency design methods, we obtain a generalized class of quadratic Volterra operators that are selective to intrinsically 2-D signals These operators can be adapted to the requirements of the signal processing task For example, one can control the "curvature tuning" by adjusting the width of the stopband for intrinsically 1-D signals, or the operators can be provided in isotropic and in orientation-selective versions We first derive the quadratic Volterra kernel involved in the computation of Gaussian curvature and then present examples of operators with other arrangements of stop and passbands Some of the resulting operators show a close relationship to the end-stopped and dot-responsive neurons of the mammalian visual cortex

A general framework for quadratic Volterra filters for edge enhancement

Abstract: An inherent problem in most image enhancement schemes is the amplification of noise, which, due to Weber's law, is mostly visible in the darker portions of an image. Using a special class of quadratic Volterra filters, we can adapt the enhancement process in a computationally efficient way to the local image brightness because these filters are approximately equivalent to the product of a local mean estimator and a highpass filter. We analyze and derive this subclass of quadratic Volterra filters by investigating the 1-D case first, and then we generalize the results to two dimensions. An important property of these filters is that they map sinusoidal inputs to constant outputs, which allows us to develop a new filter characterization that is more intuitive for our application than the 4-D frequency response. This description finally leads to a novel least-squares design methodology. Image enhancement results using our Volterra filters are superior to those obtained with standard linear filters, which we demonstrate both quantitatively and qualitatively.

Computational complexity of Volterra based nonlinear compensators

Abstract: The Volterra series can be used to represent a wide class of nonlinear systems with memory. A Volterra inverse can be used to apply post (or pre)-distortion for the purpose of nonlinear compensation. The authors determine and compare the computational complexity of two such compensators: one based on the analytical pth order Volterra inverse, the other on the adaptive Volterra inverse.

Identification of structurally constrained second-order Volterra models

Abstract: Continuing advances in industrial control hardware and software have increased interest in the identification of nonlinear dynamic models from chemical process data. Two practically important issues are those of nonlinear model structure selection and input sequence design for adequate model identification. While the "general nonlinear model identification problem" is intractably complex, we may obtain useful insights into both of these issues by restricting our attention to special cases, building incrementally on our "linear intuition". We consider the special case of second-order Volterra models, focusing on the effects of structural restrictions and non-Gaussian input sequences on the model identification problem. The results presented build on the work of Powers and his co-workers, who considered the unconstrained Gaussian problem, certain constrained special cases (e.g., the Hammerstein model), and identification using non-i.i.d. input sequences. Besides extending these results to a wider class of model structure constraints and input sequences, the results presented yield some useful insights into the issue of input sequence design.

Identification of discrete Volterra series using maximum length sequences

Abstract: 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.

