Volterra Series Research Articles

Generalized pole-residue method for dynamic analysis of nonlinear systems based on Volterra series

Generalized pole-residue method for dynamic analysis of nonlinear systems based on Volterra series

An efficient Volterra series approach for identifying nonlinear system vibration responses to an excitation

This paper focuses on the issue of nonlinear vibration responses identification of nonlinear systems. An efficient algorithm is presented, in which the nonlinear vibration system under studied is decomposed into a multiple-input/single-output (MISO) linear system with a series of power characterized inputs based on Volterra series, and the nonlinear output responses of different orders are identified by taking power spectra operation to the input and output data, revealing the contributions of each order nonlinearity to the output of the system. Compared to the existing method, the input signal of the system required in the presented approach need not remain unchanged in waveform and adjustable in magnitude. To verify the approach, a classic Duffing–Van der Pol oscillator was simulated, obtaining results very close to the fourth order Runge–Kutta method. Finally, an experiment analysis was carried out, in which the vibration transmission properties of a bolt connection were tested when the bolt was tight and loose, respectively. The results proved that the approach is effective in identifying nonlinear vibration frequency responses.

ROF-based antenna-PDM system employing DSM and DD-weight-pruning Volterra nonlinear equalization

The radio over fiber (ROF) system based on Delta-sigma modulation (DSM) can meet the requirements of the fronthaul system with its high signal fidelity and mature digital fronthaul interface specification. However, nonlinear impairments in ROF systems can seriously affect the performance of DSM signals. To reduce the impact of nonlinear effects on DSM signals with low complexity, we have designed a nonlinear equalizer based on the Volterra series. The complexity of the Volterra nonlinear equalizer (VNLE) is reduced through a weight pruning strategy. And by using directed decision instead of training sequences, we eliminate the redundancy introduced by VNLE while simultaneously ensuring its stability within the DSM system. Ultimately, we transmit one-bit single-carrier loaded DSM signals over 20 km of SMF-28 fiber and 3 m of wireless distance in an antenna polarization division multiplexing (APDM) intensity modulation and direct detection (IM/DD) system. For the first time, we add decision-directed weight-pruning Volterra nonlinear equalization (DD-PVNLE) algorithm in the APDM-IM/DD system to overcome the signal damage by the nonlinearity of the system. The BER of the Delta-sigma modulated on-off keying (DSM-OOK) signal reduces from 1 × 10−3 to 1.1 × 10−4. The high-fidelity 1024QAM, 2048QAM, and 4096QAM are recovered successfully whose BERs below the soft decision threshold 2.4 × 10−2 and hard decision threshold 3.8 × 10−3 respectively.

Оцінка точності комп’ютерного моделювання окуло-моторної системи на основі моделей Вольтерри

Integral nonlinear models are used to create mathematical models of the human oculo-motor system (OMS). These models take into account both inertial and nonlinear properties of the objects under study. To obtain empirical data for model construction, experimental studies are conducted with OMS using «input-output» data. Visual stimuli are used as test signals, displayed on a computer monitor at various distances from the starting position, which formally corresponds to the action of step signals with varying amplitudes on the object of study. In this process, the responses of the OMS are recorded using innovative eye-tracking technology. Mathematical models in the form of Volterra series and polynomials are employed for computer modeling of the OMS. The aim of this research is to analyze the accuracy of OMS identification as multidimensional transient functions based on eye-tracking data, examining the dependency of computation errors for models of different orders on the amplitudes and quantities of the test signals used. The subject of the study includes various methods for identifying OMS, algorithms, and Python-based software tools for computing the dynamic characteristics of OMS using eye-tracking technology. The research explores identification methods: compensation, approximation, and least squares methods. The accuracy of the linear, quadratic, and cubic OMS models is evaluated. The most accurate models, constructed from real experimental data, are found to be quadratic or cubic OMS models obtained using the least squares method with three test signals.

Nonlinear Langevin functionals for a driven probe.

