Second-order Volterra Research Articles

Second-order numerical method for a neutral Volterra integro-differential equation

This paper is dedicated to obtaining an approximate solution for a neutral second-order Volterra integro-differential equation. Our method is the second-order accurate finite difference scheme on a uniform mesh. The error analysis is carried out and numerical results are given to support the proposed approach.

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.

A fitted approximate method for solving singularly perturbed Volterra–Fredholm integro-differential equations with an integral boundary condition

UDC 517.9 We consider a novel numerical approach for solving boundary-value problems for the second-order Volterra–Fredholm integro-differential equation with layer behavior and an integral boundary condition. A finite-difference scheme is proposed on suitable Shishkin-type mesh to obtain the approximate solution of the presented problem. It is proven that the method is first-order convergent in the discrete maximum norm. Two numerical examples are included to show the efficiency of the method.

Optimal nonlinear system identification using fractional delay second-order volterra system

The aim of this work is to design a fractional delay second order Volterra filter that takes a discrete time sequence as input and its output is as close as possible to the output of a given nonlinear unknown system which may have higher degree nonlinearities in the least square sense. The basic reason for such a design is that rather than including higher than second degree nonlinearities in the designed system, we use the fractional delay degrees of freedom to approximate the given system. The advantage is in terms of obtaining a better approximation of the given nonlinear system than is possible by using only integer delays ( since we are giving more degrees of freedom via the fractional delays ) and simultaneously it does not require to incorporate higher degree nonlinearities than two. This work hinges around the fact that if the input signal is a decimated version of another signal by a factor of M, then fractional delays can be regarded as delays by integers less than M. Using the well known formula for calculating the discrete time Fourier transform ( TFT ) of a decimated signal, we then arrive at an expression for the DTFT of the output of a fractional delay system in terms of the unknown first and second order Volterra system coefficients and the fractional delays. The final energy function to be minimized is the norm square of the difference between the DTFT of the given output and the DTFT of the output of the fractional delay system. Minimization over the filter coefficients is a linear problem and thus the final problem is to minimize a highly nonlinear function of the fractional delays which is accomplished using search techniques like the gradient-search and nature inspired optimization algorithms. The effectiveness of the proposed method is demonstrated using two nonlinear benchmark systems tested with five different input signals. The accuracy of the stated models using the globally convergent metaheuristic, cuckoo-search algorithm ( CSA ) are observed to be superior when compared with other techniques such as real-coded genetic algorithm ( RGA ), particle swarm optimization ( PSO ) and gradient-search ( GS ) methods. Finally, statistical analysis affirms the potential of the proposed designs for its successful implementation.

Optimal nonlinear system identification using fractional delay second-order volterra system

The aim of this work is to design a fractional delay second order Volterra filter that takes a discrete time sequence as input and its output is as close as possible to the output of a given nonlinear unknown system which may have higher degree nonlinearities in the least square sense. The basic reason for such a design is that rather than including higher than second degree nonlinearities in the designed system, we use the fractional delay degrees of freedom to approximate the given system. The advantage is in terms of obtaining a better approximation of the given nonlinear system than is possible by using only integer delays U+0028 since we are giving more degrees of freedom via the fractional delays U+0029 and simultaneously it does not require to incorporate higher degree nonlinearities than two. This work hinges around the fact that if the input signal is a decimated version of another signal by a factor of M, then fractional delays can be regarded as delays by integers less than M. Using the well known formula for calculating the discrete time Fourier transform U+0028 DTFT U+0029 of a decimated signal, we then arrive at an expression for the DTFT of the output of a fractional delay system in terms of the unknown first and second order Volterra system coefficients and the fractional delays. The final energy function to be minimized is the norm square of the difference between the DTFT of the given output and the DTFT of the output of the fractional delay system. Minimization over the filter coefficients is a linear problem and thus the final problem is to minimize a highly nonlinear function of the fractional delays which is accomplished using search techniques like the gradient-search and nature inspired optimization algorithms. The effectiveness of the proposed method is demonstrated using two nonlinear benchmark systems tested with five different input signals. The accuracy of the stated models using the globally convergent metaheuristic, cuckoo-search algorithm U+0028 CSA U+0029 are observed to be superior when compared with other techniques such as real-coded genetic algorithm U+0028 RGA U+0029, particle swarm optimization U+0028 PSO U+0029 and gradient-search U+0028 GS U+0029 methods. Finally, statistical analysis affirms the potential of the proposed designs for its successful implementation.

