Volterra Series Model Research Articles

Research on Rolling Bearing Fault Diagnosis Based on Volterra Kernel Identification and KPCA

A rolling bearing fault diagnosis method based on the Volterra series and kernel principal component analysis (KPCA) is proposed. In the proposed method, first, the improved genetic algorithm (IGA) is used to identify the Volterra series model of the bearing in four states: normal, rolling element fault, inner ring fault, and outer ring fault. The Volterra time-domain kernel is used as the feature vector for kernel principal component analysis to classify and identify the faults. The feasibility of the fault diagnosis method of the Volterra level and kernel principal component analysis is verified by the experimental results.

Regularised Volterra series models for modelling of nonlinear self-excited forces on bridge decks

Volterra series models are considered an attractive approach for modelling nonlinear aerodynamic forces for bridge decks since they extend the convolution integral to higher dimensions. Optimal identification of nonlinear systems is a challenging task since there are typically many unknown variables that need to be determined, and it is vital to avoid overfitting. Several methods exist for identifying Volterra kernels from experimental data, but a large class of them put restrictions on the system inputs, making them infeasible for section model tests of bridge decks. A least-squares identification method does not restrict the inputs, but the identified model often struggles with noisy (non-smooth) kernels, which is deemed to be unphysical and a sign of overfitting. In this work, regularised least-squares identification is introduced to improve the performance of model identification using least-squares. Standard Tikhonov regularisation and other penalty techniques that impose decaying kernels are also explored. The performance of the methodology is studied using experimental data from wind tunnel tests of a twin deck section. The regularised Volterra models show equal or better results in terms of modelling the self-excited forces, and the regularisation makes the models less prone to overfitting.

Ідентифікація окуло-моторної системи людини на основі ряду Вольтерри: застосування в системі захисту інформації

The information technology of biometric identification of a person has re-ceived further development due to the use as a source of primary data of infor-mation models of the oculo-motor system (OMS) of the «input-output» type based on the Volterra series. Eye-tracking technology is used to build models. Experimental studies of the OMS of two respondents were carried out. Based on the data obtained with the Tobii Pro TX300 eye-tracker, the transient func-tions of the first, second and third orders of the OMS when applying the Volter-ra series model were determined. This makes it possible to increase the accura-cy of OMS modeling and, as a result, to increase the reliability of recognition in the space of the proposed heuristic features, which are determined using inte-gral and differential transformations of multidimensional transient functions of OMS, which greatly simplifies the identification of features and the practical implementation of the Bayesian classifier.A high variability of the transient functions of the second and third or-ders for two respondents was revealed. Thus, it seems appropriate to use multidimensional transient functions for biometric identification.A set of heuristic features are proposed, which are determined on the basis of multidimensional transient functions obtained from eye-tracking data. The informativeness of individual features and their combinations in pairswas investigated. Two-dimensional feature spaces with the maximum value of the probability of correct recognition indicator when solving the problem of biometric identification of a person were found (Pmax=0.974). The research results were obtained using the construction of Bayesian clas-sifiers in different spaces of the proposed features by means of machine learning based on the data of the formed trainingsamples

Digital models of analog circuits for musical audio production: A review of techniques and library for automated circuit solving

In music production, many recording and mixing engineers prefer to use analog equipment as a matter of perceptual preference. Digital models of analog circuits have the potential to achieve similar perceptual qualities as hardware without the drawbacks of cost, maintenance, and availability. Many different techniques of system modeling are used in music production software, ranging on a spectrum from “black-box modeling” to “white-box modeling.” In black-box modeling, the analog system is modeled as a processing block which maps an input signal to an output signal. Examples include the linear impulse response, adaptive filters, Volterra series, and Weiner-Hammerstein models. In white-box modeling, each individual component of the analog circuit is modeled as part of the overall system. Examples include wave digital filters, state-space modeling, and modified nodal analysis. Various other techniques exist on the spectrum between these two types, using some combination of each. One example is Virtual Analog Filtering based on the Topology Preserving Transform. Machine Learning techniques have also had an important role in advancing the accuracy of digital modeling. Lastly, the “Point to Point Library,” developed by the author, will be demonstrated. This MATLAB and C + + library performs automated circuit solving for modeling audio effects.