When a probe particle immersed in a fluid with nonlinear interactions is subject to strong driving, the cumulants of the stochastic force acting on the probe are nonlinear functionals of the driving protocol. We present a Volterra series for these nonlinear functionals by applying nonlinear response theory in a path integral formalism, where the emerging kernels are shown to be expressed in terms of connected equilibrium correlation functions. The first cumulant is the mean force, the second cumulant characterizes the non-equilibrium force fluctuations (noise), and higher order cumulants quantify non-Gaussian fluctuations. We discuss the interpretation of this formalism in relation to Langevin dynamics. We highlight two example scenarios of this formalism. (i) For a particle driven with the prescribed trajectory, the formalism yields the non-equilibrium statistics of the interaction force with the fluid. (ii) For a particle confined in a moving trapping potential, the formalism yields the non-equilibrium statistics of the trapping force. In simulations of a model of nonlinearly interacting Brownian particles, we find that nonlinear phenomena, such as shear-thinning and oscillating noise covariance, appear in third- or second-order response, respectively.

Development of Statistical Downscaling Model Based on Volterra Series Realization, Principal Components, Climate Classification, and Ridge Regression

This paper applied the fuzzy function approach, combined with the ridge regression model, to produce daily rainfall projections from large-scale climate variables. This study developed a statistical downscaling model based on principal components, c-means fuzzy clustering, Volterra series, and ridge regression. The model is known, hereafter as SDC2R2. In the developed downscaling model, the use of ridge regression, instead of multiple linear regression, is proposed to downscale daily rainfall with wide range (WR) predictors. The WR predictors were applied to sufficiently incorporate climate change signals. The developed model also captured the non-linear interactions of the climate variables by applying the transformation of Volterra series realization over WR predictors. This transformation was performed by applying principal components as orthogonal filters. Further, these principal components were clustered by using c-means clustering and non-linear transformations were applied on these membership functions, to improve the prediction ability of the model. The reanalysis of climate data from the National Centres for Environmental Prediction (NCEP) was used to develop the model and was validated by using the Global Climate Model (GCM) for four locations in the Manawatu River basin. The developed model was used to obtain future daily rainfall projections from three Representative Concentrative Pathways (RCP 2.6, RCP 4.5, and RCP 8.5) scenarios from the Canadian Earth System Model (CanESM2) GCM. The performance of the model was compared with a widely used statistical downscaling model (SDSM). It was observed that the model performed better than SDSM in downscaling rainfall on a daily basis. Every scenario indicated that there is a probability of obtaining high future rainfall frequency. The results of this study provide valuable information for decision-makers since climate change may potentially impact the Manawatu basin.

Overcoming the coherence time barrier in quantum machine learning on temporal data

The practical implementation of many quantum algorithms known today is limited by the coherence time of the executing quantum hardware and quantum sampling noise. Here we present a machine learning algorithm, NISQRC, for qubit-based quantum systems that enables inference on temporal data over durations unconstrained by decoherence. NISQRC leverages mid-circuit measurements and deterministic reset operations to reduce circuit executions, while still maintaining an appropriate length persistent temporal memory in the quantum system, confirmed through the proposed Volterra Series analysis. This enables NISQRC to overcome not only limitations imposed by finite coherence, but also information scrambling in monitored circuits and sampling noise, problems that persist even in hypothetical fault-tolerant quantum computers that have yet to be realized. To validate our approach, we consider the channel equalization task to recover test signal symbols that are subject to a distorting channel. Through simulations and experiments on a 7-qubit quantum processor we demonstrate that NISQRC can recover arbitrarily long test signals, not limited by coherence time.

