Second-order Volterra Series Research Articles

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.

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.

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.

Design of nonlinear predictive generalized minimum variance control for performance monitoring of nonlinear control systems

In this paper, a nonlinear predictive generalized minimum variance (NPGMV) controller is proposed and explicitly formulated for a class of nonlinear systems modeled by autoregressive second-order Volterra series, applying the polynomial approach. Hence, a new benchmark controller for performance assessment is introduced to improve the achievable control performance. Furthermore, to have an efficient control assessment, a data-driven algorithm based on the NPGMV control is presented that uses only the closed-loop operating data. In the design procedure, a multi-step cost function is defined to incorporate predictive action. Exploiting the predictive control concept enables the control scheme to handle constrained problems. Also, the proposed control algorithm utilizes an inherent integrating effect, which is essential for practical purposes. Volterra series are employed for modeling and identification of the nonlinear processes, using conventional least-squares methods. To show the effectiveness of the proposed methodology, simulation results and comparison studies are provided on a cascade Wiener model and a continuous stirred tank reactor (CSTR) chemical pilot plant. Finally, an experimental study on a pressure pilot plant is used to demonstrate the applicability of the proposed control scheme. The simulation and experimental results indicate satisfactory performance of the proposed controller.

A novel ensemble-based conceptual-data-driven approach for improved streamflow simulations

A novel ensemble-based conceptual-data-driven approach (CDDA) is developed where a data-driven model (DDM) is used to “correct” the residuals from an ensemble of hydrological model (HM) simulations. The CDDA respects hydrological processes via the HM and it benefits from the DDM's ability to simulate the complex relationship between residuals and input variables. The CDDA can accomodate any HM and DDM, allowing for different configurations to be tested. The CDDA is tested for ensemble streamflow simulation in three Swiss catchments where the HM, HBV (Hydrologiska Byråns Vattenbalansavdelning), is coupled with eight different DDMs: Multiple Linear Regression, k Nearest Neighbours Regression, Second-Order Volterra Series Model, Artificial Neural Networks, and two variants of eXtreme Gradient Boosting (XGB) and Random Forests (RF). The proposed CDDA was able to improve the mean continuous ranked probability score by 16–29% over the standalone HM. Since XGB and RF demonstrated the best performance, they are recommended for simulating the HM residuals.

Comparative Study of Statistical Analysis Methods for Nonlinear Hydrodynamic Responses of Offshore Structures

AbstractThe present study compares the statistical analysis methods for nonlinear hydrodynamic responses in offshore engineering. In particular, the Kac–Siegert and Hermite-moment methods were compared for estimating the probability distribution of the second-order responses represented via the two-term Volterra series. The Kac–Siegert method analytically formulates the probability density function (PDF) of the second-order Volterra series using an eigenvalue problem constructed with the frequency-domain transfer functions and the wave spectrum, whereas the Hermite-moment method utilizes the statistical moments to determine the coefficients of the fitting function. In addition, the probability distribution of the peak values in the second-order Volterra series with high spectral bandwidth was derived explicitly. The fatigue damage rate and the extreme response were estimated analytically. The accuracy and applicability of each method were investigated by comparing the methods with the results of the direct sampling obtained from the time series.

Adaptive Volterra Filter for Parallel MRI Reconstruction

Parallel magnetic resonance imaging (MRI) technique is able to accelerate MRI speed for reducing costs and enhancing patient’s comfortability. Parallel MRI can be categorized into two types: image-based and k-space-based methods. For k-space-based parallel MRI, missing k-space data is reconstructed by interpolating existing acquired k-space data with appropriate coefficients, which is generally considered as a linear process. However, noise cannot be suppressed or removed during the linear reconstruction process and therefore reconstructed image often suffers serious noise, especially when the acceleration factor is high. Non-linear filters are known to remove non-linear noise better. Based on the Volterra series that discovers and removes the second-order non-linear noise, we proposed a non-linear reconstruction strategy called adaptive Volterra generalized autocalibrating partial parallel acquisition (AV-GRAPPA) to reconstruct the unacquired k-space signals. For the proposed AV-GRAPPA, optimal selection of the second-order Volterra series terms is adjusted and determined for optimizing reconstruction quality. Experimental results show that the proposed method is able to better remove the reconstruction noise and suppress aliasing artifacts.