A model for the dissipation rate in linear unsteady flow through porous media

We present a model for the volume-averaged dissipation rate in linear unsteady flow through porous media. The model is derived by blending a new small-time asymptotic expression for the dissipation rate obtained from boundary layer theory with the known large-time asymptotic expression obtained from Darcy's law. The resulting model is a second-order Volterra functional of the volume-averaged acceleration. We validate the model with an analytical solution for transient flow through a porous medium composed of circular tubes and with numerical simulations of transient and oscillatory flow through a cylinder array and through a hexagonal sphere pack.

A Nonlinear Model for Evaluating Dynamic Resilience of Water Supply Hydropower Generation‐Environment Conservation Nexus System

AbstractUndesirable disturbances (e.g., extraordinary drought) are prone to cause cascading failures of water supply‐hydropower generation‐environment conservation (WHE) nexus system. Simulating a dynamic resilience of the nexus system can provide opportunities for pre‐disturbance effective prevention and post‐disturbance recovery from the cascading failures. Previous dynamic resilience simulating is often based on the linear assumption which is not applicable for the WHE nexus system that involves nonlinear processes. To improve simulation accuracy of the dynamic resilience of the WHE nexus system, a novel nonlinear dynamic resilience model that incorporates both a WHE nexus model and a second‐order Volterra (SOV) model is proposed. The WHE nexus model simulates amount of impact propagated by an original disturbance event on each subsystem, while the SOV model captures nonlinear relationship between the amount of impact and a corresponding dynamic resilience. The proposed model was applied to the Liuchong River Basin, China, under different dry year scenarios. The results show that the dynamic resilience simulated through the SOV model performs better than that through a first‐order Volterra model which is suitable for a linear system, with Nash‐Sutcliffe efficiencies of water supply, hydropower generation, and environment conservation subsystems increasing by 36.8%, 70.9%, and 17% respectively. Among the WHE nexus system, the water supply subsystem is found to be the least resilient. Moreover, the proposed nonlinear model is more suitable for simulating and predicting dynamic resilience when a severer water resources drought occurs. The proposed model helps to guide resilience management of water resource system by combining nexus and resilience thinking.

A Simplified Volterra Equalizer Based on System Characteristics for Direct Modulation Laser (DML)-Based Intensity Modulation and Direct Detection (IM/DD) Transmission Systems

The nonlinear Volterra equalizer has been proved to be able to solve the problem of nonlinear distortion, but it has high computational complexity and is difficult to implement. In this paper, a simplified second-order Volterra nonlinear equalizer designed for intensity modulation/direct detection systems based on direct modulated laser is proposed and demonstrated, taking into account the characteristics of the system. It has been proved that the received signal of direct modulation laser/direct detection system can be expressed in Volterra series form, but its form is too complex, and the device parameters should also be considered. We re-derived it and obtained a more concise form. At the same time, we proposed a method to simplify the second-order Volterra nonlinear equalizer without relying on device parameters. The performance of the proposed Volterra nonlinear equalizer is evaluated experimentally on a 56 Gb/s 4-ary pulse amplitude modulation link implemented by using a 1.55 µm direct modulation laser. The results show that, compared with the traditional Volterra nonlinear equalizer, the receiver sensitivity of the equalizer is only reduced by 0.2 dB at most, but the complexity can be reduced by 50%; compared with diagonally pruned Volterra nonlinear equalizers, the complexity of the equalizer is the same, but the reception sensitivity can be improved by 0.5 dB.