A Nonlinear Effective Connectivity Measure Based on Granger Causality and Volterra Series

Estimating effective connectivity, especially in brain networks, is an important topic to find out the brain functions. Various effective connectivity measures are presented, but they have drawbacks, including bivariate structure, the problem in detecting nonlinear interactions, and high computational cost. In this paper, we have proposed a novel multivariate effective connectivity measure based on a hierarchical realization of the Volterra series model and Granger causality concept, namely hierarchical Volterra Granger causality (HVGC). HVGC is a multivariate connectivity measure that can detect linear and nonlinear causal effects. The performance of HVGC is compared with Granger causality index (GCI), conditional Granger causality index (CGCI), transfer entropy (TE), phase transfer entropy (Phase TE), and partial transfer entropy (Partial TE) in simulated and physiological datasets. In addition to accuracy, specificity, and sensitivity, the Matthews correlation coefficient (MCC) is used to evaluate the connectivity estimation in simulated datasets. Furthermore influence of different SNRs is investigated on the estimated connectivity. The obtained results show that HVGC with a minimum MCC of 0.76 performs well in the detection of both linear and nonlinear interactions in simulated data. HVGC is also applied to a physiological dataset that was cardiorespiratory interaction signals recorded during sleep from a patient suffering from sleep apnea. The results of this dataset also demonstrate the capability of the proposed method in the detection of causal interactions. Applying HVGC on the simulated fMRI dataset led to a high MCC of 0.78. Moreover, the results indicate that HVGC has slight changes in different SNRs. The results indicate that HVGC can estimate the causal effects of a linear and nonlinear system with a low computational cost and it is slightly affected by noise.

A Frequency-Selective Digital Predistortion Method Based on a Generalized Indirect Learning Architecture

Frequency-selective (FS) digital predistortion (DPD), which focuses on suppressing the nonlinear distortion at a specific band, has gained increased interest in recent years. However, there is a lack of a general FS DPD approach that is applicable to most scenarios and is suitable for practical implementation. This paper proposes an FS DPD method based on a generalized indirect learning architecture (GILA). Our work on this GILA-based FS DPD method is composed of two contributions. First, an FS Volterra series model structure, which consists of a basic linear part and a partial-band pre-compensation part, is proposed to construct an FS predistorter. Second, a GILA-based identification method is proposed to extract coefficients of the FS predistorter with the FS Volterra series model structure. The proposed GILA-based FS DPD is a general method with low complexity for implementation. Both simulations and experiments verify that the target band of the DPD linearization is arbitrary for the proposed GILA-based FS DPD method. As a special case, when using the proposed method to focus on suppressing the out-of-band distortion, the adjacent channel power ratio performance of the proposed method can be 7.1 dB better than that of the conventional full-band DPD.

A Bivariate Volterra Series Model for the Design of Power Amplifier Digital Predistorters.

The operation of the power amplifier (PA) in wireless transmitters presents a trade-off between linearity and power efficiency, being more efficient when the device exhibits the highest nonlinearity. Its modeling and linearization performance depend on the quality of the underlying Volterra models that are characterized by the presence of relevant terms amongst the enormous amount of regressors that these models generate. The presence of PA mechanisms that generate an internal state variable motivates the adoption of a bivariate Volterra series perspective with the aim of enhancing modeling capabilities through the inclussion of beneficial terms. In this paper, the conventional Volterra-based models are enhanced by the addition of terms, including cross products of the input signal and the new internal variable. The bivariate versions of the general full Volterra (FV) model and one of its pruned versions, referred to as the circuit-knowledge based Volterra (CKV) model, are derived by considering the signal envelope as the internal variable and applying the proposed methodology to the univariate models. A comparative assessment of the bivariate models versus their conventional counterparts is experimentally performed for the modeling of two PAs driven by a 30 MHz 5G New Radio signal: a class AB PA and a class J PA. The results for the digital predistortion of the class AB PA under a direct learning architecture reveal the benefits in linearization performance produced by the bivariate CKV model structure compared to that of the univariate CKV model.

Comparison of Feature Selection Techniques for Power Amplifier Behavioral Modeling and Digital Predistortion Linearization.