Addressing the incorrect usage of wavelet-based hydrological and water resources forecasting models for real-world applications with best practices and a new forecasting framework

Many recent studies propose wavelet-based hydrological and water resources forecasting models that have been incorrectly developed and that cannot properly be used for real-world forecasting problems. The incorrect development of these wavelet-based forecasting models occurs during wavelet decomposition (the process of extracting high- and low-frequency information into different sub-time series known as wavelet and scaling coefficients, respectively) and as a result introduces error into the forecast model inputs. The source of this error is due to the boundary condition that is associated with wavelet decomposition (and the wavelet and scaling coefficients) and is linked to three main issues: 1) using ‘future data’ (i.e., data from the future that is not available); 2) inappropriately selecting decomposition levels and wavelet filters; and 3) not carefully partitioning calibration and validation data. By not addressing these boundary conditions during wavelet decomposition, incorrectly developed wavelet-based forecasting models often result in much better performance than what is realistically achievable. We demonstrate that the discrete wavelet transform (DWT) multiresolution analysis (DWT-MRA) and maximal overlap discrete wavelet transform (MODWT) multiresolution analysis (MODWT-MRA), two commonly adopted wavelet decomposition methods used in the development of hydrological and water resources wavelet-based forecasting models, suffer from these boundary conditions and cannot be used properly for real-world forecasting. However, by following a proposed set of best (correct) practices, we show that the MODWT and à trous algorithm (AT) can be used to correctly forecast target (e.g., hydrological and water resources) processes in real-world scenarios. In this vein, we contribute a set of best practices, which focusses on deriving “boundary-corrected” wavelet and scaling coefficients from time series data, overcoming the boundary condition issues and providing hydrological and water resources modellers with a justified and coherent strategy for developing wavelet-based forecasting models that may be used for real-world forecasting problems. We coalesce these best practices into a new forecasting framework named Wavelet Data-Driven Forecasting Framework (WDDFF) that uses a combination of input variable selection and data-driven models to convert “boundary-corrected” wavelet and scaling coefficients into forecasts of a target process. Through a real-world urban water demand forecasting experiment in Montreal, Canada, we demonstrate the superiority of WDDFF against benchmark forecasting models such as (non-wavelet-based) random walk, multiple linear regression, extreme learning machine, and second-order Volterra series models. For the same case study, we also show how the WDDFF provides realistic and accurate forecasts while a recently proposed wavelet-based forecasting model that adopts the (invalid) MODWT-MRA for wavelet decomposition provides incorrect and unrealistic forecasts. We conclude that WDDFF is a useful tool for forecasting real-world hydrological and water resources processes that overcomes the limitations of many earlier wavelet-based forecasting methods and should be explored further for forecasting different processes such as streamflow, rainfall, evaporation, etc.

Sparse Modeling of Nonlinear Secondary Path for Nonlinear Active Noise Control

In many practical active noise control systems, the nonlinear secondary path (NSP) model can offer broader applications than the linear secondary path (LSP) model for the widespread existence of nonlinearities in the secondary path. However, the computational burden often limits the application of the NSP algorithms. In this paper, we apply a sparse Volterra model for the LSP and NSP instead of the complete Volterra model, which may be over parameterized for the nonlinear active noise control (NANC) system. The developed new control algorithms greatly result in reduction of computational complexity. We also provide an analysis of the NANC system to show that the noise at the canceling point could be approximated by the function expansion filters when the secondary path is modeled as the second-order Volterra series. In addition, two new function expansion forms, the even mirror Fourier nonlinear filter with a linear finite-impulse response section and the Chebyshev filter, are explored to process the nonlinearities in the NANC system using the filtered-x least mean square and filtered-error least mean square algorithm structures. The computational complexity analysis and the computer simulations validate that the new control algorithms with a sparse Volterra model for the secondary path can significantly reduce computational load without sacrificing the control performance.

Design wave method for the extreme horizontal slow-drift motion of moored floating platforms

The present study introduces a design wave method for estimating the extreme horizontal slow-drift motion of moored floating offshore platforms under extreme conditions. Here, the design wave refers to an irregular incident wave of short duration that induces the extreme response of the desired return period. The present method is composed of the following four steps: linearization of the dynamic system, probabilistic analysis of the second-order Volterra series, generation of the irregular design waves, and the fully-coupled nonlinear simulations. For generating the design waves, two different conditioning methods are presented and compared: the conditioning of the extreme response amplitude and the conditioning of the most likely extreme response profile. The procedure was applied to a deep-water semi-submersible, and the results appeared to be promising compared to the full-length nonlinear simulations.

