Volterra Series Expansion Research Articles

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.

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

Truncated stochastically switching processes.

There are a large variety of hybrid stochastic systems that couple a continuous process with some form of stochastic switching mechanism. In many cases the system switches between different discrete internal states according to a finite-state Markov chain, and the continuous dynamics depends on the current internal state. The resulting hybrid stochastic differential equation(hSDE) could describe the evolution of a neuron's membrane potential, the concentration of proteins synthesized by a gene network, or the position of an active particle. Another major class of switching system is a search process with stochastic resetting, where the position of a diffusing or active particle is reset to a fixed position at a random sequence of times. In this case the system switches between a search phase and a reset phase, where the latter may be instantaneous. In this paper, we investigate how the behavior of a stochastically switching system is modified when the maximum number of switching (or reset) events in a given time interval is fixed. This is motivated by the idea that each time the system switches there is an additive energy cost. We first show that in the case of an hSDE, restricting the number of switching events is equivalent to truncating a Volterra series expansion of the particle propagator. Such a truncation significantly modifies the moments of the resulting renormalized propagator. We then investigate how restricting the number of reset events affects the diffusive search for an absorbing target. In particular, truncating a Volterra series expansion of the survival probability, we calculate the splitting probabilities and conditional MFPTs for the particle to be absorbed by the target or exceed a given number of resets, respectively.

Identification of the parametric array loudspeaker system using differential Volterra filter

The self-demodulation property of a parametric array loudspeaker (PAL) system can peel off audible sound modulated on ultrasonic wave and deliver it with high directivity, but also introduces nonlinear distortion. The PAL system can be modeled by the Volterra series expansion in any acoustic field region. However, measurements of Volterra kernel using simple frequency sweep excitation signals may not provide adequate information about the nonlinear system, and the use of wideband signals such as speech signals makes convergence difficult. Based on the Westervelt model, we propose a PAL system identification method utilizing the differential Volterra kernel. The results demonstrate superior fitting performance for real swept frequency output data and speech signal output data compared to the existing one-dimensional Volterra filter (ODVF). Additionally, the proposed method exhibits fewer parameters and faster convergence speed than the traditional volterra kernel.

A Dual Averaging Algorithm for Online Modeling of Infinite Memory Nonlinear Systems

An on-line modelling algorithm is derived from a generic stochastic Dual Averaging (DA) method. It employs a negative entropy as a distance-generating function and the Volterra series expansion as a dictionary. Assuming that the measurement data are not <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i.i.d.</i> but generated by a nonlinear dynamical system with an infinite, exponentially fading memory, the error bounds are established for both the generic DA method and for the proposed modelling algorithm. The experiments performed on a set of benchmark systems confirm the applicability of the algorithm in real-world scenarios and demonstrate its low computational complexity.

Efficient Realization for Third-Order Volterra Filter Based on Singular Value Decomposition

Nonlinear distortion in loudspeaker systems degrades sound quality and must be properly compensated for by linearization techniques. One technique to reduce nonlinear distortion is to use a Volterra Filter, which approximates the nonlinearity of the target loudspeaker using the Volterra series expansion. In general, the Volterra Filter is computationally very expensive, and the amount of computation needs to be reduced for real-time processing. In this paper, we propose an efficient implementation of the third-order Volterra filter based on singular value decomposition. The proposed method determines the necessary coefficients based on the symmetry of the third-order Volterra filter and applies singular value decomposition to them. In the filter structure consisting of singular values and their corresponding singular vector, the computational complexity of the third-order Volterra filter can be reduced by eliminating the part of the filter with small singular values. By focusing on the magnitude of the singular values, the proposed method can improve the computational efficiency of the third-order Volterra filter without decreasing its approximation accuracy. Simulation results show that the proposed method can improve the computational efficiency by 60% while maintaining the nonlinear distortion compensation performance of the micro-speaker for smartphones by about 8 dB.

Fast-Dynamic Grey Wolf Optimizer for solving model order reduction of bilinear systems based on multi-moment matching technique