The power amplifier (PA) is the most critical subsystem in terms of linearity and power efficiency. Digital predistortion (DPD) is commonly used to mitigate nonlinearities while the PA operates at levels close to saturation, where the device presents its highest power efficiency. Since the DPD is generally based on Volterra series models, its number of coefficients is high, producing ill-conditioned and over-fitted estimations. Recently, a plethora of techniques have been independently proposed for reducing their dimensionality. This paper is devoted to presenting a fair benchmark of the most relevant order reduction techniques present in the literature categorized by the following: (i) greedy pursuits, including Orthogonal Matching Pursuit (OMP), Doubly Orthogonal Matching Pursuit (DOMP), Subspace Pursuit (SP) and Random Forest (RF); (ii) regularization techniques, including ridge regression and least absolute shrinkage and selection operator (LASSO); (iii) heuristic local search methods, including hill climbing (HC) and dynamic model sizing (DMS); and (iv) global probabilistic optimization algorithms, including simulated annealing (SA), genetic algorithms (GA) and adaptive Lipschitz optimization (adaLIPO). The comparison is carried out with modeling and linearization performance and in terms of runtime. The results show that greedy pursuits, particularly the DOMP, provide the best trade-off between execution time and linearization robustness against dimensionality reduction.

Robust Flow Field Signal Estimation Method for Flow Sensing by Underwater Robotics

The flow field is difficult to evaluate, and underwater robotics can only partly adapt to the submarine environment. However, fish can sense the complex underwater environment by their lateral line system. In order to reveal the fish flow sensing mechanism, a robust nonlinear signal estimation method based on the Volterra series model with the Kautz kernel function is provided, which is named KKF-VSM. The flow field signal around a square target is used as the original signal. The sinusoidal noise and the signal around a triangular obstacle are considered undesired signals, and the predicting performance of KKF-VSM is analyzed after introducing them locally in the original signals. Compared to the radial basis function neural network model (RBF-NNM), the advantages of KKF-VSM are not only its robustness but also its higher sensitivity to weak signals and its predicting accuracy. It is confirmed that even for strong nonlinear signals, such as pressure responses in the flow field, KKF-VSM is more efficient than the commonly used RBF-NNM. It can provide a reference for the application of the artificial lateral line system on underwater robotics, improving its adaptability in complex environments based on flow field information.

Featured Cover

The cover image is based on the Research Article Nonlinear Structural Damage Detection using Output only based Volterra Series Model by Jun Li et al., https://doi.org/10.1002/stc.2802.

Multi-step ahead short-term wind speed forecasting approach coupling variational mode decomposition, improved beetle antennae search algorithm-based synchronous optimization and Volterra series model

Accurate wind speed forecasting has become an indispensable part in the dispatching and management of wind power generation. However, the volatility and intermittency of wind speed impede the improvement of forecasting accuracy. To figure out the problem and strengthen prediction performance, a novel hybrid framework consisting of variational mode decomposition (VMD), phase space reconstruction (PSR), improved beetle antenna search (BAS) and Volterra series model is established for multi-step ahead short-term wind speed forecasting. To confirm the performance of the proposed compound model, three datasets from the Sotavento Galicia wind farm are used for multi-step short-term wind speed forecasting experiment. The prediction results indicate that: (a) the prediction precision of the proposed model is significantly improved by employing VMD based on central frequency observation approach, which can weaken the non-stationarity of the original series; (b) the proposed DEBAS algorithm based on hierarchical strategy achieves the best performance compared with other comparative algorithms on the benchmark functions, because DEBAS can expand population diversity in the later iteration; (c) the forecasting performance of the hybrid model can be further enhanced by the proposed synchronous optimization strategy based on the improved DEBAS.

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.

Hysteresis Modeling of Magnetic Shape Memory Alloy Actuator Based on Volterra Series

Magnetic shape memory alloy-based actuator (MSMA-BA) has the capability to generate the micro–nano-scale precision positioning, which has unparalleled virtue in contrast with the traditional motion driving mechanism. Nevertheless, the output displacement of the MSMA-BA exhibits complex hysteresis nonlinearity, which hinders its utility in the high-precision positioning field. In this article, an online Volterra series model based on wavelet neural network (VSM-WNN) is proposed to describe the rate-dependent hysteresis of the MSMA-BA. The kernel function of the VSM is extracted with a WNN, in which the feasibility of this extraction is proved in theory via Taylor expansion of the wavelet function. Finally, the validity of the proposed model is confirmed by means of experiments. Experimental results indicate that the VSM-WNN has a remarkable performance in describing the traits of hysteresis behavior of the MSMA-BA.