A combined first- and second-order theory for the deckwetness prediction of sandglass-type floating body in irregular head waves

Deckwetness is one of the most important problems for seakeeping performance. With regard to the conventional ship-type Floating Production, Storage, and Offloading Units (FPSOs), the deckwetness depends on the relative vertical motion which is dominated usually by the first-order wave frequency loads. However, it can not apply to the deckwetness prediction of the sandglass-type FPSO since the influence of second-order slowly varying loads is remarkable. In this paper, a combined first- and second-order theory accounting for the first-order wave loads and nonlinear second-order slowly varying loads, is established to predict the deckwetness occurrence. By using the second-order Volterra series model, the relative vertical motion between the bow and incident wave surface can be described as a stationary non-Gaussian process. Then general formulas are derived for the prediction of the deckwetness occurrence for the sandglass-type floating body, based on the mathematical characteristics of the relative vertical motion process. As an application, the theory presented is discussed with reference to the deckwetness prediction of the sandglass-type floating model. Therein, a time domain analysis is employed to get the relative vertical motion response whereby the above formulas can be efficiently calculated. The influence factors including the length of the time simulation and the randomness of the wave phase are examined. Next, the convergence of the results is investigated with the increase of the order of non-Gaussian terms included in the formulas. By comparing with the experimental data, the reasonability and validity of the proposed method are proved.

Control performance assessment for a class of nonlinear systems using second-order Volterra series models based on nonlinear generalised minimum variance control

In this paper, control performance assessment for a class of nonlinear systems modelled by autoregressive second-order Volterra series with a general linear additive disturbance is presented. The proposed approach employs the nonlinear generalised minimum variance (NGMV) controller concept. The Volterra series models provide a natural extension of a linear convolution model with the nonlinearity considered in an additive term. The polynomial operator form is used throughout this paper for the description of the system input–output model. The closed form formulation of NGMV controller for autoregressive second-order Volterra series is presented in a polynomial form then a control assessment criterion based on the NGMV control is given. Simulation results and comparison studies are used to show the effectiveness of the proposed approach for a class of nonlinear systems.

Volterra prediction model for speech signal series

The given English phonemes, words and sentences are sampled and preprocessed. For these real measured speech signal series, time delay and embedding dimension are determined by using mutual information method and Cao's method, respectively, so as to perform phase space reconstruction of the speech signal series. By using small data set method, the largest Lyapunov exponent of the speech signal series is calculated and the fact that its value is greater than zero presents chaotic characteristics of the speech signal series. This, in fact, performs the chaotic characteristic identification of the speech signal series. By introducing second-order Volterra series, in this paper we put forward a type of nonlinear prediction model with an explicit structure. To overcome some intrinsic shortcomings caused by improper parameter selection when using the least mean square (LMS) algorithm to update Volterra model efficiency, by using a variable convergence factor technology based on a posteriori error assumption on the basis of LMS algorithm, a novel Davidon-Fletcher-Powell-based second of Volterra filter (DFPSOVF) is constructed and is performed to predict speech signal series of the given English phonemes, words and sentences with chaotic characteristics. Simulation results under MATLAB 7.0 environment show that the proposed nonlinear model DFPSOVF can guarantee its stability and convergence and there are no divergence problems in using LMS algorithm; for single-frame and multi-frame of the measured speech signals, when root mean square error (RMSE) is used as an evaluation criterion the prediction accuracy of the proposed nonlinear prediction model DFPSOVF in this paper is better than that of the linear prediction (LP) that is traditionally employed. The primary results of single-frame and multi-frame predictions are given. So, the proposed DFPSOVF model can substitute linear prediction model on certain conditions. Meanwhile, it can better reflect trends and regularity of the speech signal series and fully meet requirements for speech signal prediction. The memory length of the proposed prediction model may be selected by the embedding dimension of the speech signal series. The proposed model can present a nonlinear analysis and more valuable model structure for speech signal series, and opens up a new way to speech signal reconstruction and compression coding so as to improve complexity and process effect of speech signal processing method.

Non-linear generalised minimum variance control state space design for a second-order Volterra series model

This paper presents a non-linear generalised minimum variance (NGMV) controller for a second-order Volterra series model with a general linear additive disturbance. The Volterra series models provide a natural extension of a linear convolution model with the nonlinearity considered in an additive term. The design procedure is entirely carried out in the state space framework, which facilitates the application of other analysis and design methods in this framework. First, the non-linear minimum variance (NMV) controller is introduced and then by changing the cost function, NGMV controller is defined as an extended version of the linear cases. The cost function is used in the simplest form and can be easily extended to the general case. Simulation results show the effectiveness of the proposed non-linear method.

