Showing papers on "Volterra series published in 2019"

Deep Convolutional Networks in System Identification

Abstract: Recent developments within deep learning are relevant for nonlinear system identification problems. In this paper, we establish connections between the deep learning and the system identification communities. It has recently been shown that convolutional architectures are at least as capable as recurrent architectures when it comes to sequence modeling tasks. Inspired by these results we explore the explicit relationships between the recently proposed temporal convolutional network (TCN) and two classic system identification model structures; Volterra series and block-oriented models. We end the paper with an experimental study where we provide results on two real-world problems, the well-known Silverbox dataset and a newer dataset originating from ground vibration experiments on an F-16 fighter aircraft.

Differentiable reservoir computing

Abstract: Much effort has been devoted in the last two decades to characterize the situations in which a reservoir computing system exhibits the so-called echo state (ESP) and fading memory (FMP) properties. These important features amount, in mathematical terms, to the existence and continuity of global reservoir system solutions. That research is complemented in this paper with the characterization of the differentiability of reservoir filters for very general classes of discrete-time deterministic inputs. This constitutes a novel strong contribution to the long line of research on the ESP and the FMP and, in particular, links to existing research on the input-dependence of the ESP. Differentiability has been shown in the literature to be a key feature in the learning of attractors of chaotic dynamical systems. A Volterra-type series representation for reservoir filters with semi-infinite discrete-time inputs is constructed in the analytic case using Taylor's theorem and corresponding approximation bounds are provided. Finally, it is shown as a corollary of these results that any fading memory filter can be uniformly approximated by a finite Volterra series with finite memory.

Deep Neural Network Equalization for Optical Short Reach Communication

Abstract: Nonlinear distortion has always been a challenge for optical communication due to the nonlinear transfer characteristics of the fiber itself. The next frontier for optical communication is a second type of nonlinearities, which results from optical and electrical components. They become the dominant nonlinearity for shorter reaches. The highest data rates cannot be achieved without effective compensation. A classical countermeasure is receiver-side equalization of nonlinear impairments and memory effects using Volterra series. However, such Volterra equalizers are architecturally complex and their parametrization can be numerical unstable. This contribution proposes an alternative nonlinear equalizer architecture based on machine learning. Its performance is evaluated experimentally on coherent 88 Gbaud dual polarization 16QAM 600 Gb/s back-to-back measurements. The proposed equalizers outperform Volterra and memory polynomial Volterra equalizers up to 6th orders at a target bit-error rate (BER) of 10 − 2 by 0.5 dB and 0.8 dB in optical signal-to-noise ratio (OSNR), respectively.

Volterra Kernels Assessment via Time-Delay Neural Networks for Nonlinear Unsteady Aerodynamic Loading Identification

Abstract: Reduced-order modeling using the Volterra series approach has been successfully applied in the past decades to weakly nonlinear aerodynamic and aeroelastic systems. However, aspects regarding the i...

Modeling and compensating dynamic nonlinearities in LED photon-emission rates to enhance OWC

Abstract: LEDs can be modulated at relatively high speeds to support wireless optical data communication (OWC). Yet, particularly LEDs optimized for illumination act as a non-linear low-pass communication channel. It has become clear in recent literature that their non-linearity and low-pass behavior cannot be seen as two separable, cascaded mechanisms. Although standard nonlinear equalizer schemes, e.g. based on Volterra Series, have been proposed and tested before, our recent research results show that a more dedicated approach in which we specifically analyze the hole-electron recombination mechanisms, yield a very effective and computationally-efficient compensation approach. In this manuscript, we will review the non-linear differential equations for photon emissions, its electrical equivalent circuit and a discrete-time variant with delays and non-linearities. This can be inverted, in the sense that we can actively eliminate or mitigate the non-linear dynamic LED distortion by adequate signal processing. We propose an aggressive simplification of the compensation circuit that allows us to use a relatively simple structure with only a couple of parameters.

High-speed underwater wireless optical communications: from a perspective of advanced modulation formats [Invited]

Abstract: In this paper, recent advances in underwater wireless optical communication (UWOC) are reviewed for both LED- and LD-based systems, mainly from a perspective of advanced modulation formats. Volterra series-based nonlinear equalizers, which can effectively counteract the nonlinear impairments induced by the UWOC system components, are discussed and experimentally demonstrated. Both the effectiveness and robustness of the proposed Volterra nonlinear equalizer in UWOC systems under different water turbidities are validated. To further approach the Shannon capacity limit of the UWOC system, the probabilistic constellation shaping technique is introduced, which can overcome the inherent gap between a conventional regular quadrature amplitude modulation (QAM) format and the Shannon capacity of the channel. The experimental results have shown a significant system capacity improvement compared to the cases using a regular QAM.