Nonlinear structural damage detection using output‐only Volterra series model

Volterra series is a promising technique with great potential for nonlinear system identification. The conventional Volterra series model computes the output responses by performing multiple convolutions between the input excitation and Volterra kernels function. However, the difficulty in acquiring the excitation forces of civil engineering structures under operating conditions greatly limits the application of using Volterra series-based method for system identification. This paper proposes an output-only-based approach using Volterra series model for nonlinear structural damage detection, by quantifying the nonlinear behavior of structures without the prior knowledge of external excitations. The proposed approach uses the structural responses measured at two different locations to identify the kernel function parameters and evaluate the contribution of nonlinear components in the measured responses. The ratio between the standard deviation of the nonlinear components and that of the overall structural response is adopted as damage-sensitive index to quantify the contributions from these two adjacent sensors for performing nonlinear structural damage detection. Numerical studies on a beam structure with a breathing crack under different levels of white noise excitations and experimental studies on a precast segmental concrete column subjected to ground motions with different peak ground acceleration (PGA) values are conducted to validate the capability and accuracy of using the proposed approach for nonlinear structural damage detection. The results demonstrate that the proposed approach is capable of performing nonlinearity quantification effectively and locating structural nonlinear damage. The increasing damage index value can also be used to register the increasing damage severity.

Laguerre–Volterra Feed-Forward Neural Network for Modeling PAM-4 High-Speed Links

In this article, we present a PAM-4 IBIS-AMI model derived from machine learning for time-domain simulation. More specifically, we report a Laguerre–Volterra-expanded feed-forward neural network (LVFFN) approach with one hidden layer and ten neurons to model the 28-Gb/s PAM-4 high-speed link buffer. The proposed LVFFN model reduces the model size and improves the computational efficiency dramatically compared with the Volterra series model and other transitional artificial neural network models. The LVFFN model is implemented in IBIS-AMI, an industrial standard, and is simulated in existing software platforms for eye-diagram analysis. This work has two innovations: 1) we propose a method that dramatically reduces the neural network model complexity through a Laguerre–Volterra expansion when modeling weakly nonlinear systems with memory and 2) we implement an LVFFN model into IBIS-AMI to enhance the model interoperability and transportability.

A Code-Domain, In-Band, Full-Duplex Wireless Communication Link With Greater Than 100-dB Rejection

This article presents a CMOS-based, code-domain (CD), full-duplex (FD) transceiver operating in a link at 1 GHz. The CD FD link rejects in-band transmitter self-interference (TX SI) by more than 100 dB through a combination of pseudo-noise (PN) code orthogonality, circulator, and digital cancellation algorithms. A nonmagnetic CMOS circulator based on switched transmission lines is used as an antenna interface with >40-dB maximum rejection. Then, transmitter (TX) and receiver (RX) modulators spread the TX SI and correlate the desired RX signal in the RF domain. Orthogonality between the PN codes allows an additional >40-dB maximum rejection relaxing the FD transceiver linearity requirements. Finally, digital self-interference cancellation (SIC) eliminates any residual TX SI in the digital domain using a least mean squares (LMS) estimation for the SI channel based on a nonlinear truncated Volterra series model. The digital SIC can have almost 40 dB of rejection depending on the SI power level. An implemented FD node leverages these techniques to achieve an overall rejection that is around 104 dB bringing the TX signal from 20 dBm to the noise floor of the RX at -85 dBm.

Sparse Identification of Volterra Models for Power Amplifiers Without Pseudoinverse Computation

We present a new formulation of the doubly orthogonal matching pursuit (DOMP) algorithm for the sparse recovery of Volterra series models. The proposal works over the covariance matrices by taking advantage of the orthogonal properties of the solution at each iteration and avoids the calculation of the pseudoinverse matrix to obtain the model coefficients. A detailed formulation of the algorithm is provided along with a computational complexity assessment, showing a fixed complexity per iteration compared with its previous versions in which it depends on the iteration number. Moreover, we empirically demonstrate the reduction in computational complexity in terms of runtime and highlight the pruning capabilities through its application to the digital predistortion of a class J power amplifier operating under 5G-NR signals with the bandwidth of 20 and 30 MHz, concluding that this proposal significantly outperforms existing techniques in terms of computational complexity.