Grey Wolf Optimizer (GWO) has been known as one of the most popular and powerful metaheuristic search algorithms. It has great advantages such as simplicity, accuracy, and good convergence rate, but its performance degrades when the global optimum is not zero. To address these issues, in this paper, a new formulation is introduced to change its searching and hunting mechanisms, while the updating structure of wolves is also modified. With these changes, a proper compromise is achieved between exploitation and exploration. Therefore, the proposed algorithm, called Fast-Dynamic GWO (FDGWO), is more accurate with a faster convergence rate compared to GWO. Thirty-eight typical benchmark functions are optimized with FDGWO and compared with a set of efficient metaheuristic search algorithms. The Wilcoxon rank-sum test is used to statistically assess the results. With an overall efficacy of 84.2%, FDGWO is the most effective among competing algorithms. Then, the performance of FDGWO for ten real-world constrained engineering problems is compared with some highly accurate recent algorithms such as BP-ϵMAg-ES and COLSHADE. The results show that while FDGWO performs almost as accurate as these algorithms, but its computational complexity is quite less. Next, a new hybrid method based on FDGWO is proposed for model order reduction (MOR) of bilinear systems. First, the order of reduced system is determined via an iterative approach based on H2 norm of the error system. The unknown parameters of reduced model are then determined by minimizing a multi-objective fitness function aiming to match the multi-moments of original and reduced bilinear systems. Some stability constraints based on the Volterra series expansion of the states of the bilinear system are added to the optimization problem, which is solved by FDGWO algorithm. Finally, two bilinear test systems are reduced by the proposed hybrid method and compared with the most common bilinear MOR methods.

Nonlinear Dynamic System Identification in the Spectral Domain Using Particle-Bernstein Polynomials

System identification (SI) is the discipline of inferring mathematical models from unknown dynamic systems using the input/output observations of such systems with or without prior knowledge of some of the system parameters. Many valid algorithms are available in the literature, including Volterra series expansion, Hammerstein–Wiener models, nonlinear auto-regressive moving average model with exogenous inputs (NARMAX) and its derivatives (NARX, NARMA). Different nonlinear estimators can be used for those algorithms, such as polynomials, neural networks or wavelet networks. This paper uses a different approach, named particle-Bernstein polynomials, as an estimator for SI. Moreover, unlike the mentioned algorithms, this approach does not operate in the time domain but rather in the spectral components of the signals through the use of the discrete Karhunen–Loève transform (DKLT). Some experiments are performed to validate this approach using a publicly available dataset based on ground vibration tests recorded from a real F-16 aircraft. The experiments show better results when compared with some of the traditional algorithms, especially for large, heterogeneous datasets such as the one used. In particular, the absolute error obtained with the prosed method is 63% smaller with respect to NARX and from 42% to 62% smaller with respect to various artificial neural network-based approaches.

Experimental Comparison of PAM and CAP Modulation for Visible Light Communication Under Illumination Constraints

In this paper, we study different modulation techniques for visible light communication (VLC), taking illumination constraints into account. Two modulation schemes are compared, namely pulse amplitude modulation (PAM) and carrierless amplitude and phase (CAP) modulation, through both simulations and experimental measurements. The data link under study is based on low-cost components comprising a white light-emitting diode (LED) and a silicon PIN photodiode. Moreover, the proposed VLC system complies with illumination standards and limits the brightness level to that of a typical office room. The impact of the roll-off factor parameter, which is directly related to the total occupied bandwidth, and the maximum achievable throughput for PAM and CAP are studied. Moreover, adaptive postdistortion based on a Volterra series expansion is implemented to mitigate the effects of LED nonlinearity in practice. We also demonstrate that 8-PAM can outperform 64-CAP and discrete multitone (DMT) when the LED nonlinearity is adequately compensated.

Unsteady and lineal translation of a sphere through a viscoelastic fluid

Unsteady translation of a sphere in a viscoelastic fluid is studied analytically. General, fully invertible relationships between an imposed time-dependent velocity and the resultant force are described in terms of a Volterra series expansion. These results are used to analyze particle motion in fluids described by the Johnson-Segalman and Giesekus models and to define a general framework for analyzing weakly nonlinear microrheology measurements.

Power spectral density analysis for nonlinear systems based on Volterra series

A consequence of nonlinearities is a multi-harmonic response via a mono-harmonic excitation. A similar phenomenon also exists in random vibration. The power spectral density (PSD) analysis of random vibration for nonlinear systems is studied in this paper. The analytical formulation of output PSD subject to the zero-mean Gaussian random load is deduced by using the Volterra series expansion and the conception of generalized frequency response function (GFRF). For a class of nonlinear systems, the growing exponential method is used to determine the first 3rd-order GFRFs. The proposed approach is used to achieve the nonlinear system’s output PSD under a narrow-band stationary random input. The relationship between the peak of PSD and the parameters of the nonlinear system is discussed. By using the proposed method, the nonlinear characteristics of multi-band output via single-band input can be well predicted. The results reveal that changing nonlinear system parameters gives a one-of-a-kind change of the system’s output PSD. This paper provides a method for the research of random vibration prediction and control in real-world nonlinear systems.