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

Abstract: 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.

Analysis and Compensation of Nonidealities in Frequency Multiplier-Based High-Frequency Vector Signal Generators

Abstract: This paper expounds on the application of frequency multipliers for the frequency upconversion of amplitude- and vector-modulated signals to high frequencies. First, a complex baseband equivalent Volterra series behavioral model is derived for representing the envelope transformation of frequency multipliers. Based on this model, a low-complexity baseband digital predistortion (DPD) scheme is developed to mitigate unwanted distortions exhibited by the frequency multipliers. The proposed DPD scheme consists of three modules: 1) a polynomial predistorter (PD) is first employed to mitigate the memoryless (ML) distortions associated with the frequency conversion; 2) a $D$ th order root module is included to correct for the phase modulation (PM)–PM distortions exhibited by the frequency multiplier with a frequency multiplication factor of $D$ ; and 3) a pruned Volterra-based PD is added to correct for residual distortions attributed to the output stage. As a proof of concept using a low-complexity experimental setup, the proposed DPD scheme is used to linearize several frequency multipliers, namely, a frequency doubler with an output at 12.5 GHz, a frequency quadrupler to produce an output at 25 GHz, and a frequency tripler with an output at 63 GHz. The proposed DPD scheme showed excellent capacity to linearize these frequency multipliers when driven by orthogonal frequency division multiplexing signals with modulation bandwidths up to 400 MHz with less than 56 coefficients.

Non-linear process convolutions for multi-output Gaussian processes

Abstract: The paper introduces a non-linear version of the process convolution formalism for building covariance functions for multi-output Gaussian processes. The non-linearity is introduced via Volterra series, one series per each output. We provide closed-form expressions for the mean function and the covariance function of the approximated Gaussian process at the output of the Volterra series. The mean function and covariance function for the joint Gaussian process are derived using formulae for the product moments of Gaussian variables. We compare the performance of the non-linear model against the classical process convolution approach in one synthetic dataset and two real datasets.

A Behavioral Dynamic Nonlinear Model for Time-Interleaved ADC Based on Volterra Series

Abstract: The non-ideal circuit implementations cause a significant degradation in the performance of the time-interleaved analog-to-digital converter (TIADC) system. In this paper, a behavioral model for the TIADC based on Volterra series is proposed to model the dynamic nonlinearities in TIADC. The time-domain and frequency-domain expressions of the behavioral model based on hybrid Volterra series are derived first. Then, the discrete-time equivalent model is proposed by transforming the hybrid TIADC system to a discrete time one based on discrete-time Volterra series only. The derivations give a theoretical foundation to use discrete-time Volterra series to model the mixed-domain TIADC system, which makes it possible to make full use of the related existing derivations, conclusions, and methodologies on discrete-time Volterra series. We also summarize some common special cases of Volterra series to provide practical guidelines for ADC and TIADC practitioners. We present the main features of these models and their relationship with the Volterra series. The simulation and experimental results show the effectiveness of the proposed model.

Differentiable reservoir computing

Abstract: Much effort has been devoted in the last two decades to characterize the situations in which a reservoir computing system exhibits the so-called echo state (ESP) and fading memory (FMP) properties. These important features amount, in mathematical terms, to the existence and continuity of global reservoir system solutions. That research is complemented in this paper with the characterization of the differentiability of reservoir filters for very general classes of discrete-time deterministic inputs. This constitutes a novel strong contribution to the long line of research on the ESP and the FMP and, in particular, links to existing research on the input-dependence of the ESP. Differentiability has been shown in the literature to be a key feature in the learning of attractors of chaotic dynamical systems. A Volterra-type series representation for reservoir filters with semi-infinite discrete-time inputs is constructed in the analytic case using Taylor's theorem and corresponding approximation bounds are provided. Finally, it is shown as a corollary of these results that any fading memory filter can be uniformly approximated by a finite Volterra series with finite memory.

Volterra-series approach to stochastic nonlinear dynamics: The Duffing oscillator driven by white noise

Abstract: The Duffing oscillator is a paradigm of bistable oscillatory motion in physics, engineering, and biology. Time series of such oscillations are often observed experimentally in a nonlinear system excited by a spontaneously fluctuating force. One is then interested in estimating effective parameter values of the stochastic Duffing model from these observations---a task that has not yielded to simple means of analysis. To this end we derive theoretical formulas for the statistics of the Duffing oscillator's time series. Expanding on our analytical results, we introduce methods of statistical inference for the parameter values of the stochastic Duffing model. By applying our method to time series from stochastic simulations, we accurately reconstruct the underlying Duffing oscillator. This approach is quite straightforward---similar techniques are used with linear Langevin models---and can be applied to time series of bistable oscillations that are frequently observed in experiments.