Frequency-domain Volterra kernel-based adaptation: Formulations and algorithms

For the correlated input, the Volterra kernel-based least mean-square (LMS) algorithm in the time-domain exhibits a slow learning rate caused by the large eigenvalue spread of the input covariance matrix. To tackle such an issue, this paper develops a novel frequency-domain Volterra kernel-based filter, resulting in the periodic update constrained frequency-domain second-order Volterra normalized LMS (named as P-CFDSOV-NLMS1) algorithm. Subsequently, by using one- and two-dimensional discrete Fourier transforms (DFTs) simultaneously, another frequency-domain implementation and corresponding P-CFDSOV-NLMS2 algorithm are constructed. In contrast, the P-CFDSOV-NLMS1 scheme only requires one-dimensional DFT operations and takes advantage of the joint information between the block input vectors. Then, the mean and mean-square convergence behaviors of the P-CFDSOV-NLMS1 algorithm are investigated. Furthermore, the designed frequency-domain method is extended to three different widely complex-valued Volterra kernel-based models. Finally, computer simulations reveal that the suggested algorithms outperform the previously reported frequency-domain techniques in terms of convergence speed and tracking ability.

ОБ АСИМПТОТИЧЕСКОЙ УСТОЙЧИВОСТИ РЕШЕНИЙ ЛИНЕЙНОГО ВОЛЬТЕРРОВА ИНТЕГРО-ДИФФЕРЕНЦИАЛЬНОГО УРАВНЕНИЯ ВТОРОГО ПОРЯДКА В СЛУЧАЕ НЕДИФФЕРЕНЦИРУЕМОСТИ СРЕЗАННЫХ ФУНКЦИЙ

Sufficient conditions are established for the asymptotic stability of any solution of a linear second-order Volterra integro-differential equation in the case when the truncated functions may be non-differentiable at some points of the semiaxis. To do this, first, using a non-standard method of reduction to a system, the equation under consideration is reduced to an equivalent system, and the method of transforming equations and the method of cutting functions are applied to this system. The truncated kernel and free term are associated with the coefficient of the new unknown function using the inequality between the geometric and arithmetic means of two functions. Using the method of integral inequalities, sufficient conditions are established for the boundedness on the semiaxis of all solutions of the system. Then the method of squaring equations is used to the first equation of the system and the Lyusternik-Sobolev lemma is applied. Finally, based on a non-standard substitution, we complete the proof of the asymptotic stability of solutions to a given second-order linear integro-differential equation. An illustrative example is given.

Contrast-specific imaging of histotripsy: Chirp-coded subharmonic imaging combined with Volterra Filtering

Histotripsy is a non-thermal focused ultrasound therapy under development to ablate tissue mechanically via bubble cloud activity. Real-time ultrasound imaging is used for treatment guidance. Bubble cloud hyperechogenicity is reduced in deep abdominal targets, making contrast-specific imaging an active area of research. Subharmonic imaging with chirp-coded excitation can improve bubble cloud detection, though the contrast-to-tissue ratio (CTR) was still limited to nearly 6 dB. Nonlinear components of a signal can be delineated with Volterra filtering, including those associated with bubble oscillations. In this study, we tested Volterra filtering as a means to enhance bubble cloud detection. Histotripsy bubble clouds were generated in scattering tissue-mimicking phantoms. Imaging of the bubble clouds was performed with a chirped pulse (1.9-μs duration, 7–12 MHz bandwidth). The scattered signals were processed with a subharmonic matched filter, followed by a tuned second-order Volterra filter. Volterra filtering improved the CTR for bubble cloud detection two-fold relative to matched filtering alone (12.5 dB vs 5.6 dB). Further improvement in bubble contrast was observed for third-order Volterra filtering (CTR of 20.3 dB) but at the cost of underestimating the bubble cloud area. Overall, these findings indicate the utility of Volterra filtering as a means to improve histotripsy image guidance.