Input/output reduced model of a damped nonlinear beam based on Volterra series and modal decomposition with convergence results

This paper addresses the model reduction and the simulation of a damped Euler–Bernoulli–von Kármán pinned beam excited by a distributed force. This nonlinear problem is formulated as a PDE and reformulated as a well-posed state-space system. The model order reduction and simulation are derived by combining two approaches: a Volterra series expansion and truncation and a pseudo-modal truncation defined from the eigenbasis of the linearized problem. The interest of this approach lies in the large class of input waveshapes that can be considered and in the simplicity of the simulation structure. This structure only involves cascades of finite-dimensional decoupled linear systems and multilinear functions. Closed-form bounds depending on the model coefficients and the truncation orders are provided for the Volterra convergence domain and the approximation error. These theoretical results are generalized to a large class of nonlinear models, and refinement of bounds are also proposed for a large sub-class. Numerical experiments confirm that the beam model is well approximated by the very first Volterra terms inside the convergence domain.

Discrete-Time Signatures and Randomness in Reservoir Computing.

A new explanation of the geometric nature of the reservoir computing (RC) phenomenon is presented. RC is understood in the literature as the possibility of approximating input-output systems with randomly chosen recurrent neural systems and a trained linear readout layer. Light is shed on this phenomenon by constructing what is called strongly universal reservoir systems as random projections of a family of state-space systems that generate Volterra series expansions. This procedure yields a state-affine reservoir system with randomly generated coefficients in a dimension that is logarithmically reduced with respect to the original system. This reservoir system is able to approximate any element in the fading memory filters class just by training a different linear readout for each different filter. Explicit expressions for the probability distributions needed in the generation of the projected reservoir system are stated, and bounds for the committed approximation error are provided.

Identification of Wiener–Hammerstein systems by [formula omitted]–constrained Volterra series

In the paper the Wiener–Hammerstein (LNL) system is identified with the help of Volterra series and ℓ1-constrained least squares. The nonparametric approach to the problem ensures wide applicability of the proposed method, as it does not require an extensive a priori knowledge about the underlying system. In the main part of the paper, the convergence of the identification algorithm is shown for a general class of systems with smooth nonlinearities and stable dynamics, under the assumption that the input is a bounded i.i.d.random signal. The rate of convergence is examined as well. Moreover, the large error of the standard least squares-based models, inherently associated with high dimensional representations (such as those based on Volterra series expansion), is reduced by the imposed ℓ1-constraint and hence the method allows for a scarcer data set. In addition, the results of numerical simulations are provided to illustrate the superiority of the proposed method over the standard one.

Medium amplitude parallel superposition (MAPS) rheology. Part 1: Mathematical framework and theoretical examples

A new mathematical representation for nonlinear viscoelasticity is presented based on the application of the Volterra series expansion to the general nonlinear relationship between shear stress and shear strain history. This theoretical and experimental framework, which we call medium amplitude parallel superposition (MAPS) rheology, reveals a new material property, the third-order complex modulus, which describes completely the weakly nonlinear response of a viscoelastic material in an arbitrary simple shear flow. In this first part, we discuss several theoretical aspects of this mathematical formulation and new material property. For example, we show how MAPS measurements can be performed in strain- or stress-controlled contexts and provide relationships between the weakly nonlinear response functions measured in each case. We show that the MAPS response function is a superset of the response functions that have been previously reported in medium amplitude oscillatory shear and parallel superposition rheology experiments. We also show how to exploit inherent symmetries of the MAPS response function to reduce it to a minimal domain for straightforward measurement and visualization. We compute this material property for a few constitutive models to illustrate the potential richness of the datasets generated by MAPS experiments. Finally, we discuss the MAPS framework in the context of some other nonlinear, time-dependent rheological probes and explain how the MAPS methodology has a distinct advantage over these others because it generates data embedded in a very high-dimensional space without driving fluid mechanical instabilities, and is agnostic to the flow protocol.

Volterra Expansion Based Intra Channel Nonlinear Effect Equalization Method for Optical OFDM/OQAM Systems

Optical orthogonal frequency-division multiplexing offset quadrature amplitude modulation (O-OFDM/OQAM) system relaxes the orthogonal condition of the sub-carriers from the complex domain to the real field. Inter-symbol-interference (ISI) and inter-carrier-interference (ICI) could be suppressed by using filter banks with promising time-frequency-localization (TFL) properties. Therefore, cyclic prefix (CP) inserted between consecutive OFDM blocks could be removed for O-OFDM/OQAM system to improve system spectral efficiency. When passing through fiber channel, O-OFDM/OQAM faces serious intrinsic imaginary interference (IMI) induced by chromatic dispersion (CD), and fiber nonlinear effect, which would deteriorate system performance evidently. Fiber nonlinear effect induced interference is a great impairment for long haul transmission O-OFDM/OQAM system, and can not be equalized directly by using nonlinear equalization method designed for traditional optical OFDM. There is still much room for improvement of nonlinear equalization method for O-OFDM/OQAM. In this paper, we systematically study Volterra expansion based nonlinear equalization method (VENE) for O-OFDM/OQAM. We theoretically deduce simplified Volterra series expansion nonlinear transmission matrix (SVEM) for O-OFDM/OQAM based on mathematic deduction. With SVEM, intra channel nonlinear effect induced distortions could be modeled and estimated. By using specially designed pilot blocks, we obtain approximate solution of SVEM and perform effective VENE with very limited complexities. As shown in multiple Montel Carlo simulation results, nonlinear robustness for O-OFDM/OQAM has been improved significantly thanks to VENE, with various transmission distances and system parameters.

