Showing papers on "Volterra series published in 1997"

A new Volterra predistorter based on the indirect learning architecture

Abstract: Nonlinear compensation techniques are becoming increasingly important. We present a new Volterra-based predistorter, which utilizes the indirect learning architecture to circumvent a classical problem associated with predistorters, namely that the desired output is not known in advance. We utilize the indirect learning architecture and the recursive least square (RLS) algorithm. Specifically, we propose an indirect Volterra series model predistorter which is independent of a specific nonlinear model for the system to be compensated. Both 16-phase shift keying (PSK) and 16-quadrature amplitude modulation (QAM) are used to demonstrate the efficacy of the new approach.

Volterra series transfer function of single-mode fibers

Abstract: A nonrecursive Volterra series transfer function (VSTF) approach for solving the nonlinear Schrodinger (NLS) wave equation for a single-mode optical fiber is presented. The derivation of the VSTF is based on expressing the NLS equation In the frequency domain and retaining the most significant terms (Volterra kernels) in the resulting transfer function. Due to its nonrecursive property and closed-form analytic solution, this method can excel as a tool for designing optimal optical communication systems and lumped optical equalizers to compensate for effects such as linear dispersion, fiber nonlinearities and amplified spontaneous emission (ASE) noise from optical amplifiers. We demonstrate that a third-order approximation to the VSTF model compares favorably with the split-step Fourier (recursive) method in accuracy for power levels used in current optical communication systems. For higher power levels, there is a potential for improving the accuracy by including higher-order Volterra kernels at the cost of increased computations. Single-pulse propagation and the interaction between two pulses propagating at two different frequencies are also analyzed with the Volterra method to verify the ability to accurately model nonlinear effects. The analysis can be easily extended to include inter-channel interference in multi-user systems like wavelength-division multiple-access (WDM), time-division multiplexed (TDM), or code-division multiplexed (CDM) systems.

Linear multichannel blind equalizers of nonlinear FIR Volterra channels

Abstract: Truncated Volterra expansions model nonlinear systems encountered with satellite communications, magnetic recording channels, and physiological processes. A general approach for blind deconvolution of single-input multiple-output Volterra finite impulse response (FIR) systems is presented. It is shown that such nonlinear systems can be blindly equalized using only linear FIR filters. The approach requires that the Volterra kernels satisfy a certain coprimeness condition and that the input possesses a minimal persistence-of-excitation order. No other special conditions are imposed on the kernel transfer functions or on the input signal, which may be deterministic or random with unknown statistics. The proposed algorithms are corroborated with simulation examples.

Modeling Methodology for Nonlinear Physiological Systems

Abstract: A general modeling approach for a broad class of nonlinear systems is presented that uses the concept of principal dynamic modes (PDMs) These PDMs constitute a filter bank whose outputs feed into a multi-input static nonlinearity of multinomial (polynomial) form to yield a general model for the broad class of Volterra systems Because the practically obtainable models (from stimulus-response data) are of arbitrary order of nonlinearity, this approach is applicable to many nonlinear physiological systems heretofore beyond our methodological means Two specific methods are proposed for the estimation of these PDMs and the associated nonlinearities from stimulus-response data Method I uses eigendecomposition of a properly constructed matrix using the first two kernel estimates (obtained by existing methods) Method II uses a particular class of feedforward artificial neural networks with polynomial activation functions The efficacy of these two methods is demonstrated with computer-simulated examples, and their relative performance is discussed The advent of this approach promises a practicable solution to the vexing problem of modeling highly nonlinear physiological systems, provided that experimental data be available for reliable estimation of the requisite PDMs

Volterra models and three-layer perceptrons

Abstract: This paper proposes the use of a class of feedforward artificial neural networks with polynomial activation functions (distinct for each hidden unit) for practical modeling of high-order Volterra systems. Discrete-time Volterra models (DVMs) are often used in the study of nonlinear physical and physiological systems using stimulus-response data. However, their practical use has been hindered by computational limitations that confine them to low-order nonlinearities (i.e., only estimation of low-order kernels is practically feasible). Since three-layer perceptrons (TLPs) can be used to represent input-output nonlinear mappings of arbitrary order, this paper explores the basic relations between DVMs and TLPs with tapped-delay inputs in the context of nonlinear system modeling. A variant of TLP with polynomial activation functions-termed "separable Volterra networks" (SVNs)-is found particularly useful in deriving explicit relations with DVM and in obtaining practicable models of highly nonlinear systems from stimulus-response data. The conditions under which the two approaches yield equivalent representations of the input-output relation are explored, and the feasibility of DVM estimation via equivalent SVN training using backpropagation is demonstrated by computer-simulated examples and compared with results from the Laguerre expansion technique (LET). The use of SVN models allows practicable modeling of high-order nonlinear systems, thus removing the main practical limitation of the DVM approach.