ОБ ОЦЕНКЕ СНИЗУ РЕШЕНИЙ СЛАБО НЕЛИНЕЙНОГО ВОЛЬТЕРРОВА ИНТЕГРО-ДИФФЕРЕНЦИАЛЬНОГО УРАВНЕНИЯ ВТОРОГО ПОРЯДКА С ЗАПАЗДЫВАНИЯМИ

Sufficient conditions are established for lower bounds and tending to infinity as the argument of solutions to a weakly non-linear second-order Volterra integro-differential equation with delays on the half-axis grows unbounded. For this purpose, a non-standard method of reduction to a system, the method of transforming the equations of V. Volterra, the method of partial cutting, the method of integral inequalities with delays and the method of estimating solutions from below Yu.A. Ved are developed. It is shown that solutions with initial data from a certain manifold have lower bounds, and such solutions tend to infinity according to the exponential law with unlimited growth of the argument. This article proposes a technique for solving the problem of estimating from below on the half-line solutions of a weakly nonlinear Volterra-type integro-differential equation of the second order with delays, while the main role is played by the non-standard method of reduction to the system and the method of estimating solutions from below Yu.A.Ved (1965).

Offshore tethered platform springing response statistics

This paper demonstrates the validity of the Naess–Gadai method for extrapolating extreme value statistics of second-order Volterra series processes through application on a representative model of a deep water small size tension leg platform (TLP), with specific focus on wave sum frequency effects affecting restrained modes: heave, roll and pitch. The wave loading was estimated from a second order diffraction code WAMIT, and the stochastic TLP structural response in a random sea state was calculated exactly using Volterra series representation of the TLP corner vertical displacement, chosen as a response process. Although the wave loading was assumed to be a second order (non-linear) process, the dynamic system was modelled as a linear damped mass-spring system. Next, the mean up-crossing rate based extrapolation method (Naess–Gaidai method) was applied to calculate response levels at low probability levels. Since exact solution was available via Volterra series representation, both predictions were compared in this study, namely the exact Volterra and the approximate one. The latter gave a consistent way to estimate efficiency and accuracy of Naess–Gaidai extrapolation method. Therefore the main goal of this study was to validate Naess–Gaidai extrapolation method by available analytical-based exact solution. Moreover, this paper highlights limitations of mean up-crossing rate based extrapolation methods for the case of narrow band effects, such as clustering, typically included in the springing type of response.

Widely nonlinear quaternion-valued second-order Volterra recursive least squares filter

Recently, a family of the quaternion-valued second-order Volterra LMS (QSOV-LMS) algorithms were proposed to deal with 3d and 4d signals. However, it should be noted that the QSOV-LMS algorithms suffer from low convergence speed and high steady-state properties. To this end, a widely nonlinear quaternion recursive least square algorithm is proposed to train the second-order Volterra filter (WNL-QSOVRLS) for enhancing the performance of QSOV-LMS algorithms. Furthermore, to avoid the immense computation complexity, we also present a novel widely nonlinear quaternion Volterra recursive least square dichotomous coordinate descent filtering model (WNL-QSOVRLS-DCD) which introduce the quaternion dichotomous coordinate descent (QDCD) algorithm for the first time. Moreover, based on the augmented model of quaternion second-order Volterra filter, the two proposed algorithms can process quaternion-valued circular and non-circular signals well. Finally, experiments verify the performance of the proposed algorithms.

