Showing papers on "Volterra series published in 2011"

Nonlinear Electrical Compensation for the Coherent Optical OFDM System

Abstract: A main drawback of Coherent Optical Orthogonal Frequency Division Multiplexing (CO-OFDM) system is its sensitivity to fiber nonlinearity. Nonlinear electrical equalizer based on Volterra model has been demonstrated capable of compensating fiber nonlinear distortion in an OOK or PSK optical communication system. However, the implementation complexity of a Volterra model based electrical equalizer prohibits its deployment in a real-life CO-OFDM system. In this paper, we demonstrate that the number of kernels of a Volterra model based equalizer can be significantly reduced using the modified Gram-Schmidt method with reorthogonalization techniques. The resulting “sparse” Volterra model based electrical equalizer and the electrical equalizer based on the “full” Volterra model have comparable performance and can compensate intra-channel nonlinearity of a 16-QAM 100 Gbit/s CO-OFDM System.

Algebraic and Geometric Methods in Nonlinear Control Theory

Abstract: Controllability, Observability, Realization and other Structural Properties- Realization Theory for Nonlinear Systems Three Approaches- The Local Realization of Generating Series of Finite Lie Rank- Realizations of Polynomial Systems- Symmetries and Local Controllability- The Intrinsic Geometry of Dynamic Observations- Design of Nonlinear Observers by a Two-Step-Transformation- Feedback Synthesis and Linearization Techniques- On the Input-Output Decoupling of Nonlinear Systems- Control of Nonlinear Systems Via Dynamic State-Feedback- A Classification of Nonlinear Systems Based on the Invariant Subdistribution Algorithm- Asymptotic Expansions, Root-Loci and the Global Stability of Nonlinear Feedback Systems- Everything You Always Wanted to Know About Linearization- Feedback Linearization and Simultaneous Output Block Decoupling of Nonlinear Systems- Global Feedback Linearizability of Locally Linearizable Systems- Global Aspects of Linearization, Equivalence to Polynomial Forms and Decomposition of Nonlinear Control Systems- The Extended-Linearization Approach for Nonlinear Systems Problems- About the Local Linearization of Nonlinear Systems- Optimal Control- Envelopes, Conjugate Points, and Optimal Bang-Bang Extremals- Geometry of the Optimal Control- Volterra Series and Optimal Control- Optimal Control and Hamiltonian Input-Output Systems- Discrete-Time Systems- Nonlinear Systems in Discrete Time- Local Input-Output Decoupling of Discrete Time Nonlinear Systems- Orbit Theorems and Sampling- Various other Theoretical Aspects- An Infinite Dimensional Variational Problem Arising in Estimation Theory- Iterated Stochastic Integrals in Nonlinear Control Theory- Approximation of Nonlinear Systems by Bilinear Ones- Applications- Feedback Linearization Techniques in Robotics and Power Systems- CAD for Nonlinear Systems Decoupling, Perturbations Rejection and Feedback Linearization with Applications to the Dynamic Control of a Robot Arm- A Nonlinear Feedback Control Law for Attitude Control- Identification of Different Discrete Models of Continuous Non-linear Systems Application to Two Industrial Pilot Plants- Bang-Bang Solutions for a Class of Problems Arising in Thermal Control

Simplified dynamic deviation reduction-based Volterra model for Doherty power amplifiers

Abstract: A simplified 2nd-order dynamic deviation reduction-based Volterra series model is proposed for characterizing wideband multi-carrier Doherty power amplifiers. By removing several redundant terms, this model has much less complexity but maintains excellent modeling performance, compared to the full-size model. Experimental results show that both nonlinear distortion and memory effects in the Doherty PA can be almost completely removed by employing digital predistortion with proposed model, when excited by 2- and 4-carrier WCDMA signals.

