Volterra Series Expansion Research Articles

Nonlinear analysis of VCO jitter generation using Volterra series

PurposeCharge pump phase locked loops (CPPLLs) are nonlinear systems as a result of the nonlinear behavior of voltage-controlled oscillators (VCO). This paper aims to specify jitter generation of voltage controlled oscillator phase noise in CPPLLs, by considering approximated practical model for VCO.Design/methodology/approachCPPLL, in practice, shows nonlinear behavior, and usually in LC-VCOs, it follows second-degree polynomial function behavior. Therefore, the nonlinear differential equation of the system is obtained which shows the CPPLLs are a nonlinear system with memory, and that Volterra series expansion is useful for such systems.FindingsIn this paper, by considering approximated practical model for VCO, jitter generation of voltage controlled oscillator phase noise in CPPLLs is specified. Behavioral simulation is used to validate the analytical results. The results show a suitable agreement between analytical equations and simulation results.Originality/valueThe proposed method in this paper has two advantages over the conventional design and analysis methods. First, in contrast to an ideal CPPLL, in which the characteristic of the VCO’s output frequency based on the control voltage is linear, in the present paper, a nonlinear behavior was considered for this characteristic in accordance with the real situations. Besides, regarding the simulations in this paper, a behavior similar to the second-degree polynomial was considered, which caused the dependence of the produced jitter’s characteristic corner frequency on the jitter’s amplitude. Second, some new nonlinear differential equations were proposed for the system, which ensured the calculation of the produced jitter of the VCO phase noise in CPPLLs. The presented method is general enough to be used for designing the CPPLL.

Nonlinear degradation of a visible-light communication link: A Volterra-series approach

Visible light communications can be used to provide illumination and data communication at the same time. In this paper, a reverse-engineering approach is presented for assessing the impact of nonlinear signal distortion in visible light communication links. The approach is based on the Volterra series expansion and has the advantage of accurately accounting for memory effects in contrast to the static nonlinear models that are popular in the literature. Volterra kernels describe the end-to-end system response and can be inferred from measurements. Consequently, this approach does not rely on any particular physical models and assumptions regarding the individual link components. We provide the necessary framework for estimating the nonlinear distortion on the symbol estimates of a discrete multitone modulated link. Various design aspects such as waveform clipping and predistortion are also incorporated in the analysis. Using this framework, the nonlinear signal-to-interference is calculated for the system at hand. It is shown that at high signal amplitudes, the nonlinear signal-to-interference can be less than 25 dB.

Difference Frequency Response of a Detector Diode Under Dual W-Band Frequency Illumination

In this paper, a GaAs beam lead diode is illuminated by two fixed W-band millimeter wave frequencies to assess the difference frequency response at X-band. Measured results are compared to an expected mixed frequency response, utilizing a Volterra series expansion solution, which considers the reactive component influences under nonconjugate matching conditions. A nonlinear model is assessed with the solution approach to predict potential performance. A nonconjugate matched condition Volterra series expansion provides good agreement with less than 1-dB variations across the central slice of the response.

Sparse Functional Dynamical Models—A Big Data Approach

ABSTRACTNonlinear dynamical systems are encountered in many areas of social science, natural science, and engineering, and are of particular interest for complex biological processes like the spiking activity of neural ensembles in the brain. To describe such spiking activity, we adapt the Volterra series expansion of an analytic function to account for the point-process nature of multiple inputs and a single output (MISO) in a neural ensemble. Our model describes the transformed spiking probability for the output as the sum of kernel-weighted integrals of the inputs. The kernel functions need to be identified and estimated, and both local sparsity (kernel functions may be zero on part of their support) and global sparsity (some kernel functions may be identically zero) are of interest. The kernel functions are approximated by B-splines and a penalized likelihood-based approach is proposed for estimation. Even for moderately complex brain functionality, the identification and estimation of this sparse functional dynamical model poses major computational challenges, which we address with big data techniques that can be implemented on a single, multi-core server. The performance of the proposed method is demonstrated using neural recordings from the hippocampus of a rat during open field tasks. Supplementary materials for this article are available online.