Comparison of nonlinear equalizers for high-speed visible light communication utilizing silicon substrate phosphorescent white LED.

Compared with multicolor-chip integrated white LEDs, phosphor-based white LEDs are more attractive for daily illumination due to lower cost and complexity, and thus they are preferable for future commercial use of visible light communication (VLC) systems. However, the application of phosphorescent white LEDs has a lower data rate than multicolor-chip integrated LEDs because of severe nonlinear impairments and limited bandwidth caused by the slow-responding phosphor. In this paper, for the first time we propose to employ phosphorescent white LEDs based on silicon substrate with adaptive bit-loading discrete multitone (DMT) modulation and a memoryless polynomial based nonlinear equalizer to achieve a high-speed VLC system. We also present a comprehensive comparison among nonlinear equalizers based on the Volterra series model, memory polynomial model, memoryless polynomial model and deep neural network (DNN) with experimental results utilizing a silicon substrate phosphorescent white LED, and provide detailed suggestions on how to choose the most suitable nonlinear mitigation scheme considering different practical conditions and the tradeoff between complexity and performance. Beyond 3.00 Gb/s DMT VLC transmission over 1-m indoor free space is successfully demonstrated with bit error rate (BER) under the 7% forward error correction (FEC) limit of 3.8×10-3. As far as we know, this is the highest data rate ever reported for VLC systems based on a single high-power phosphorescent white LED.

Behavior Modeling of the Amplifier Nonlinearity With the AC Josephson Voltage Standard

Behavior models of electronic components are critical in many metrological and measurement areas, e.g., time-domain simulations for analog-integrated circuit design and predistortion of the amplifier to reduce the distortion. This paper describes the use of pulse-driven ac Josephson voltage standard (ACJVS) to synthesize quantum accurate waveforms with ultralow distortion and noise floor to model the amplifier nonlinearity, which is a promising way to increase the modeling accuracy. Four Volterra series models including the memory polynomial model and three submodels of the dynamic deviation reduction model are analyzed and compared in detail. Normalized mean square error is calculated to evaluate the modeling accuracy. The difference of the measured and predicted outputs of the amplifier are compared in both time and frequency domains. The result shows that high-performance models can be achieved, which demonstrates the feasibility of modeling nonlinear amplifier behavior with the ACJVS.

Multi-variable Volterra kernels identification using time-delay neural networks: application to unsteady aerodynamic loading

In the last decades, the Volterra series theory has been used to construct reduced-order models of nonlinear systems in engineering and applied sciences. For the particular case of weakly nonlinear aerodynamic and aeroelastic systems, the Volterra series theory has been tested as an alternative to the high computing costs of CFD methods. The Volterra series model determination depends on identifying the kernels associated with the respective convolution integrals. The Volterra kernels identification has been tried in many ways, but the majority of them addresses only the direct kernels of single-input, single-output nonlinear systems. However, multiple-input, multiple-output relations are the most typical case for many dynamic systems. In this case, the so-called Volterra cross-kernels represent the internal couplings between multiple inputs. Not many generalizations of the single-input kernel identification methods to multi-input Volterra kernels are available in the literature. This work proposes a methodology for the identification of Volterra direct kernels and cross-kernels, which is based on time-delay neural networks and the relationship between the kernels functions and the internal parameters of the network. Expressions to derive the pth-order Volterra direct kernels and cross-kernels from the internal parameters of a trained time-delay neural network are derived. The method is checked with a two-degree-of-freedom, two-input, one-output nonlinear system to demonstrate its capabilities. The application to a mildly nonlinear unsteady aerodynamic loading due to pitching and heaving motions of an airfoil is also evaluated. The Volterra direct kernels and cross-kernels of up to third order are successfully identified using training datasets computed with CFD simulations of the Euler equations. Comparisons between CFD simulations and Volterra model predictions are presented, thereby ensuring the potential of the method to systematically extract kernels from neural networks.