Enhanced Hammerstein Behavioral Model for Broadband Wireless Transmitters

Abstract: A novel enhanced Hammerstein behavior model consisting of a weighted memoryless polynomial followed by a Volterra filter is proposed. The weighted polynomial is used for predicting the strong static nonlinear behaviors of the power amplifiers (PAs). Since the Volterra filter is employed only for the mild dynamic nonlinearities, the filter can be implemented with low nonlinear order. Thus, this proposed model is capable of predicting both the static and dynamic nonlinearities of RF PAs with the acceptable complexity. The modeling performance of the proposed model is assessed in terms of in-band and out-of-band errors, such as normalized mean square error and adjacent channel error power ratio, and it is compared with a conventional Hammerstein, an augmented Hammerstein, and a Volterra series with respect to computation complexities such as the number of floating point operations and coefficients. The excellent estimation capability of the enhanced Hammerstein model is validated by two kinds of PAs: Si lateral diffusion metal-oxide-semiconductor and GaN high electron-mobility transistor amplifiers. Furthermore, the proposed scheme is applied to the digital predistortion (DPD) to cancel the nonlinearities of the PAs. The modeling performances and DPD experimental results clearly demonstrate the superiority of the enhanced Hammerstein scheme: the computational complexity is comparable with the augmented Hammerstein behavioral model, but the modeling performance is similar to the Volterra filter, which is the most accurate model.

Empirical Volterra-Series Modeling of Commercial Light-Emitting Diodes

Abstract: Light-emitting diodes (LEDs) constitute a low-cost alternative for optical data transmission of up to ~ 1 Gb/s. What differentiates such applications from, e.g., backhaul optical networks, is the fact that apart from their data throughput, LEDs are generally not as well characterized by the manufacturer as, for example, optical fiber amplifiers. While for simple modulation formats, this lack of knowledge is not a severe impediment; in any other situation, one may face rather complex behaviors of commercial LEDs. In this paper, the main electro-optical characteristics of LEDs are discussed, and it is shown that some popular simple nonlinear models available in the literature are inadequate in describing their dynamics. As a way out of this malady, we present a reverse-engineering approach that is based on Volterra expansions of the electro-optical characteristic function of LEDs, enabling the introduction of a realistic empirical model for commercial devices.

Digital Postcompensation Using Volterra Series Transfer Function

Abstract: We propose a noniterative digital backward propagation technique, based on an inverse modified Volterra series transfer function to postcompensate transmission linear and nonlinear impairments in the presence of optical noise. Using a single-channel 40-Gb/s nonreturn-to-zero quadrature phase-shift-keying optical signal propagated over 20 × 80 km of standard single-mode fiber, and performing digital postcompensation around the Nyquist rate, our compensation algorithm is able to surpass the maximum accuracy obtained with a symmetric split-step Fourier method, enabling us to increase the nonlinear tolerance by approximately 2 dB.

Intermodulation Distortion in Multicarrier Satellite Systems: Analysis and Turbo Volterra Equalization

Abstract: The urgent objective of transmitting high data rates over satellite, coupled with the challenge to maximize satellite mass efficiency, has necessitated that multiple carriers share the same transponder high-power amplifier (HPA). This paper presents analytical framework that characterizes the resulting intermodulation distortion (IMD) by utilizing Volterra series representation to account for the memory within the carrier itself and those associated with other carriers. Also provided is analytical evaluation of nonlinear IMD which involves computing statistical averages of higher-order products of Volterra series containing complex-valued symbols from multiple carriers. Using this theoretical characterization, novel algorithms are developed to overcome IMD in highly distortion-limited environments by employing the powerful Turbo equalization method with linear minimum mean-squared error criterion. Further, the solution is adaptive so compensation does not require prior knowledge of the HPA characteristics and can be rapidly responsive to variations in the environment. Through extensive simulations, it is shown that the proposed multicarrier analysis and Turbo Volterra techniques can be used to substantially remove IMD resulting from operating the nonlinear transponder HPA, shared by multiple carriers, near saturation. By applying more iterations of joint equalization and decoding, the solution can approach the ideal performance when feeding back correct decisions.

Mitigation of intra-channel nonlinearities using a frequency-domain volterra series equalizer

Abstract: We apply a frequency-domain Volterra series nonlinear equalizer to a 20 Gbaud NRZ-QPSK signal propagated over 1600 km. Using 2 samples/symbol we obtain a 2 dB improvement on the nonlinear tolerance over backward propagation split-step Fourier method.