On Convergence of Volterra Series Expansion of a Class of Nonlinear Systems

AbstractA fundamental issue in conducting the analysis and design of a nonlinear system via Volterra series theory is how to ensure the excitation magnitude and/or model parameters will be in the appropriate range such that the nonlinear system has a convergent Volterra series expansion. To this aim, parametric convergence bounds of Volterra series expansion of nonlinear systems described by a NARX model, which can reveal under what excitation magnitude or within what parameter range a given NARX system is able to have a convergent Volterra series expansion subject to any given input signal, are investigated systematically in this paper. The existing bound results often are given as a function of the maximum input magnitude, which could be suitable for single‐tone harmonic inputs but very conservative for complicated inputs (e.g. multi‐tone or arbitrary inputs). In this study, the output response of nonlinear systems is expressed in a closed form, which is not only determined by the input magnitude but also related to the input energy or waveform. These new techniques result in more accurate bound criteria, which are not only functions of model parameters and the maximum input magnitude but also consider a factor reflecting the overall input energy or wave form. This is significant to practical applications, since the same nonlinear system could exhibit chaotic behavior subject to a simple single‐tone input but might not with respect to other different input signals (e.g. multi‐tone inputs) of the same input magnitude. The results provide useful guidance for the application of Volterra series‐based theory and methods from an engineering point of view. The Duffing equation is used as a benchmark example to show the effectiveness of the results.

Dual Frequency Millimeter-Wave Illumination of a Detector Diode

In this letter, a GaAs beam lead diode is illuminated by two fixed W-band millimeter-wave frequencies to assess the difference frequency response at the X-band. Measured results are compared with an expected mixed frequency response, utilizing a Volterra series expansion solution, which considers the reactive component influences under nonconjugate matching conditions. A nonlinear model is assessed with the solution approach to predict potential performance. A nonconjugate matched condition provides good agreement with less than 2-dB variation across the central slice of the response.

Modeling and study of nonlinear effects in electrodynamic shakers

An electrodynamic shaker is inherently a nonlinear electro-mechanical system. In this work, we have developed a lumped parameter model for the entire electromechanical system, developed an approach to non-destructively determine these parameters, and predict the nonlinear response of the shaker. This predicted response has been validated using experimental data. Through such an approach, we have been able to accurately predict the resulting distortions in the response of the shaker and other nonlinear effects like DC offset in the displacement response. Our approach offers a key advantage vis-à-vis other approaches which rely on techniques involving Volterra Series expansions or techniques based on blackbox models like neural networks, which is that in our approach, apart from predicting the response of the shaker, the model parameters obtained have a physical significance and changes in the parameters can be directly mapped to modification in key design parameters of the shaker. The proposed approach is also advantageous in one more way: it requires measurement of only four parameters, voltage, current, displacement and acceleration for estimating shaker model parameters non-destructively. The proposed model can be used for the design of linearization controllers, prototype testing and simulation of new shaker designs as well as for performance prediction of shakers under testing conditions.

Laguerre Filter Analysis with Partial Least Square Regression Reveals a Priming Effect of ERK and CREB on c-FOS Induction.

Signaling networks are made up of limited numbers of molecules and yet can code information that controls different cellular states through temporal patterns and a combination of signaling molecules. In this study, we used a data-driven modeling approach, the Laguerre filter with partial least square regression, to describe how temporal and combinatorial patterns of signaling molecules are decoded by their downstream targets. The Laguerre filter is a time series model used to represent a nonlinear system based on Volterra series expansion. Furthermore, with this approach, each component of the Volterra series expansion is expanded by Laguerre basis functions. We combined two approaches, application of a Laguerre filter and partial least squares (PLS) regression, and applied the combined approach to analysis of a signal transduction network. We applied the Laguerre filter with PLS regression to identify input and output (IO) relationships between MAP kinases and the products of immediate early genes (IEGs). We found that Laguerre filter with PLS regression performs better than Laguerre filter with ordinary regression for the reproduction of a time series of IEGs. Analysis of the nonlinear characteristics extracted using the Laguerre filter revealed a priming effect of ERK and CREB on c-FOS induction. Specifically, we found that the effects of a first pulse of ERK enhance the subsequent effects on c-FOS induction of treatment with a second pulse of ERK, a finding consistent with prior molecular biological knowledge. The variable importance of projections and output loadings in PLS regression predicted the upstream dependency of each IEG. Thus, a Laguerre filter with partial least square regression approach appears to be a powerful method to find the processing mechanism of temporal patterns and combination of signaling molecules by their downstream gene expression.

Drought class transition analysis through different models: a case study in North China

The standardized precipitation index (SPI) and standardized runoff index (SRI) are computed for several gauge stations in Panjiakou Reservoir catchment of Luanhe Basin, a drought prone region of North China. Based on the SPI and SRI time series, two different models, a weighted Markov chain model and a Volterra adaptive filter model for chaotic time series, were established to predict drought classes and achieve both short- and long-term drought forecasting. These approaches were compared with a three-dimensional (3D) loglinear model, reported in our previous work. It was observed that all the three models have pros and cons when applied to drought prediction in Panjiakou Reservoir catchment. The 3D loglinear model is able to forecast drought class within 1 month. However, its predicting accuracy declines with the increase of prediction time scale, and this confines its application. The weighted Markov chain model is a useful tool for drought early warning. Its precision, which is significantly related to the stable condition of drought classes, is highest for Non-drought, followed by Moderate and Severe/Extreme drought, and lowest for Near-normal. The Volterra adaptive filter model for chaotic time series combined the phase space reconstruction technique, Volterra series expansion technique and adaptive filter optimization technique, and was for the first time used in a drought class transition study. This model is effective and highly precise in long-term drought prediction (for example, 12 months). It is able to provide reliable information for the medium- and long-term decisions and plans for water resources systems.

