Showing papers on "Volterra series published in 2008"

Open-Loop Digital Predistorter for RF Power Amplifiers Using Dynamic Deviation Reduction-Based Volterra Series

Abstract: In this paper, we propose an efficient open-loop digital predistorter (DPD) derived from the dynamic deviation reduction-based Volterra series that allows compensation for both nonlinear distortion and memory effects induced by RF power amplifiers in wireless transmitters. In this approach, the parameters of the predistorter can be directly extracted from an offline system identification process. This eliminates the usual requirement for a closed-loop real-time parameter adaptation, which dramatically reduces the implementation complexity of the system. It is shown that a further reduction in system complexity can be achieved by applying under-sampling theory in the model extraction and utilizing parameter interpolation in the DPD implementation. Experimental results show that by utilizing this technique with only a small number of parameters, nonlinear distortion induced by the PA can be significantly reduced, as evaluated by both adjacent channel power ratio reduction and normalized root mean square error improvement. A comparison with a memoryless polynomial function based predistorter and an analysis of the impact of decresting are also presented.

Digital Predistortion for Envelope-Tracking Power Amplifiers Using Decomposed Piecewise Volterra Series

Abstract: Due to dynamic changes of supply voltage, envelope-tracking (ET) power amplifiers (PAs) exhibit very distinct characteristics in different power regions. It is very difficult to compensate the distortion induced by these amplifiers by employing conventional digital predistortion techniques. In this paper, by introducing a new piecewise Volterra model based on a vector threshold decomposition technique, we first set several thresholds in the input power level according to the PA characteristics, and decompose the input complex envelope signal into several sub-signals by using these thresholds. We then process each sub-signal separately by employing the dynamic deviation reduction-based Volterra series, and finally recombine them together to produce the predistorted output. Experimental results show that by using this new decomposed piecewise digital predistorter model, the distinct characteristics of the ET system at different signal power levels can be accurately modeled, and thus, the distortion, including both static nonlinearities and memory effects, caused by the amplifier nonlinear behavior can be effectively compensated.

Multilayer Perceptrons: Approximation Order and Necessary Number of Hidden Units

Abstract: This paper considers the approximation of sufficiently smooth multivariable functions with a multilayer perceptron (MLP). For a given approximation order, explicit formulas for the necessary number of hidden units and its distributions to the hidden layers of the MLP are derived. These formulas depend only on the number of input variables and on the desired approximation order. The concept of approximation order encompasses Kolmogorov-Gabor polynomials or discrete Volterra series, which are widely used in static and dynamic models of nonlinear systems. The results are obtained by considering structural properties of the Taylor polynomials of the function in question and of the MLP function.

Frequency-Selective Predistortion Linearization of RF Power Amplifiers

Abstract: This paper presents a frequency-selective RF vector predistortion linearization system for RF multicarrier power amplifiers (PAs) affected by strong differential memory effects. Differential memory effects can be revealed in two-tone experiment by the divergence for increasing tone-spacing of the vector Volterra coefficients associated with the lower and upper intermodulations tones. Using large-signal vector measurement with a large-signal network analyzer, a class-AB LDMOS RF PA is demonstrated to exhibit a strong differential memory effect for modulation bandwidth above 0.3 MHz. New frequency-selective RF and baseband predistortion linearization algorithms are proposed to separately address the linearization requirements of the interband and inband intermodulation products of both the lower and upper sidebands. Theoretical verification of the algorithms are demonstrated with Matlab simulations using a Volterra/Wiener PA model with memory effects. The baseband linearization algorithm is next implemented in a field-programmable gate array and experimentally investigated for the linearization of the class-AB LDMOS PA for two carrier wideband code-division multiple-access signals. The ability of the algorithm to selectively linearize the two interband and four inband intermodulation products is demonstrated. Adjacent channel leakage ratio of up to 45 dBc for inband and interband are demonstrated experimentally at twice the typical fractional bandwidth.

Application of Multi-Input Volterra Theory to Nonlinear Multi-Degree-of-Freedom Aerodynamic Systems

Abstract: This paper presents a reduced-order-modeling approach for nonlinear, multi-degree-of-freedom aerodynamic systems using multi-input Volterra theory. The method is applied to a two-dimensional, 2 degree-of-freedom transonic airfoil undergoing simultaneous forced pitch and heave harmonic oscillations. The so-called Volterra cross kernels are identified and shown to successfully model the aerodynamic nonlinearities associated with the simultaneous pitch and heave motions. The improvements in accuracy over previous approaches that effectively ignored the cross kernels by using superposition are demonstrated.