Data-Driven Modeling Based on Volterra Series for Multidimensional Blast Furnace System

Abstract: The multidimensional blast furnace system is one of the most complex industrial systems and, as such, there are still many unsolved theoretical and experimental difficulties, such as silicon prediction and blast furnace automation. For this reason, this paper is concerned with developing data-driven models based on the Volterra series for this complex system. Three kinds of different low-order Volterra filters are designed to predict the hot metal silicon content collected from a pint-sized blast furnace, in which a sliding window technique is used to update the filter kernels timely. The predictive results indicate that the linear Volterra predictor can describe the evolvement of the studied silicon sequence effectively with the high percentage of hitting the target, very low root mean square error and satisfactory confidence level about the reliability of the future prediction. These advantages and the low computational complexity reveal that the sliding-window linear Volterra filter is full of potential for multidimensional blast furnace system. Also, the lack of the constructed Volterra models is analyzed and the possible direction of future investigation is pointed out.

Detecting nonlinear behaviour using the Volterra series to assess damage in beam-like structures

Abstract: Using the example of a cracked cantilever beam, this paper illustrates a means of identifying damage in structures using the so-called higher order Frequency Response Function (FRFs) which are based on the Volterra series. It is well known that, when a beam subject to a dynamic excitation vibrates, a transverse ``breathing''crack present in the beam can change the state (from open to closed and vice-versa), causing nonlinear dynamic behaviour. A simple model of a cracked cantilever beam vibrating in its first mode is proposed. Across the frequency range which encompasses the first mode of vibration, it is possible to model the response characteristics of a cracked beam using a relatively simple asymmetric bilinear oscillator. As described in this article, it is possible to use these higher order FRFs to characterise the nonlinear behaviour of the cantilever beam and investigate the qualitative relation with the parameters of the fault such as entity and location. In this study, the case of single harmonic excitation has been considered initially. Then, a new characteristic function, again based on the higher order FRFs, is proposed for detecting the crack by exploiting the fact that due to the second-order nonlinear behaviour, two harmonic inputs combine to excite the sum of their frequencies. Comparisons are made between results derived using the simple model described and those obtained from a FE model simulating some experimental tests on the beam.

Computation of Convergence Bounds for Volterra Series of Linear-Analytic Single-Input Systems

Abstract: In this paper, the Volterra series decomposition of a class of single-input time-invariant systems, analytic in state and affine in input, is analyzed. Input-to-state convergence results are obtained for several typical norms (L∞ ([0,T]), L∞ (R+) as well as exponentially weighted norms). From the standard recursive construction of Volterra kernels, new estimates of the kernel norms are derived. The singular inversion theorem is then used to obtain the main result of the paper, namely, an easily computable bound of the convergence radius. Guaranteed error bounds for the truncated series are also provided. The relevance of the method is illustrated in several examples.

A Nonlinear Frequency Domain Model for Limit Cycles in Thermoacoustic Systems with Modal Coupling

Abstract: For prediction of limit cycle oscillations of linearly unstable thermo-acoustic systems, a frequency-domain, low-order system with explicit modal coupling is developed. To this purpose, a model for the nonlinear heat source dynamics is obtained from unsteady computational fluid dynamics in combination with feed-forward neural network identification. From the neural network, an equivalent representation for the input-output relation in Volterra series form is derived, where Volterra kernels are computed in terms of the weights of the neural network. Then the kernels are transformed into the frequency domain to obtain the higher order transfer functions, through which the modes are coupled. In this way nonlinear energy exchange among the modes can be described explicitly. Comparison with a Galerkin time domain simulation shows that deviations from purely sinusoidal behaviour in the limit cycle are captured correctly, while the computational cost is drastically reduced.

A Discrete-Time Polynomial Model of Single Channel Long-Haul Fiber-Optic Communication Systems

Abstract: To mitigate various physical impairments of long-haul dense wavelength division multiplexing (DWDM) systems and exploit their system capacity, there is a need to develop a two-dimensional (time and wavelength) discrete-time input-output model which can become the foundation of signal processing for optical communications. As the first step, this paper develops a model for single channel multipulse multispan systems based on the Volterra series transfer function (VSTF) method. This model is suitable for high-bit-rate time-division multiplexing (TDM) transmission in the pseudo-linear regime and is easily extendable to the multichannel case. We overcome the well-known triple integral problem and reduce it to a simple integral. This model takes into account fiber losses, frequency chirp and photodetection, which are ignored in the literature. Furthermore, with this model we introduce the intersymbol interference (ISI), self phase modulation (SPM), intrachannel cross phase modulation (IXPM) and intrachannel four wave mixing (IFWM) coefficients to characterize the impact of these effects on the system performance. The model is in excellent agreement with SSF (split-step Fourier) simulation. To illustrate its application, we develop a constrained coding scheme based on the system model to suppress the impact of various impairments.