On the Analysis and Emulation of Nonlinear Component Characteristics

Abstract: We present theoretical background and implementation considerations of Volterra series based nonlinear component models. We demonstrate the advantage of combining frequency and time domain representations and present an efficient way to find high-order filter taps.

L 0 -Norm Adaptive Volterra Filters

Abstract: The paper addresses adaptive algorithms for Volterra filter identification capable of exploiting the sparsity of nonlinear systems. While the l 1 -norm of the coefficient vector is often employed to promote sparsity, it has been shown in the literature that superior results can be achieved using an approximation of the l 0 -norm.Thus, in this paper, the Geman-McClure function is adopted to approximate the l 0 -norm and to derive l 0 -norm adaptiveVolterra filters. It is shown through experimental results, also involving a real-world system, that the proposed adaptive filters can obtain improved performance in comparison with classical approaches and l 1 -norm solutions.

Practical realization of discrete-time Volterra series for high-order nonlinearities

Abstract: Full realization of all versions of Volterra series like pure, truncated, and doubly finite Volterra series, and especially, the realization of their high orders is an intractable problem. Hence, practical implementation of Volterra series for high-order nonlinearities is not feasible with reasonable computational cost. For this reason, mathematicians, neuroscientists, and especially, biomedical and electrical engineers are forced to use only the low-order Volterra series. In this paper, we provide a full realization of off-repetitive discrete-time Volterra series (ORDVS) by departure from a traditional approach in favor of choosing a hierarchical structure. The proposed method is named fast full tantamount of off-repetitive discrete-time Volterra series (FFT-ORDVS). We have proven that the proposed off-repetitive discrete-time Volterra series approximates the basic discrete-time Volterra series very well and with much less computational complexity. In a conventional method, if $${M} +1$$ is considered as the memory length of the ORDVS, around $$2^{M}$$ math operations are needed for the full realization of it. In most cases, M is a large number and consequently, $$2^{M}$$ is too large. To solve this problem, we have proposed a simple polynomial time solution and using the proposed method, the same task is done only by 6M math operations. It means that we have found a shortcut to change an intractable problem ($${O}(2^{M}))$$ to a simple P problem (O(M)). This achievement enables researchers to use high-order kernels and consequently covers high-order nonlinearities with the lowest possible computational load. We have proven our claims mathematically and validated the performance of the proposed method using two numerical examples and a real problem.

Medium Amplitude Parallel Superposition (MAPS) Rheology, Part 1: Mathematical Framework and Theoretical Examples

Abstract: A new mathematical representation for nonlinear viscoelasticity is presented based on 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 super-set 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 data sets 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-Assisted Optical Phase Conjugation: A Hybrid Optical-Digital Scheme for Fiber Nonlinearity Compensation

Abstract: Digital nonlinearity compensation (NLC) schemes such as digital backpropagation and Volterra equalization are well known to be effective techniques in mitigating optical fiber nonlinearity, thus offering improved transmission performance. Alternatively, optical NLC, and specifically optical phase conjugation (OPC), has been proposed to relax the digital signal processing complexity. In this paper, a novel hybrid optical-digital NLC scheme combining OPC and a Volterra equalizer is proposed, termed Volterra-Assisted OPC (VAO). It has a twofold advantage: it overcomes the OPC limitation in asymmetric links and substantially enhances the performance of Volterra equalizers. When NLC is operated over the entire transmitted optical bandwidth, the proposed scheme is shown to outperform both OPC and Volterra equalization alone by up to 4.2 dB in a five-channel, 32 GBaud PM-16QAM transmission over a 1000 km EDFA-amplified fiber link. Moreover, VAO is also demonstrated to be very robust when applied to long-transmission distances, with a 2.5-dB gain over OPC-only systems at 3000 km. VAO combines the advantages of both optical and digital NLC offering a promising tradeoff between performance and complexity for future high-speed optical communication systems.

A novel Multiplicative Polynomial Kernel for Volterra series identification

Abstract: Volterra series are especially useful for nonlinear system identification, also thanks to their capability to approximate a broad range of input-output maps. However, their identification from a finite set of data is hard, due to the curse of dimensionality. Recent approaches have shown how regularized kernel-based methods can be useful for this task. In this paper, we propose a new regularization network for Volterra models identification. It relies on a new kernel given by the product of basic building blocks. Each block contains some unknown parameters that can be estimated from data using marginal likelihood optimization. In comparison with other algorithms proposed in the literature, numerical experiments show that our approach allows to better select the monomials that really influence the system output, much increasing the prediction capability of the model.