Advanced Concepts in Adaptive Signal Processing

Abstract: Preface. 1: Introduction and Background. 1.1. Common Adaptive Concepts from Different Disciplines. 1.2. Generic Applications of Adaptive Methods. 1.3. Performance Measures in Adaptive Systems. 1.4. The Minimum Mean Squared Error Solution. 1.5. Adaptive Algorithms for FIR Systems. 1.6. Adaptive Algorithms for IIR Systems. 1.7. New Horizons in Adaptive Signal Processing. 1.8. Notation and Conventions. 2: Advanced Algorithms for 1-D Adaptive Filtering. 2.2. Data- Reusing LMS Algorithms. 2.3. Orthogonalization by PR Modulation. 2.3. The Gauss-Newton Adaptive Filtering algorithm. 2.4. Block Adaptive IIR Filters Using the PCG Method. 3: Structures and Algorithms for Two-Dimensional Adaptive Signal Processing. 3.1. Applications of Two-Dimensional Adaptive Filtering. 3.2. Two- Dimensional FIR Adaptive Filtering. 3.3. Two-Dimensional IIR Adaptive Filters. 3.4. Two-Dimensional IIR Adaptive Filtering Experiments. 3.5. Uniqueness Characteristics of the 2-D IIR MSE Minimization. 4: Adaptive Fault Tolerance. 4.1. Application of AFT to FIR Adaptive Filters. 4.2. Adaptive Filter Structures. 4.3. A Simple Fault Tolerant FIR Adaptive Filter. 4.4. The Transform Domain FTAF. 4.5. The DFT-Based TDFTAF with the Conjugate Gradient. 4.6. Robust and Practical TDFTAFs. 4.7. Full Fault Tolerance Transforms. 4.8. Discussion. 5: Adaptive Polynomial Filters. 5.1. The Volterra Series. 5.2. Gradient Based Adaptive Volterra Filters. 5.3. RLSSecond-Order Volterra Adaptive Filter. 5.4. LS Lattice Second-Order Volterra Adaptive Filter. 5.5. QR-Based LS Lattice Second Order Volterra Filter. 5.6. The Adaptive Volterra Filter for Gaussian Signals. 5.7. Other Polynomial-Based Nonlinear Adaptive Filters. 5.8. Discussion. Appendix. Subject Index.

Adaptive Volterra filters using orthogonal structures

Abstract: This paper presents an adaptive Volterra filter that employs a recently developed orthogonalization procedure of Gaussian signals for Volterra system identification. The algorithm is capable of handling arbitrary orders of nonlinearity P as well as arbitrary lengths of memory M for the system model. The adaptive filter consists of a linear lattice predictor of order N, a set of Gram-Schmidt orthogonalizers for N vectors of size P+1 elements each, and a joint process estimator in which each coefficient is adapted individually. The complexity of implementing this adaptive filter is comparable to the complexity of the system model when N is much larger than P, a condition that is true in many practical situations. Experimental results demonstrating the capabilities of the algorithm are also presented in the paper.

Behavioral modeling of narrowband microwave power amplifiers with applications in simulating spectral regrowth

Abstract: A new behavioral model for narrowband microwave power amplifiers is proposed. Analytic expressions for the gain compression (AM-AM) and amplitude dependent phase distortion (AM-PM) of a nonlinear amplifier are derived from a third-order Volterra series model. The cases of a single-tone and of a two-tone signal are explored. We show that the gain compression characteristics of nonlinear amplifiers depend on the amplitude modulation characteristics of the signal. Furthermore, we show that the time-averaged phase deviation is independent of the modulation envelope. This justifies the new model proposed for obtaining the envelope transfer characteristics by applying the Bessel-Fourier technique only to the AM-AM characteristic. This model is verified by comparing spectral regrowth simulations of digitally-modulated signals to those measured in a 1.9 GHz GaAs FET power amplifier.

High-order system identification with an adaptive recursive second-order polynomial filter

Abstract: In this letter, an adaptive recursive nonlinear filter based on the Volterra series and an infinite impulse response (IIR) structure is considered. For certain types of nonlinear systems where high-order nonlinearities are recursively generated, we show that the adaptive recursive second-order polynomial filter has improved performance over the well-known (nonrecursive) adaptive second-order Volterra filter and a third-order Volterra filter. This filter represents an alternative to using a traditional Volterra filter whose order has been increased to match that of the system being modeled.

Theory and applications of adaptive second order IIR Volterra filters

Abstract: An adaptive nonlinear filter based on a second order Volterra series and on an IIR filter structure is presented. This filter is able to model higher than second order nonlinearities for systems where the nonlinearities are harmonically related. This solution represents an alternative to using higher than second order Volterra filters. We present a full derivation of this gradient search based adaptive nonlinear filter and also highlight the various assumptions and simplifications which require to be made in order to produce a practical algorithm. A comparison is made in terms of the performance and computational complexity between an adaptive second order IIR Volterra filter and an adaptive second and third order Volterra filters.