Nonlinear Modeling of Causal Interrelationships in Neuronal Ensembles

Abstract: The increasing availability of multiunit recordings gives new urgency to the need for effective analysis of ldquomultidimensionalrdquo time-series data that are derived from the recorded activity of neuronal ensembles in the form of multiple sequences of action potentials-treated mathematically as point-processes and computationally as spike-trains. Whether in conditions of spontaneous activity or under conditions of external stimulation, the objective is the identification and quantification of possible causal links among the neurons generating the observed binary signals. A multiple-input/multiple-output (MIMO) modeling methodology is presented that can be used to quantify the neuronal dynamics of causal interrelationships in neuronal ensembles using spike-train data recorded from individual neurons. These causal interrelationships are modeled as transformations of spike-trains recorded from a set of neurons designated as the ldquoinputsrdquo into spike-trains recorded from another set of neurons designated as the ldquooutputsrdquo. The MIMO model is composed of a set of multiinput/single-output (MISO) modules, one for each output. Each module is the cascade of a MISO Volterra model and a threshold operator generating the output spikes. The Laguerre expansion approach is used to estimate the Volterra kernels of each MISO module from the respective input-output data using the least-squares method. The predictive performance of the model is evaluated with the use of the receiver operating characteristic (ROC) curve, from which the optimum threshold is also selected. The Mann-Whitney statistic is used to select the significant inputs for each output by examining the statistical significance of improvements in the predictive accuracy of the model when the respective inputs is included. Illustrative examples are presented for a simulated system and for an actual application using multiunit data recordings from the hippocampus of a behaving rat.

An Approximation of Volterra Series Using Delay Envelopes, Applied to Digital Predistortion of RF Power Amplifiers With Memory Effects

Abstract: In this letter, a new model for digital predistortion (DPD) of radio frequency power amplifiers for wide-band applications is proposed. The model is based on a spline approximation of Volterra series by considering second-order cross-terms. The advantage of the spline cross-term model is a reduction in the number of model parameters. We further reduce the model order by operating on delayed envelope samples, rather than the complex baseband samples. A block of wide-band code-division multiple access signal is sent through a 90 W class-AB power amplifier, based on the freescale LDMOS output device, and the input/output baseband samples were used to identify the DPD parameters.

Evaluation of the potential of periodically operated reactors based on the second order frequency response function

Abstract: A new, fast and easy method for analysing the potential for improving reactor performance by replacing steady state by forced periodic operation is presented. The method is based on Volterra series, generalized Fourier transform and the concept of higher-order frequency response functions (FRFs). The second order frequency response function, which corresponds to the dominant term of the non-periodic (DC) component, G2(ω, −ω), is mainly responsible for the average performance of the periodically operated processes. Based on that, in order to evaluate the potential of periodic reactor operation, it is enough to derive and analyze G2(ω, −ω). The sign of this function defines the sign of the DC component and reveals whether a performance improvement by cycling is possible compared to optimal steady state process. The method is used to analyze the periodic performance of a continuous stirred tank reactor (CSTR), a plug flow tubular reactor (PFTR) and a dispersive flow tubular reactor (DFTR), after introducing periodic changes of the input concentrations. A homogeneous, n-th order reaction is studied under isothermal conditions.

Nonlinear control of the air feed of a fuel cell

Abstract: This document presents the design of a hierarchical control to regulate the oxygen excess ratio of a fuel cell. The master controller calculates the necessary air flow to stabilize the oxygen excess ratio at a fixed set point. A nonlinear model based predictive controller (NMPC) using a Volterra series model is used as a master controller. The slave controller, a nonlinear PI, uses the reference of the air flow calculated by the master controller to stabilize the air flow in the compressor and allows reference tracking. The proposed control strategy is applied to full nonlinear model of a fuel cell in which simulations are carried out.

Turbo Equalization of Non-Linear Satellite Channels Using Soft Interference Cancellation

Abstract: Satellite communication channels can be well described as non-linear functions with memory. The Volterra series representation provides accurate modeling of satellite channel dynamics, and thus, it constitutes a widely used approach to mathematically describe them. In this work, iterative correction of the non-linear distortion introduced by such channels is considered, by employing a soft interference canceller operating in a turbo equalization framework.

Applications of Multilinear and Waveform Relaxation Methods for Efficient Simulation of Interconnect-Dominated Nonlinear Networks

Abstract: Applications of multilinear and waveform relaxation methods are presented for efficient transient analysis of interconnect-dominated nonlinear networks. In this paper, two procedures that realize these well-known and fundamental theories in conventional circuit simulation tools are developed by taking advantage of the unique characteristics of interconnect networks. The multilinear theory uses the Volterra functional series to decompose the nonlinear network into multiple linear networks. Then, the solutions of the mildly nonlinear network are obtained from the linear combinations of sequences of responses of the decomposed linear networks. On the other hand, the waveform relaxation technique is used to solve networks with strong nonlinearity. The networks are partitioned into linear and nonlinear subnetworks and each subnetwork is solved iteratively using the waveform relaxation technique. Simplified analysis steps that give good insight into these techniques are also derived analytically. Finally, the accuracy and efficiency of the methods are verified with two examples.