Performance enhancement of CAP-VLC system utilizing GRU neural network based equalizer

In this paper, a gated recurrent unit (GRU) neural network based equalizer is proposed for the first time to compensate for linear and nonlinear distortions in carrier-less amplitude phase (CAP) band-limited visible light communication (VLC) systems. The equalization scheme is mainly based on the GRU algorithm, which captures the dependencies between sequences within the memory size through a gating mechanism. Moreover, we experimentally compare the performance of traditional finite impulse response (FIR) based equalizer using real-valued GRU neural network (RV-GRUNN) with and without memory as well as separate complex-valued GRU neural network (SCV-GRUNN) with and without memory. Compared with the memoryless networks, the networks with memory improve the convergence speed and fitting accuracy at the expense of complexity. Compared with real-valued networks, the separate complex-valued networks have lower complexity and better generalization ability. By deploying SCV-GRUNN (M=3), a 1.8-m 560 Mb/s CAP128 VLC system is successfully demonstrated with Q factor improvement of 3.81 dB, 2.92 dB, 1.42 dB and 1.05 dB over the second-order Volterra Series, RV-GRUNN(M=1), SCV-GRUNN (M=1) and RV-GRUNN (M=3), respectively.

Collocation Approach Based on an Extended Cubic B -Spline for a Second-Order Volterra Partial Integrodifferential Equation

This paper focuses on an efficient spline-based numerical technique for numerically addressing a second-order Volterra partial integrodifferential equation. The time derivative is discretized using a finite difference scheme, while the space derivative is approximated using the extended cubic B -spline basis. The scheme is also tested for stability study to ensure that the errors do not accumulate. The convergence of the proposed scheme is also investigated. The scheme’s key benefit is that the approximate solution is produced as a smooth piecewise continuous function allowing us to approximate the solution at any location in the domain. Numerical study is performed, and the comparison of results is made to previously reported results in the literature to show the efficiency of the suggested scheme.

A family of quaternion-valued pipelined second-order Volterra adaptive filters for nonlinear system identification

This paper primarily proposes a family of quaternion Volterra filters based on the feedforward pipelined structure (QPSOVAFs) for nonlinear quaternion system identification to reduce the computational complexity. Then, the strictly nonlinear QPSOVAF (SNL-QPSOVAF), semi-widely nonlinear QPSOVAF(SWNL-QPSOVAF) and widely nonlinear QPSOVAF (WNL-QPSOVAF) are proposed. This architecture consists of several quaternion-valued second-order Volterra (SOV) modules. The structure’s nonlinear subsection executes a nonlinear mapping from the input space to an intermediate space using the feedforward SOV; the linear combiner subsection performs a linear mapping from the intermediate space to the output space. Moreover, the theoretical analysis expresses the effectiveness of the proposed QPSOVAFs in a specific condition. Finally, simulation results further prove that the proposed QPSOVAFs have good performance in identifying the quaternion-valued nonlinear system.

Recursive Maximum Correntropy Algorithms for Second-Order Volterra Filtering

As a special case of the Volterra system, the second-order Volterra (SOV) filter is very efficient for nonlinear system identification. The improved correntorpy based on the generalized Gaussian density function has been proven robust against impulsive noise. In this brief, we propose several SOV filters based on a recursive maximum correntropy (RMC) algorithm for nonlinear system identification. We first introduce a basic RMC algorithm, which faces a trade-off between filtering accuracy and tracking capability due to the use of a fixed forgetting factor (FFF). Two RMCs with variable FF (VFF) are further proposed to enhance the tracking ability. Simulation results demonstrate that our proposed algorithms outperform existing ones in impulsive noise environments and/or in time-varying systems.

Accelerated Gradient Descent Estimation for Rational Models by Using Volterra Series: Structure Identification and Parameter Estimation

This brief studies a rational model identification problem using a novel gradient descent algorithm. The rational model is approximated by a second-order Volterra system, whose parameters and structure are estimated through an accelerated gradient descent algorithm. This algorithm improves the convergence rates by computing an optimal step-size and introducing a preconditioning method. In addition, it can avoid the eigenvalue calculation if the preconditioning matrix is equal to the inverse of the information matrix. A numerical experiment is given to illustrate the performances of the proposed algorithm.

Steady-State Mean-Square Error Analysis for Adaptive Joint Process Filter Using Pipelined Feedforward Second-Order Volterra Architecture

Steady-State Mean-Square Error Analysis for Adaptive Joint Process Filter Using Pipelined Feedforward Second-Order Volterra Architecture