Experimental validation of Volterra series nonlinear modelling for microwave subcarrier optical systems

Abstract: A detailed comparative experimental and theoretical study of nonlinear distortion in laser diodes is undertaken to provide practical proving of the effectiveness and utility of the Volterra series method. The observed harmonic and intermodulation distortion levels are compared with the results given by the Volterra series nonlinear laser model using intrinsic parameters extracted from reflection and small-signal frequency response measurements. Second-harmonic, two-tone and three-tone intermodulation distortion data are all found to be in good agreement with the theoretical results, providing endorsement of the Volterra series approach for the study of distortion in microwave subcarrier optical systems.

Adaptive Volterra filtering with complete lattice orthogonalization

Abstract: The article presents a new recursive least squares (RLS) adaptive nonlinear filter, based on the Volterra series expansion. The main approach is to transform the nonlinear filtering problem into an equivalent multichannel, but linear, filtering problem. Then, the multichannel input signal is completely orthogonalized using sequential processing multichannel lattice stages. With the complete orthogonalization of the input signal, only scalar operations are required, instability problems due to matrix inversion are avoided and good numerical properties are achieved. The avoidance of matrix inversion and vector operations reduce the complexity considerably, making the filter simple, highly modular and suitable for VLSI implementations. Several experiments demonstrating the fast convergence properties of the filter are also included.

A modified Volterra series approach for the characterization of non-linear dynamic systems

Abstract: The paper presents a theoretical study on the convergence properties of the new modified Volterra-like modeling approach for non-linear dynamic systems in comparison with the classical Volterra series representation. In particular, the assumptions which enable a single-fold non-linear convolution integral to be adopted also in the presence of strong non-linearities are pointed out. Experimental and simulation results which confirm the theoretical considerations are also provided.

Modeling the Nonlinearities in Horn Loudspeakers

Abstract: Nonlinearities in horn loudspeakers are modeled by electromechanical-acoustical analogous circuits with lumped parameters as well as by block-oriented systems composed of linear dynamic subsystems and nonlinear static subsystems. The nonlinear parameters are developed by a power series expansion, and the second- and third-order system functions based on a Volterra series approach are presented. These higher order system functions allow a quantitative prediction of the harmonic and intermodulation distortion in the radiated sound pressure signal and show in principle the characteristic steady-state response of the nonlinear distortion for each nonlinearity. This information is helpful for interpreting the results of nonlinear distortion measurements and determining the dominant nonlinearities in horn loudspeakers. The structure of the derived system models is valuable a priori information for performing system identification to verify the modeling and to determine relevant loudspeaker parameters.

Parallel-cascade realizations and approximations of truncated Volterra systems

Abstract: This paper introduces parallel-cascade realizations of truncated Volterra systems with arbitrary, but finite order of nonlinearity. Parallel-cascade realizations implement higher-order Volterra systems using parallel and multiplicative combinations of lower-order Volterra systems. Such realizations are very modular and therefore well-suited for VLSI implementation. A systematic way of approximating higher-order Volterra systems in parallel-cascade form using a reduced number of branches and a bound on the mean-square error in the output signals caused by such approximate realizations are derived. A variation of the parallel-cascade structure in which a pth order Volterra filter is implemented as a parallel combination of linear filters whose outputs are raised to the pth power is also described.

The computational complexity of nonlinear compensators based on the Volterra inverse

Abstract: The performance of many analogue and digital signal processing systems is limited by nonlinear distortion mechanisms which can be modelled with a Volterra series. The nonlinear distortion can be compensated by the application of post (or pre)-distortion based on a Volterra inverse. The computational complexity associated with this type of compensation can be very high, particularly for systems with high nonlinearity order and long memory. We determine the 3rd and 5th order analytical Volterra inverses, and examine their associated computational complexity. We show how the analytical Volterra inverse can be used to determine the memory span of the kernels of an adaptive Volterra inverse, leading to computational complexity expressions. We then compare the computational complexity of the analytical and adaptive Volterra inverse. The results show that the analytical inverse has a much lower complexity than the adaptive inverse.