Comparison of Nonlinearity Measures based on Time Series Analysis for Nonlinearity Detection

Abstract: The main purpose of this paper is a study of the eciency of dierent nonlinearity detection methods based on time-series data from a dynamic process as a part of system identication. A very useful concept in measuring the nonlinearity is the denition of a suitable index to measure any deviation from linearity. To analyze the properties of such an index, the observed time series is assumed to be the output of Volterra series driven by a Gaussian input. After reviewing these methods, some modications and new indices are proposed, and a benchmark simulation study is made. Correlation analysis, harmonic analysis and higher order spectrum analysis are selected methods to be investigated in our simulations. Each method has been validated with its own advantages and disadvantages.

Experimental identification of beams with localized nonlinearities

Abstract: The idea underlying time-frequency identification techniques is that, for certain classes of structural response signals, the availability of a limited number of experimental data can be partially mitigated by taking into account the localization in time of the frequency components of the signals. This paper aims to assess the efficacy of time-frequency and time-scale estimators in the identification of weakly nonlinear systems. The example described refers to a beam characterized by a concentrated nonlinearity, whose first mode and related super-harmonics were seen to simulate a Duffing oscillator. A parametric time-frequency identification was conducted under the assumption that the beam’s input/output relationship could be approximated by a certain number of terms of the Volterra series representation, this resulting in a set of diagrams of instantaneous estimators. Though a substantial stability over time was observed only for the estimates associated with linear parameters, the identified model showe...

A Linearization Strategy for Undersampling Analog-to-Digital Converters

Abstract: The ability of high performance Radar and Broadband Systems to detect weak targets in presence of strong interferers or clutter is given by their Spurious Free Dynamic Range (SFDR). Although the Signal-to- Noise-Ratio (SNR) necessary for detection may be improved by well-known system processing gains, the dynamic range is ultimately limited by distortion terms caused by nonlinear behaviour of receiver components. The Software Defined Radio (SDR) paradigm assigns the Analog-to-Digital Converter a key role in receiver design. For systems using IF- Subsampling, linearity requirements place a heavy burden on the ADC, as SFDR signifcantly degrades with increasing input frequency. As a consequence, the ADC can only be used at input frequencies fairly below its intrinsic full power bandwidth, restricting the systems IF placement. This contribution discusses the possibility of processing ADC output data in the digital domain to achieve improved linearity. The Volterra series approach of nonlinear systems and its constrained variants are discussed. We will show in detail that for higher input frequencies, dynamic errors cause the harmonic terms to loose their in-phase ability; in higher Nyquist zones a frequency-dependend dynamic phase error has to be considered. Assumptions are backed by an evaluation of coherent data from the LTC2208 (16 Bit, 120 MSPS). A specific correction algorithm incorporating the dynamic phase error will be presented, which yielded 25 dB SFDR improvement in the 7th Nyquist Zone (360-420 MHz). The reproducibility of correction results is considered in some detail.

A Volterra Series Nonlinear Model of the Sampling Distortion in Flash ADCs Due to Substrate Noise Coupling

Abstract: This brief presents a Volterra series model to evaluate the impact of substrate noise on Flash analog-to-digital converters. The proposed approach first relates substrate noise to the induced timing uncertainty of the comparator by means of an analytical linear model. The analysis then expresses the resulting sampling distortion power by also considering the comparator's nonlinearities and thus models the signal-to-noise-and-distortion ratio (SNDR) of the converter. In particular, it is shown that substrate noise causes an FM modulation of the clock, and the resulting timing error is a joint effect of AM-FM modulation of the input signal and the coupled noise. The derivation of the analytical expression is also valid for any multitone ground perturbation and is validated on a high-resolution Flash converter. The developed nonlinear model for the SNDR is found to almost double the noise amplitude range compared with the linear model, thus predicting the impact of realistically large noise signals.

Low-complexity FPGA implementation of Volterra predistorters for power amplifiers

Abstract: In this paper we present a FPGA design of a digital predistorter (DP) for power amplifiers (PAs) regarding memory effects. As model description the baseband Volterra series are utilized. A reduction of Volterra coefficients can be achieved using their symmetry properties. An advantage of our DP approach is a direct offline model identification, without need to analytically or iteratively calculate a PA model inverse. The DP implementation is very flexible and saves FPGA ressources. Our simulation and measurement results show a good linearization performance applying simple Volterra model structures. DP with and without memory are compared.