Investigation of the accuracy of integral models of the oculo-motor system

For mathematical modeling of the human oculomotor system (OMS), integral nonlinear models are used, which simultaneously take into account the nonlinear and inertial properties of the research object. Based on the data of experimental studies of the OMS "input-output", transient and diagonal intersections of transient functions of the second and third orders are determined. To obtain experimental data, an innovative eye tracking technology is used, which allows recording eye responses to test visual stimuli. Thus test signals are displayed on the computer monitor at different distances from the starting position in the horizontal direction. The aim of the research is to study the accuracy of OMS identification using eye-tracking data by evaluating the calculation errors of multidimensional transient functions when using methods of nonlinear dynamic identification based on models in the form of Volterra series and polynomials. The object of the study is the process of nonparametric identification of the OMS based on Volterra models in the time domain. The subject of the research is algorithmic and software tools for calculating the dynamic characteristics of OMS based on eye-tracking data, analyzing the accuracy of the obtained models using two identification methods: the approximation method and the least squares method (LSM). The means of nonlinear dynamic identification of the human OMS based on Volterra series and polynomials were developed in the Python programming environment. The accuracy estimates of various OMS (linear, quadratic, and cubic) models were obtained based on data from three responses to test signals of different amplitudes. For the same test signals, the same models in the form of the Volterra series and polynomials were obtained, as these models coincide in the region of convergence of the Volterra series. The analysis of errors in the assessment of the dynamic characteristics of the OMS demonstrated that the model in the form of an integral polynomial of the second degree, which was built using the LSM based on three responses, has an accuracy twice as high as the accuracy of similar models built using the data of two responses. Thus, in further studies of the psychophysiological state of a person based on nonlinear dynamic models of the OMS according to the data of three responses, it is advisable to use a model in the form of a quadratic Volterra polynomial. Figs.: 17. Tabl. 3. Refs.: 10 titles.

Balanced truncation for quadratic-bilinear control systems

We discuss model order reduction (MOR) for large-scale quadratic-bilinear (QB) systems based on balanced truncation. The method for linear systems mainly involves the computation of the Gramians of the system, namely reachability and observability Gramians. These Gramians are extended to a general nonlinear setting in Scherpen (Systems Control Lett. 21, 143-153 1993). These formulations of Gramians are not only challenging to compute for large-scale systems but hard to utilize also in the MOR framework. This work proposes algebraic Gramians for QB systems based on the underlying Volterra series representation of QB systems and their Hilbert adjoint systems. We then show their relation to a certain type of generalized quadratic Lyapunov equation. Furthermore, we quantify the reachability and observability subspaces based on the proposed Gramians. Consequently, we propose a balancing algorithm, allowing us to find those states that are simultaneously hard to reach and hard to observe. Truncating such states yields reduced-order systems. We also study sufficient conditions for the existence of Gramians, and a local stability of reduced-order models obtained using the proposed balanced truncation scheme. Finally, we demonstrate the proposed balancing-type MOR for QB systems using various numerical examples.

Nonautonomous Volterra Series Expansion of the Variable Phase Approximation and its Application to the Nucleon-Nucleon Inverse Scattering Problem

Abstract In this paper, the nonlinear Volterra series expansion is extended and used to describe certain types of nonautonomous differential equations related to the inverse scattering problem in nuclear physics. The nonautonomous Volterra series expansion lets us determine a dynamic, polynomial approximation of the variable phase approximation (VPA), which is used to determine the phase shifts from nuclear potentials through first-order nonlinear differential equations. By using the first-order Volterra expansion, a robust approximation is formulated to the inverse scattering problem for weak potentials and/or high energies. The method is then extended with the help of radial basis function neural networks by applying a nonlinear transformation on the measured phase shifts to be able to model the scattering system with a linear approximation given by the first-order Volterra expansion. The method is applied to describe the ${}^1S_0$ NN potentials in neutron+proton scattering below 200 MeV laboratory kinetic energies, giving physically sensible potentials and below $1\%$ averaged relative error between the recalculated and the measured phase shifts.

Nonlinear behavioral analysis of diode model using Volterra series

Nonlinear behavioral analysis of diode model using Volterra series

Research on laser communication ranging Integration and VLC nonlinear compensation technology

Abstract Visible light communication (VLC) is a new spectrum communication technology that provides a new direction to solve spectrum resource problems. The nonlinearity problem in VLC will lead to a significant reduction in the performance of the communication system. To solve this problem, this paper studies the integration of laser communication ranging and VLC nonlinear compensation technology based on analyzing the principles of ranging methods and factors affecting ranging performance. Aiming at the problem that the complexity of the Random Fourier Features Kernel adaptive filter (RFF-KAF) method will increase significantly as the dimension of the mapping space increases, we introduce some second-order Volterra series square terms into the input vector of RFF-KAF and propose an improved prediction method. Experimental simulation results show that the method proposed in this article can greatly improve the performance of the RFF-KAF method while maintaining the same mapping space dimension. Compared with obtaining similar performance by increasing the mapping space dimension, the cost of this method is greatly reduced.