IIR Volterra filtering with application to bilinear systems

Abstract: 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 sepa...

Exponential nonlinear Volterra filters for contrast sharpening in noisy images

Abstract: This paper presents a preliminary study of a new class of nonlinear (Volterra) filters which reduce the noise in an image while simultaneously enhancing the contrast. The design of these filters draws heavily from the theory of generalized Fock (GF) spaces of Volterra series. The structure of these filters is obtained by an orthogonal projection in a GF space under the data constraints. As a consequence, even though this structure embodies an infinite Volterra series, it is represented in a closed form as a linear combination of exponentials, the exponents of which are linear functions of the input image vector.

Use of Quadratic Transfer Functions to Predict Response of Tension Leg Platforms

Abstract: Higher-order nonlinear transfer functions are applied to model the computed nonlinear responses obtained from the dynamic analysis of a tension leg platform (TLP). Under the nonlinear wave-loading condition considered in the present study, the horizontal motion of TLP exhibits a significant amount of response components at frequencies that are outside the range of the excitation frequencies, but which are near the natural frequency of TLP. Higher-order nonlinear transfer functions based on a Volterra series representation are used to model these nonlinear responses that cannot be properly represented with a linear transfer function only. The transfer function model clearly shows the degrees of nonlinearity of these responses as strongly quadratic. To examine the applicability of the nonlinear transfer functions, both the quadratic transfer functions obtained from a wave spectrum and the corresponding responses were applied to other wave spectra with different magnitude of wave power, and the results are discussed.

Application of Volterra series to the problem of self-oscillating mixer

Abstract: A new approach to the nonlinear problem of self-oscillating mixer has been investigated using Volterra series. The circuit under consideration is first converted into a one-port network. The input and coupling impedances of various ports are represented by Volterra kernels generated by nonlinear current method. Advantage of this approach is that the phase relationships among signals are not required for the analysis. Also, no stability criterion testing is needed to ensure convergence to the correct solution numerically. It is computationally efficient and mathematically simple, yet reasonably accurate. Measured results with respect to RF frequency and power show good agreement with that calculated.

Accurate performance modeling of subcarrier multiplexed fiber/radio systems: implications of laser nonlinear distortion and wide dynamic range

Abstract: In this paper, we present an analytical approach to accurately assessing the performance of subcarrier multiplexed optical systems for the delivery of future mobile radio broadband services. The existence of a wide dynamic range of power levels in the return link, together with laser nonlinear distortion, represents a major problem and makes performance evaluation difficult. By modeling laser nonlinear distortion using Volterra series analysis, we show that the performance assessment of these systems becomes not only tractable, but also that they can be can be optimized based on the worst case condition. This analysis is performed here for the case of continuous phase frequency shift keying (CPFSK) modulation, for which we calculate the power spectral density (PSD) of the intermodulation products. The overall system performance is that investigated, and its dependence on laser bias and RIN is examined. As an illustrative example we consider a specific 50 channel CPFSK system occupying the bandwidth of 2 to 2.1 GHz.

Intermodulation analysis of FET resistive mixers using Volterra series

Abstract: We have implemented the Volterra-series method to analyze intermodulation (IM) distortion in FET resistive mixers. The nonlinearities of the channel conductance of a NE71000 MESFET and a NE32400 HFET were first characterized using a low-frequency harmonic power measurement. The data was then used in a simulation program and results for two-tone IM distortion in X-band were compared with the measured data for both MESFET and HFET resistive mixer circuits. Very good agreement was achieved in each case. We have also shown by simulation that the two separate contributions to the third-order IM distortion from the mixing between the input signals themselves and the mixing between the input signals and the second-order mixing products have a very strong cancellation, which results in the low IM distortion in the FET resistive mixers observed in measurements.