Third-Order Volterra Kernel Identification Technique in Aerodynamics

Abstract: In this paper, we will extend a Volterra identification technique of nonlinear systems. In reality there exists a large class of weakly Nonlinear System which can be well defined by the first few kernels of the Volterra series. In general, Engineers believe that identifying high-order Volterra kernels is a big problem and hope for the advent of better identification techniques. However, with the extensive development of the Volterra kernels’ identification technique, the situation may improve. The formulas used to calculate kernels up to the third-order are given.

Fast orthogonal decomposition of Volterra cubic kernels using oblique unfolding

Abstract: Discrete-time Volterra modeling is a central topic in many application areas and a large class of nonlinear systems can be modeled using high-order Volterra series. The problem with Volterra series is that the number of parameters grows very rapidly with the order of the nonlinearity and the memory in the system. In order to efficiently implement this model, kernel eigen-decomposition can be used in the context of a Parallel-Cascade realization of a Volterra system. So, using the multilinear SVD (HOSVD) for decomposing high-order Volterra kernels seems natural. In this paper, we propose to drastically reduce the computational cost of the HOSVD by (1) considering the symmetrized Volterra kernel and (2) exploiting the column-redundancy of the associated mode by using an oblique unfolding of the Volterra kernel. Keeping in mind that the complexity of the full HOSVD for a cubic (I × I × I) unstructured Volterra kernel needs 12I4 flops, our solution allows reducing the complexity to 2I4 flops, which leads to a gain equal to six for a sufficiently large size I.

Linearization of ADCs via digital post processing

Abstract: Volterra series offers a practical tool for mathematical modeling of nonlinear systems such as Analog to Digital Converters (ADCs). In this paper, we report of ADC linearization by employing Volterra model and digital post processing. We first obtain the Volterra model from calibration data by using a Vandermonde matrix. Then, we employ an efficient method for data collection, which are employed to build an ADC module. The model is then used to estimate and compensate the nonlinearity.

A Compressed Sensing Approach for Modeling the Super-Resolution Near-Field Structure Disc System With a Sparse Volterra Filter

Abstract: In this paper, we investigate the compressed sensing (CS) algorithms for modeling a super-resolution near-field structure (super-RENS) disc system with a sparse Volterra filter. It is well known that the super-RENS disc system has severe nonlinear inter-symbol interference (ISI). A nonlinear system with memory can be well described with the Volterra series. Furthermore, CS can restore sparse or compressed signals from measurements. For these reasons, we employ the CS algorithms to estimate a sparse super-RENS read-out channel. The evaluation results show that the CS algorithms can efficiently construct a sparse Volterra model for the super-RENS read-out channel and that observable nonlinear interactions take place among restricted components in the read-out channel.

Dynamic deviation volterra predistorter designed for linearizing power amplifiers

Abstract: Polynomial models of predistorter combined by the “black box” principle have been considered. A Volterra model using one-dimensional dynamic deviation was proposed. An adaptive predistorter was synthesized for linearizing the Wiener-Hammerstein model of power amplifiers. Estimates of the linearization accuracy and a comparative analysis of predistorter models were also presented.

Analytical approach to single memristor circuits

Abstract: In this paper, we provide a novel approach to describe and analyze memristive circuits. This method is based on a Volterra series representation of the essential time functions of the circuit. This does not only provide the possibility of calculating voltages and currents over time in specific memristive networks, but describes the behavior of a general single memristor circuit, e.g. a circuit with an arbitrary linear two-port which is coupled to a memristor and a current source. We consider this approach as a first step in the direction of a comprehensive memristive network theory.

Systematic pruning of Volterra series using Wiener G-functionals for power amplifier and predistorter modeling

Abstract: The Volterra series is a well-established technique for modeling nonlinear dynamic system. Their wide adoption in predicting the behavior of Radio Frequency Power Amplifiers has been significantly hindered by the large number of coefficients involved especially when they exhibit high order of nonlinearity and strong memory effects. In this paper, we present a new pruning approach based on Wiener G-functionals which is sought for the selection of reduced number of dominant kernels in the Volterra series.