Low-pass equivalent feedback topology for Power Amplifier modeling

Abstract: In [1], Pedro et al. have presented an RF feedback model that was conceived to match the physical behavior of a general Power Amplifier (PA) circuit. Unfortunately, a procedure for the extraction of this model’s parameters has not yet been presented, because of the difficulties introduced by its recursive topology and the limited frequency band that is accessible in both the PA input and output ports. Nevertheless, this model has been used by various research groups as a design basis of new PA behavioral models, which generally approximate the feedback structure by a non-recursive Volterra series topology. This paper presents a new model whose parameters are easy to extract, and that keeps the original topological information of [1] by maintaining a feedback structure.

The cube coefficient subspace architecture for nonlinear digital predistortion

Abstract: In this paper, we present the cube coefficient subspace (CCS) architecture for linearizing power amplifiers (PAs), which divides the overparametrized Volterra kernel into small, computationally efficient subkernels spanning only the portions of the full multidimensional coefficient space with the greatest impact on linearization. Using measured results from a Q-band solid state PA, we demonstrate that the CCS predistorter architecture achieves better linearization performance than state-of-the-art memory polynomials and generalized memory polynomials.

Numerical Solution of Multiple Nonlinear Volterra Integral Equations

Abstract: We analyze a discretization method for solving nonlinear integral equations that contain multiple integrals. These equations include integral equations with a Volterra series, instead of a single integral term, on one side of the equation. We prove existence and uniqueness of solutions, and convergence and estimates of the order of convergence for the numerical methods of solution.

On the convergence of Volterra series of finite dimensional quadratic MIMO systems

Abstract: In this paper, the Volterra series decomposition of a class of quadratic, time invariant single-input finite dimensional systems is analyzed. The kernels are given by a recursive sequence of linear PDEs in the time domain, and an equivalent algebraic recursion in the Laplace domain. This is used to prove the convergence of the Volterra series to a (possibly weak) trajectory of the system, to provide a practicable value for the radius of convergence of the input in L ∞(ℝ+) and to compute a guaranteed error bound in L ∞(ℝ+) for the truncated series. The result is then extended to MIMO systems. A numerical simulation is performed on an academic SISO example, to illustrate how easily the truncated Volterra series can be implemented.

Analytical investigation and evaluation of pulse broadening factor propagating through nonlinear optical fibers (traditional and optimum dispersion compensated fibers)

Abstract: In this paper, analytical relation for pulse width evolution and broadening in fiber systems using the Volterra series transfer function (VSTF) in linear and nonlinear cases are derived. This evaluation is done for traditional and optimum dispersion compensated fibers. Effects of group velocity dispersion (GVD) and self-phase modulation (SPM) are taken into account. It is shown that the analytical formulation can be applied to design and analysis the long hauls practical systems, and is helpful in understanding the pulse distortion caused by the interaction between SPM and GVD. The proposed relations are extracted analytically and for the first time pulse broadening factor in general case is derived.

An optimal expansion of Volterra models using independent Kautz bases for each kernel dimension

Abstract: A new solution for the problem of selecting poles of the two-parameter Kautz functions in Volterra models is proposed. In general, a large number of parameters are required to represent the Volterra kernels, although this difficulty can be overcome by describing each kernel using a basis of orthonormal functions, such as the Kautz basis. This representation has a Wiener structure consisting of a linear dynamic generated by the orthonormal basis followed by a non-linear static mapping represented by the Volterra series. The resulting Wiener/Volterra model can be truncated into fewer terms if the Kautz functions are properly designed. The underlying problem is how to select the arbitrary complex poles that fully parametrize these functions. This problem has been approached in previous research by minimizing an upper bound for the error resulting from the truncation of the kernel expansion. The present paper goes even further in that each multidimensional kernel is decomposed into a set of independent Kautz bases, each of which is parametrized by an individual pair of conjugate Kautz poles intended to represent the dominant dynamic of the kernel along a particular dimension. An analytical solution for one of the Kautz parameters, valid for Volterra models of any order, is derived. A simulated example is presented to illustrate these theoretical results. The same approach is then used to model a real non-linear magnetic levitation system with oscillatory behaviour.