Volterra series truncation and kernel estimation of nonlinear systems in the frequency domain

The Volterra series model is a direct generalisation of the linear convolution integral and is capable of displaying the intrinsic features of a nonlinear system in a simple and easy to apply way. Nonlinear system analysis using Volterra series is normally based on the analysis of its frequency-domain kernels and a truncated description. But the estimation of Volterra kernels and the truncation of Volterra series are coupled with each other. In this paper, a novel complex-valued orthogonal least squares algorithm is developed. The new algorithm provides a powerful tool to determine which terms should be included in the Volterra series expansion and to estimate the kernels and thus solves the two problems all together. The estimated results are compared with those determined using the analytical expressions of the kernels to validate the method. To further evaluate the effectiveness of the method, the physical parameters of the system are also extracted from the measured kernels. Simulation studies demonstrates that the new approach not only can truncate the Volterra series expansion and estimate the kernels of a weakly nonlinear system, but also can indicate the applicability of the Volterra series analysis in a severely nonlinear system case.

An SIMO Nonlinear System Approach to Analysis and Design of Vehicle Suspensions

Vehicle suspension (or vibration control) systems are usually inherently nonlinear and can be modeled as single input multiple output (SIMO) system. In this paper, parametric convergence bounds for Volterra series expansion of nonlinear systems described by a SIMO nonlinear auto-regressive with exogenous inputs model are studied in the frequency domain, which can clearly indicate the parametric range in which a given nonlinear system has a convergent Volterra series expansion, referred to as parametric bound of convergence (PBoC). With the resulting PBoC of characteristic parameters, nonlinear systems with a nonlinear multiobjective performance (MOP) function can then be analyzed in the frequency domain using a nonlinear characteristic output spectrum method based on the Volterra series expansion. To demonstrate the results and method above, a vehicle suspension system, which is taken as a typical SIMO nonlinear system with a MOP function to optimize, is investigated. The results demonstrate a systematic and novel method for nonlinear analysis and design.

Nonlinearities Originated in a Red RC-LED and Their Impact on Spectrally Efficient Modulation

Nonlinear signal distortions originated in a resonant-cavity light emitting diode (RC-LED) are investigated. Using laboratory measurements and offline processing, the RC-LED’s nonlinear behavior is investigated for multilevel pulse amplitude modulation (M-PAM) and carrierless amplitude and phase modulation with the help of a Volterra-series expansion. Dynamic nonlinearities are identified which are strongly dependent on the modulation format used. The impact of these nonlinearities is then investigated for a short-range gigabit transmission link, consisting of a RC-LED and 30 m of step-index polymer optical fiber. It is shown that dynamic nonlinearities are a dominat source of signal impairments when using M-PAM at lower baud rates and that the performance of M-PAM can be considerably improved with a Volterra-series extension of a decision feedback equalizer (DFE) compared with a regular DFE.

Analysis and Design of a Precise Voltage Buffer

An open loop voltage buffer with an exact unity gain using a positive local feedback technique with a conventional source follower is proposed. Stability of the buffer is determined by evaluating the location of the poles and zeros and its linearity is studied using Volterra series expansion. The proposed buffer is laid out in 0.35-μm standard CMOS technology. Post layout simulations demonstrate that the buffer gain is close to unity with less than 0.2% error. The power consumption is 10 mw from a 3.3 V power supply and the achieved total harmonic distortion is -78 dB for a 10 MHz input frequency. Also Monte-Carlo simulations are carried out to investigate effects of random mismatches on the circuit operation.

Application of Rank-Constrained Optimisation to Nonlinear System Identification

Nonlinear System Identification has a rich history spanning at least 5 decades. A very flexible approach to this problem depends upon the use of Volterra series expansions. Related work includes Hammerstein models, where a static nonlinearity is followed by a linear dynamical system, and Wiener models, where a static nonlinearity is inserted after a linear dynamical model. A problem with these methods is that they inherently depend upon series type expansions and hence it is dificult to know which terms should be included. In this paper we present a possible solution to this problem using recent results on rank-constrained optimization. Simulation results are included to illustrate the eficacy of the proposed strategy.