Volterra filter equalization: a fixed point approach

Abstract: One important application of Volterra filters is the equalization of nonlinear systems. Under certain conditions, this problem can be posed as a fixed point problem involving a contraction mapping. We generalize the previously studied local inverse problem to a very broad class of equalization problems. We also demonstrate that subspace information regarding the response behavior of the Volterra filters can be incorporated to improve the theoretical analysis of equalization algorithms. To this end, a new "windowed" signal norm is introduced. Using this norm, we show that the class of allowable inputs is increased and the upper bounds on the convergence rate are improved when subspace information is exploited.

Identification of a class of nonlinear systems under stationary non-Gaussian excitation

Abstract: This paper provides new solutions to the nonlinear system identification problem when the input to the system is a stationary non-Gaussian process. We propose the use of a model called the Hammerstein series, which leads to significant reductions in both the computational requirements and the mathematical tractability of the nonlinear system identification problem. We show that unlike the Volterra series, one can obtain closed-form expressions for the Hammerstein series kernels and the quadratic coherence function in the non-Gaussian case. Estimation of the kernels and quadratic coherence function is discussed. A comparison with a nonlinear system identification approach that uses the Volterra series is provided. An automotive engineering application illustrates the usefulness of the proposed method.

Partially decoupled Volterra filters: formulation and LMS adaptation

Abstract: The adaptation of Volterra filters by one particular method-the method of least mean squares (LMS)-while easily implemented, is complicated by the fact that upper hounds for the values of step sizes employed by a parallel update LMS scheme are difficult to obtain. In this paper, we propose a modification of the Volterra filter in which the filter weights of a given order are optimized independently of those weights of higher order. Using this approach, we then solve the minimum mean square error (MMSE) filtering problem as a series of constrained optimization problems, which produce a partially decoupled normal equation for the Volterra filter. From this normal equation, we are able to develop an adaptation routine that uses the principles of partial decoupling that is similar in form to the Volterra LMS algorithm but with important structural differences that allow a straightforward derivation of bounds on the algorithm's step sizes; these bounds can be shown to depend on the respective diagonal blocks of the Volterra autocorrelation matrix. This produces a reliable set of design guidelines that allow more rapid convergence of the lower order weight sets.

Volterra analysis of spectral regrowth

Abstract: The author presents an analysis of the broadening of the bandwidth of narrowband modulated signals, commonly called spectral regrowth, caused by weak nonlinearities. The analysis uses Volterra methods and operates entirely in the frequency domain. It is applicable to signals having either discrete or continuous spectra.

Equivalent Statistical Cubicization for System and Forcing Nonlinearities

Abstract: This technical note outlines the treatment of a single-degree-of-freedom (SDOF) system containing statistically symmetric nonlinearities in both its stiffness characteristics and its excitation. Via equivalent statistical cubicization, the nonlinearities that are not in polynomial form are cast as polynomials containing first- and third-order terms. This allows the use of a Volterra series approach that yields a system of two differential equations for the first- and third-order components of the response. Transforming this system into the frequency domain produces transfer functions from which power spectral density and statistics up to the fourth order may be obtained for the system response. Finally, using the statistics within the framework of a moment-based Hermite transformation model yields an estimate of the non-Gaussian probability-density function (PDF) for the system response.

A constrained optimisation approach to the blind estimation of Volterra kernels

Abstract: A novel approach is taken for the estimation of the parameters of a Volterra model, which is based on constrained optimisation. The equations required for the determination of the Volterra kernels are formed entirely from the second and higher order statistical properties of the "output" signal to be modelled and can therefore be classed as blind in nature. These equations are highly nonlinear and their solution is achieved through a judicious use of reliably measured statistical features of the signal to be modelled, in conjunction with appropriate constraints and penalty functions. Examples are given to illustrate the method and it is evident from those that this novel approach is producing useful results in contexts that have been hitherto unattainable.