Nonlinear Dynamic Compensation of Sensors Using Inverse-Model-Based Neural Network

Abstract: Many sensors (such as low-cost sensors), in essence, display strongly nonlinear dynamic behavior that cannot be calibrated by well-developed linear dynamic compensation methods. So far, no general nonlinear dynamic compensation (NLDC) method exists, although there are some approaches based on nonlinear models (including Volterra series expansion, Wiener kernels, the Hammerstein model, and finite impulse response) that were developed to compensate some special kinds of nonlinear sensors. In this paper, we suggest a general framework for NLDC, in which removal of the influence of disturbance by using an auxiliary sensor is significantly studied and presented. The inverse model and differential-estimation-filter arrays are embedded in this general framework, where a neural network is applied to approximate the inverse mapping, and differential-filter arrays are used to estimate signal differentials up to a certain order. We also discuss the existence conditions of the general framework. The detailed design procedure of this general method is given as well. Simulation and experiments are presented to illustrate the proposed general NLDC method.

Volterra-Series-Based Distortion Analysis for Optimization of Out-of-Band Terminations in GaN HEMT Devices

Abstract: This letter focuses on the critical device-level linearity issues resulting from out-of-band terminations for reliable distortion characterization in future Universal Mobile Telecommunications System-Long Term Evolution (UMTS-LTE). Using Volterra series technique, the key distortion sources arising from the envelope and harmonic components in 0.5-mm GaN HEMT were identified using commercial and in-house bias tees. With the designed in-house bias tee, the baseband performance, in comparison with the commercial bias tee, is tested through drain-bias sensing. In reference to the commercial bias tee, up to 99.3% reduction in drain modulation is achieved using the in-house bias tee. Memory-effect characterization of GaN HEMT exemplified the implications of baseband and second-harmonic load terminations, which was theoretically confirmed through Volterra series technique. Using the in-house bias tee, under two-carrier wideband code-division multiple-access excitation, up to 47-dBc 3rd-order intermodulation ratio (IMR3) is achieved at 13.5-dB backoff. This has resulted in a 5-dB IMR suppression together with the minimization of intermodulation-distortion asymmetry, confirming the possibility to achieve the 3rd Generation Partnership Project linearity specification at the device level.

State-of-the-Art in Volterra Series Modeling for ADC Nonlinearity

Abstract: The Volterra series behavioral approach has been largely applied in recent years for modeling of nonlinear dynamic systems. Only recently it was used to mathematically model time-invariable systems such as analog to digital devices (ADCs). In this paper, Volterra series modeling methods for ADC nonlinearity are briefly introduced according to a classification criterion based on the main research trends. Some possible directions in this area are also given.

Simulation of the weakly nonlinear propagation in a straight pipe: application to a real-time brassy audio effect

Abstract: Nonlinear propagation in acoustic pipes is responsible for the brightness of the brass sounds at fortissimo nuances. Using the formalism of Volterra series, an acoustic model of this phenomenon is solved as an input-output system. An identification of the Volterra kernels leads to a structure composed of linear filters, sums, and instantaneous products of signals. A digital simulation which guarantees no aliasing is derived, from which real-time versions have been implemented.

Memory Behavioral Modeling of RF Power Amplifiers

Abstract: Power amplifiers (PAs) are the most extended and widely used nonlinear devices in communications. Their non- linearities generate distortion both in amplitude and phase of the PA's output signal. The use of wideband efficient modulations in the last generation systems, such as wideband code-division multiple access (WCDMA) or orthogonal frequency-division multiplexing (OFDM), adds new problems to the use of PAs, due to the memory effects associated with the bandwidth and the increase of distortion by the non-constant envelope. In this paper it is shown the need of taking into account the memory effects in order to get a more accurate model of PA operation. We compare the performance of the memoryless model with some well-known memory models based on memory polynomials and Volterra series using a novel calibration signal.

Large and small signal distortion analysis using modified Volterra series

Abstract: A new approach to the Volterra analysis of analog circuits is presented. Volterra analysis is widely used for the calculation of harmonic and intermodulation distortion products. However, the analysis is limited to circuits experiencing small signal excitations and becomes inaccurate when the input signal amplitude increases, especially when MOS transistors are involved. In this paper, we analyze the cause of this drawback, which is no other than the Taylor series' convergence properties. Moreover, we propose a solution, by calculating the nonlinearity coefficients using a different type of polynomial expansion, the Chebyshev series. This replacement improves significantly the capabilities of Volterra analysis. We also present results comparing Chebyshev series with other types of polynomial expansions. Finally, we apply the proposed method to analyze the intermodulation distortion (IMD) of a CMOS RF power amplifier, both in the small and the large signal regimes.