Transfer Function with Nonlinear Characteristics Definition Based on Multidimensional Laplace Transform and its Application to Forced Response Power Systems

In this research, the concept of nonlinear transfer function with nonlinear characteristics is introduced through the multidimensional Laplace transform and modal series (MS) method. The method of modal series is applied to the power systems dynamics analysis in order to consider nonlinear oscillations and modal interactions, which contribute to the response of the system's dynamic following disturbances. The method of MS allows the inclusion of input excitation functions obtained as Laplace domain kernels superposed to obtain a transfer function. Applying the Volterra series expansion through kernels decomposition, a transfer function with nonlinear characteristics is obtained which incorporates some of the main modal characteristics of the nonlinear system. Following the same schematic procedure, it is possible to determine second and higher order transfer functions. Once the transfer functions both linear and with nonlinear characteristics are determined, a time domain and frequency response analyses can be performed. The methodology is exemplified by denoting the numerical and analytical properties with the application to a synchronous machine-infinite busbar test power system and to a three synchronous machines–nine buses test power system. Bode and Nyquist analysis are utilized to demonstrate the transfer functions accuracy and frequency response.

Robust Optical Spatial Localization Using a Single Image Sensor

A novel 3-D spatial localization method using a single camera with temporal-difference image processing is proposed. The proposed active localization method uses a ring of light-emitting diodes embedded on the target. The diameter and the central location of the ring's image on the image sensor are used to estimate the target location using a Volterra series expansion of the target coordinates. No knowledge of the camera hardware parameters is needed. Instead, Volterra series parameters are obtained through a prior training that needs to be performed only once. The proposed method can be implemented with low computational complexity and storage. The performance of the proposed method is compared against an earlier method that relies on prior knowledge of the camera hardware parameters. The proposed method demonstrates excellent performance even when the target is located far away from the axial direction of the camera lens. The proposed method can be applied in low-power outdoor unmanned aerial vehicle localization and indoor robotic navigation.

On the physical definition of dynamic impedance: How to design an optimal strategy for data extraction

Dynamic electrochemical impedance spectroscopy (DEIS) has attracted the interest of researchers due to its capability of acquiring impedance spectra of non-stationary systems in a broad range of frequencies. Although developed in the 70's, a physical definition of dynamic impedance, and in general of dynamic transfer function, is not present in literature. In this manuscript the general concept of dynamic frequency response is given in correlation to the Volterra series expansion: this can be used to define the ideal dynamic impedance response of an electrochemical system. It is clarified that DEIS is the intermodulation of the ac signal with the dc signal. Starting from this, different strategies for the extraction of the dynamic impedance spectra are compared, which are based on different heuristic definitions of DEIS. There are two main methodologies, one based on the use of window functions and the other on quadrature filters. Limits and merits of the different methodologies are discussed in the frame of the data analysis and data handling and show that window functions suffers at recovering low-frequencies impedances while quadrature filters work best in experiments which have a periodic or quasi-periodic trend.

Design and analysis of high linearity mixer using subharmonic technique

SummaryIn this article, a new technique for reducing second‐order and third‐order intermodulation (IM2 and IM3) for an output current of the subharmonic mixer is presented. First, to reduce second‐order and third‐order of the Volterra's kernel, IM2 current injection technique has been used. By the auxiliary path, the IM2 current which has been generated in the doubler with the phase difference of 180° relative to inherent IM2 of the output is injected into the output node. Also, this produced IM2 in the frequency doubler has been injected into the circuit by another auxiliary path, so that, in the output, an IM3 with opposite phase of inherent IM3 of the output mixer's current is presented. Besides, to easier design, the Volterra series expansion is used and equations related to Volterra kernels of the output current are extracted. Furthermore, the noise of the auxiliary path transistors connected to the output can be neglected because of the small gm of their transistors compared to the main path. The results of simulations by ADS simulator in 0.18‐μm TSMC technology show that the proposed technique has simultaneously improved the values of IIP3 and IIP2 up to 21 and 92 dBm, compared to conventional subharmonic mixers.