A novel derivation for modal derivatives based on Volterra series representation and its use in nonlinear model order reduction

Abstract: This paper presents a novel derivation for modal derivatives based on the Volterra series representation of nonlinear structural systems. After reviewing the classical derivation, new modal derivatives are proposed based on the employment of the Volterra theory and the variational equation approach. It turns out that the gained new derivatives are almost identical to the conventional ones, except for the fact that a sum/subtraction of eigenfrequencies results in our definition. In addition to the novel derivation, some possible impacts and applications of the new derivatives are presented and discussed, pursuing the aim that the conceptual results are also useful for practical purposes.

A Deep Learning Approach for Volterra Kernel Extraction for Time Domain Simulation of Weakly Nonlinear Circuits

Abstract: Volterra kernels are well known to be the multidimensional extension of the impulse response of a linear time invariant (LTI) system. It can be used to accurately model weakly nonlinear, specifically, polynomial nonlinearity systems. It has been used in the past for white-box model order reduction (MOR) to model frequency-domain performance metric quantities such as distortion in power amplifiers (PA). In this paper, we train a neural network from time-domain response of high-speed link buffers to extract multiple high-order kernels at once. Once the kernels are extracted, they can fully characterize the dynamics of the buffers of interest. Using the kernels, we demonstrate that time-domain response is straight-forward to obtain using super-, or multi-dimensional convolution. Previous work has used a shallow feed-forward neural network to train the system by using Gaussian noise as the identification signal. This is not convenient for the method to be compatible with existing computer-aided design tools. In this work, we directly use a pseudo random bit sequence (PRBS) to train the network. The proposed technique is more challenging because the PRBS has flat regions which have highly rich frequency spectrum and requires longer memory length, but allows the method to be compatible with existing simulation programs. We investigate different topologies including feed-forward neural network and recurrent neural network. Comparisons between training phase, inference phase, convergence are presented using different neural network topologies. The paper presents a numerical example using a 28Gbps data rate PAM4 transceiver to validate the proposed method against traditional simulation methods such as IBIS or SPICE level simulation for comparison in speed and accuracy. Using Volterra kernels promises a novel way to perform accurate nonlinear circuit simulation in the LTI system framework which is already well known and well developed. It can be conveniently incorporated into existing EDA frameworks.

Overestimation problem with ANN and VSTF in optical communication systems

Abstract: The authors investigated the problem of overestimation with the Volterra series transfer function (VSTF) and an artificial neural network (ANN), which are used for non-linear equalisers in optical communication systems. The results revealed that the risk of predicting a pseudo-random binary sequence (PRBS) pattern, which causes overestimation of the equaliser performance, occurs not only with an ANN but also with the VSTF. When using PRBS9, PRBS11 and PRBS15, the number of taps of a feedforward tapped delay line, which is required in the VSTF to predict the PRBS pattern, was the same as that with the ANN. When the second-order Volterra kernels were omitted, a larger number of taps was required in the VSTF to observe the overestimation.

Quantum Calculus-based Volterra LMS for Nonlinear Channel Estimation

Abstract: A novel adaptive filtering method called q-Volterra least mean square (q-VLMS) is presented in this paper. The q-VLMS is a nonlinear extension of conventional LMS and it is based on Jackson's derivative also known as q-calculus. In Volterra LMS, due to large variance of input signal, the convergence speed is very low. With proper manipulation, we successfully improved the convergence performance of the Volterra LMS. The proposed algorithm is analyzed for the step-size bounds and results of analysis are verified through computer simulations for nonlinear channel estimation problem.

Integrating Volterra Series Model and Deep Neural Networks to Equalize Nonlinear Power Amplifiers

Abstract: The nonlinearity of power amplifiers (PAs) has been one of the severe constraints to the performance of modern wireless transceivers. This problem is even more challenging for the fifth generation (5G) cellular system since 5G signals have extremely high peak to average power ratio. This paper develops nonlinear equalizers that exploit both deep neural networks (DNNs) and Volterra series models to mitigate PA nonlinear distortions. The DNN equalizer architecture consists of multiple one-dimension convolutional layers. The input features are designed according to the Volterra series model of nonlinear PAs. This enables the DNN equalizer to mitigate nonlinear PA distortions more effectively while avoiding over-fitting under limited training data. Experiments are conducted with both simulated data based on a Doherty nonlinear PA model and real measurement data obtained from a highly nonlinear cable TV PA. The results demonstrate that the proposed DNN equalizer has superior performance over conventional nonlinear equalization approaches.