Estimating second-order Volterra system parameters from noisy measurements based on an LMS variant or an errors-in-variables method

This paper deals with the identification of a nonlinear SISO system modelled by a second-order Volterra series expansion when both the input and the output are disturbed by additive white Gaussian noises. Two methods are proposed. Firstly, we present an unbiased on-line approach based on the LMS. It includes a bias correction scheme which requires the variance of the input additive noise. Secondly, we suggest solving the identification problem as an errors-in-variables issue, by means of the so-called Frisch scheme. Although its computational cost is high, this approach has the advantage of estimating the Volterra kernels and the variances of both the additive noises and the input signal, even if the signal-to-noise ratios at the input and the output are low.

The Two-Dimensional Power Spectral Density: A Connection Between 2-D Rational Functions and Linear Systems

A derivation of the power spectral density (PSD) matrix of a continuous-time, linear, constant-coefficient system with white Gaussian noise input is given. The assumption of wide-sense stationarity is not made. By transforming the two-dimensional (2-D) autocorrelation function (ACF), a 2-D spectral density is obtained that takes the system's transient behavior into account. The 2-D PSD is a non-separable rational function. The results are applicable to second-order Volterra series representations of nonlinear systems. Simulations are presented for a fourth order system that compare the 2-D PSD amplitude (both theoretical and empirical) with its 1-D counterpart on 50 realizations of a short data segment containing an initial transient.

Nonlinear MPC based on a Volterra series model for greenhouse temperature control using natural ventilation

Suitable environmental conditions are a fundamental issue in greenhouse crop growth and can be achieved by advanced climate control strategies. In different climatic zones, natural ventilation is used to regulate both the greenhouse temperature and humidity. In mild climates, the greatest problem faced by far in greenhouse climate control is cooling, which, for dynamical reasons, leads to natural ventilation as a standard tool. This work addresses the design of a nonlinear model predictive control (NMPC) strategy for greenhouse temperature control using natural ventilation. The NMPC strategy is based on a second-order Volterra series model identified from experimental input/output data of a greenhouse. These models, representing the simple and logical extension of convolution models, can be used to approximate the nonlinear dynamic effect of the ventilation and other environmental conditions on the greenhouse temperature. The developed NMPC is applied to a greenhouse and the control performance of the proposed strategy will be illustrated by means of experimental results.

The asymptotic behaviour of second-order stochastic Volterra series models of slow drift response

The paper presents a detailed study of the structure and asymptotic behaviour of a second-order stochastic Volterra series model of the slow drift response of large volume compliant offshore structures subjected to random seas. A long standing challenge has been to develop efficient and accurate methods for calculating the response statistics of compliant offshore structures to random seas. Recent work has revealed that the statistical properties of the response process, which consist of a linear, first-order component and a nonlinear, second-order component, is surprisingly complex. The goal of the research work presented here is to complement efforts to develop numerical procedures to calculate the statistics of the response process.

Volterra series representation of time-frequency distributions

This paper addresses a Volterra series representation of bilinear (or quadratic) time-frequency distributions that belong to Cohen's class, whereby the analogy of the bilinear class with a second-order double Volterra series is utilized. In addition, a different viewpoint for the bilinear kernel and a complementary interpretation concerning the quadratic time-frequency distributions are provided.

Volterra series modelling and compensation of non-linear distortions caused by susceptibility difference artefacts related to the presence of ferromagnetic implants in magnetic resonance imaging

Magnetic resonance imaging is popular in medical settings due to its unique technical characteristics. However, its full potential has been limited by imaging artefacts caused by various phenomena. Previously, a methodology was proposed to characterize and reduce artefacts caused specifically by magnetic susceptibility differences. In the present work, the Volterra series approach is suggested as an alternative method for describing non-linear distortions induced by susceptibility artefacts. A second-order Volterra series is utilized for characterizing the image non-linearities using a block-by-block processing approach. Subsequently, a corresponding second-order inverse Volterra series is applied to compensate for the quantified distortions. In addition, a technique for automatic demarcation of recoverable and non-recoverable regions in magnetic resonance images is proposed to ameliorate the developed methodology. Experimental results indicate that this approach offers improved accuracy and flexibility in reducing non-linear distortions caused by magnetic susceptibility artefacts.