Learning Nonlinear Systems via Volterra Series and Hilbert-Schmidt Operators

Learning Nonlinear Systems via Volterra Series and Hilbert-Schmidt Operators

Discontinuous Galerkin finite element method for dynamic viscoelasticity models of power‐law type

AbstractLinear viscoelasticity can be characterized by a stress relaxation function. We consider a power‐law type stress relaxation to yield a fractional order viscoelasticity model. The governing equation is a Volterra integral problem of the second kind with a weakly singular kernel. We employ spatially discontinuous Galerkin methods, symmetric interior penalty Galerkin method (SIPG) for spatial discretization, and the implicit finite difference schemes in time, Crank–Nicolson method. Further, in order to manage the weak singularity in the Volterra kernel, we use a linear interpolation technique. We present a priori stability and error analyses without relying on Grönwall's inequality, and so provide high quality bounds that do not increase exponentially in time. This indicates that our numerical scheme is well‐suited for long‐time simulations. Despite the limited regularity in time, we establish suboptimal fractional order accuracy in time as well as optimal convergence of SIPG. We carry out numerical experiments with varying regularity of exact solutions to validate our error estimates. Finally, we present numerical simulations based on real material data.

Dominant subspaces of high-fidelity polynomial structured parametric dynamical systems and model reduction

In this work, we investigate a model order reduction scheme for high-fidelity nonlinear structured parametric dynamical systems. More specifically, we consider a class of nonlinear dynamical systems whose nonlinear terms are polynomial functions, and the linear part corresponds to a linear structured model, such as second-order, time-delay, or fractional-order systems. Our approach relies on the Volterra series representation of these dynamical systems. Using this representation, we identify the kernels and, thus, the generalized multivariate transfer functions associated with these systems. Consequently, we present results allowing the construction of reduced-order models whose generalized transfer functions interpolate these of the original system at pre-defined frequency points. For efficient calculations, we also need the concept of a symmetric Kronecker product representation of a tensor and derive particular properties of them. Moreover, we propose an algorithm that extracts dominant subspaces from the prescribed interpolation conditions. This allows the construction of reduced-order models that preserve the structure. We also extend these results to parametric systems and a special case (delay in input/output). We demonstrate the efficiency of the proposed method by means of various numerical benchmarks.

Quantification of nonlinear output frequency responses for a general input based on volterra series and conditioned spectral analysis

Quantification of nonlinear output frequency responses for a general input based on volterra series and conditioned spectral analysis

Color image sparse adversarial samples generation via color channel Volterra expansion

Color image sparse adversarial samples generation via color channel Volterra expansion

Distributed reservoir computing based nonlinear equalizer for VCSEL based optical interconnects

We propose a distributed reservoir computing based nonlinear equalizer for 106 Gbps VCSEL based optical interconnects. By comparing with neural network and Volterra series based equalizers, the computational complexity of this proposed nonlinear equalizer is significantly reduced in the training, prediction and inference process. 106 Gbps PAM4 short-range optical transmission is demonstrated by experiment, using a VCSEL-based co-packaged optical transmitter to verify the performance of the equalizer and key parameters that affect the nonlinear compensation performance are carefully investigated. We believe that by incorporating such simplified nonlinear equalizers, it is possible to compensate for nonlinear impairments while preserving the inherent advantages of VCSEL-based optical interconnects. This optical interconnect will become a promising paradigm to overcome I/O performance penalty when communication leaving packages.

Inverse Modelling Techniques as Dynamic Digital Pre-Distortion for Broadband Systems for CS-DACs Based on Volterra Series Expansion

Inverse Modelling Techniques as Dynamic Digital Pre-Distortion for Broadband Systems for CS-DACs Based on Volterra Series Expansion

Local Independence Testing for Point Processes.

Constraint-based causal structure learning for point processes require empirical tests of local independence. Existing tests require strong model assumptions, e.g., that the true data generating model is a Hawkes process with no latent confounders. Even when restricting attention to Hawkes processes, latent confounders are a major technical difficulty because a marginalized process will generally not be a Hawkes process itself. We introduce an expansion similar to Volterra expansions as a tool to represent marginalized intensities. Our main theoretical result is that such expansions can approximate the true marginalized intensity arbitrarily well. Based on this, we propose a test of local independence and investigate its properties in real and simulated data.