Approximate observability of infinite dimensional bilinear systems using a Volterra series expansion

In this paper, the approximate observability of a class of infinite dimensional bilinear systems is investigated. The observability conditions are discussed based on a Volterra series representation of this class of systems. A testable observability criterion is derived for the case where the infinitesimal generator is self-adjoint or Riesz-spectral operator. The theoretical results are illustrated by examples.

A novel nonlinear macromodeling using time piecewise volterra series representation

Accurate and effective nonlinear system-level macromodeling has played a very important role in solving electromagnetic compatibility problems in modern circuit and system design. The technique based on Volterra series expansions provides the possibility to approximate complex nonlinear circuit models. However, the traditional Volterra series macromodel has limited ability to illustrate strong nonlinearities, resulting from the inconsistency between the complex nature and the required computing cost. In this paper, a novel nonlinear macromodeling scheme using time piecewise Volterra series representations is proposed. In this method, the whole training time series is first divided into several piecewise time series according to the nonlinear characteristics and the model fidelity required. We then process each piecewise time series by the lower order Volterra series, and collect the piecewise Volterra kernels by the Volterra kernel set. Finally, we obtain the output result by combining the Volterra kernel set and the weight function. This approach, which combines the piecewise idea with Volterra series methods, strives to deliver a macromodel that can describe strong and weak nonlinearities simultaneously. Computational results and performance data are presented for the examples of a memristor and selected nonlinear circuits. These examples demonstrate that the proposed method is efficient and precise. © 2014 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.

Synthesis of multi-variate Volterra systems by a topological assemblage scheme

The Volterra series expansion is widely employed to represent the input–output relationship of nonlinear dynamical systems. Such a representation is based on the Volterra frequency-response functions (VFRF), which can be calculated from the governing equations of the system by the harmonic probing method. This operation is straightforward for simple systems, but may become very complicated for multi-variate or high-order systems. An alternative technique for the evaluation of the VFRFs of multi-variate systems is presented here generalizing a previously reported technique that focused on uni-variate cases. Additionally, it is extended to derive, within the same framework, a set of associated linear equations (ALE). Examples of a 2-dof system with polynomial nonlinearities, a wind-excited cable and a 2-dof system subjected to ocean waves are utilized to demonstrate applications of the proposed technique.

Coupled flap and edge wise blade motion due to a quadratic wind force definition

The wind force on turbine blades, consisting of a drag and lift component, depends nonlinearly on the relative wind velocity. This relative velocity comprises mean wind speed, wind speed fluctuations and the structural response velocity. The nonlinear wind excitation couples the flap wise and edge wise response of a turbine blade. To analyze this motion coupling, an isolated blade is modelled as a continuous cantilever beam and corresponding nonlinear expressions for the drag and lift force are defined. After reduction of the model on the basis of its principal modes, the nonlinear response up to the second order is derived with the help of a Volterra series expansion and the harmonic probing technique. This technique allows for response analysis in the frequency domain, via which the combined flap and edge wise responses can easily be visualized. As a specific case, the characteristics of the NREL5 turbine blades are adopted. For both non-operating and operating conditions, blade responses in a turbulent wave field, based on a Kaimal spectrum, are determined. The second-order responses are shown to cause additional motion coupling, and moreover, are proven not to be negligible straightforwardly.

Robust nonlinear MPC based on Volterra series and polynomial chaos expansions

A robust nonlinear model predictive controller (NMPC) based on a Volterra series is proposed. Polynomial chaos expansions (PCE) are used to represent the uncertainty in the Volterra series coefficients and this uncertainty is then propagated onto the output predictions. The key advantage of the PCE is that it provides an analytical expression to compute the L2-norm of the output prediction error resulting in computational savings, compared to previously proposed techniques, which are essential for real time implementation. Terminal and input constraints based on Structured Singular Value based-norms are used to ensure convergence to a set-point and compliance with constraints in manipulated variables. The algorithm is applied to a multivariable pH neutralization system. A comparative study shows superior closed loop performance and computational efficiency of the proposed technique as compared to previously proposed algorithms.

Introducing a Green–Volterra series formalism to solve weakly nonlinear boundary problems: Application to Kirchhoff's string

This paper introduces a formalism which extends that of “Green's function” and that of “the Volterra series”. These formalisms are typically used to solve, respectively, linear inhomogeneous space–time differential equations in physics and weakly nonlinear time-differential input-to-output systems in automatic control. While Green's function is a space–time integral kernel which fully characterizes a linear problem, the Volterra series expansions involve a sequence of multi-variate time integral kernels (of convolution type for time-invariant systems). The extension proposed here consists in combining the two approaches, by introducing a series expansion based on multi-variate space–time integral kernels. This series allows the representation of the space–time solution of weakly nonlinear boundary problems excited by an “input” which depends on space and time.