Multidimensional nonlinear myopic maps, Volterra series, and uniform neural-network approximations

Abstract: Our main result is a theorem which gives necessary and sufficient conditions under which discrete-space multidimensional myopic input-output maps with vector-valued inputs drawn from a certain large set can be uniformly approximated arbitrarily well using a structure consisting of a linear preprocessing stage followed by a memoryless nonlinear network. Noncausal as well as causal maps are considered. Approximations for noncausal maps for which inputs and outputs are functions of more than one variable are of current interest in connection with, for example, image processing.

A block LMS algorithm for third-order frequency-domain Volterra filters

Abstract: The objective of this paper is to present an adaptive algorithm to identify third-order frequency-domain Volterra filter coefficients, which correspond to the discrete Fourier transform (DFT) of the time-domain Volterra filter coefficients. The approach rests upon the block least mean square (LMS) algorithm based on the overlap-save method.

An efficient adaptive predistorter for nonlinear high power amplifier in satellite communication

Abstract: This paper presents an efficient adaptive predistortion technique for compensation of linear and nonlinear distortion caused by high-power amplifier with memory in satellite communication channels. The previous adaptive predistortion techniques, based on Volterra series modeling, are not suitable for real-time implementation due to high computational burden and slow convergence rate. In this paper, the memoryless HPA preceded by linear dynamic system is modeled by the Wiener system which is then precompensated by the proposed adaptive predistorter structured by the Hammerstein model. An adaptive algorithm for adjusting the parameters of the predistorter is derived using the stochastic gradient method. The validity of the proposed approach is confirmed via computer simulation by applying it to 16-QAM satellite communication channel where the HPA is preceded by a linear filter.

Identification of nonlinear cascade systems using paired multisine signals

Abstract: The identification of nonlinear cascade models has been widely studied, as they often reflect the physical structure of practical nonlinear systems. The problem when using such models is establishing their structure and then identifying their linear subsystems. Both can be obtained from measured Volterra kernels. By performing tests with a pair of input signals, specially designed in order to measure these kernels, enough information can be gathered to separate the linear systems. A brief introduction is given to the measurement of Volterra kernels using periodic multisine signals. A method using combined tests is then proposed to estimate the nonparametric and parametric models of the linear subsystems. An example is given for a simulated system with a second-order nonlinearity.

Nonlinear dynamical system identification using reduced Volterra models with generalised orthonormal basis functions

Abstract: Volterra models can be used to describe a wide class of nonlinear systems. However their practical use is limited due to the huge number of coefficients that need to be estimated even for simple SISO systems. Orthonormal basis functions, like distorted sine functions and Laguerre functions, have been proposed as a means to reduce the number of parameters. In linear system identification generalized orthonormal basis functions have been widely used to reduce the number of parameters one needs to estimate with very promising results. In this paper, we extend the use of generalized orthonormal basis functions to cover the nonlinear system identification and discuss the merits of such use. Finally, we give two examples on which we implement the proposed method, a CSTR system (SISO case) and a model IV fluid catalytic cracking unit (MIMO case).

Identification of cubically nonlinear systems using undersampled data

Abstract: A practical technique for identification of cubically nonlinear systems using higher order spectra of the discrete data samples of the system input and output is proposed. This technique differs from the conventional one in that it only requires the sampling frequency for the system output to be equal to twice the bandwidth of the system input, instead of six times the bandwidth of the system input. This means the demand for high speed processing and a large amount of data in the conventional approach can be greatly relieved. Two methods are developed: one is suitable for systems with a Gaussian random input, the other is suitable for systems with a non-Gaussian random input. The advantages of the two methods over their conventional counterparts are demonstrated via computer simulation.

A mixed-domain method for identification of quadratically nonlinear systems

Abstract: A new mixed-domain method for identifying Volterra transfer functions of a nonlinear system, which can be represented by a second-order truncated Volterra series, is presented in this paper. This method is built on a discrete mixed-domain Volterra model derived from analyzing both discrete time- and frequency-domain Volterra models of quadratically nonlinear systems. It is shown that the conventional discrete frequency-domain Volterra model can be derived from the discrete mixed-domain Volterra model by making certain approximations. In this sense, the frequency-domain model can be considered to be a cough version of the mixed-domain model and thus cannot outperform the mixed-domain model in terms of modeling capability. In addition, the new method is shown to be able to properly identify the Volterra transfer functions even when the output of the quadratically nonlinear system is aliased. Based on this insight, a new anti-aliasing frequency-domain method, which is immune from the output aliasing problem, is developed. The superiority of these new methods over the conventional method is demonstrated by using them to analyze known quadratically nonlinear systems.

A complex pi-sigma network and its application to equalizaton of nonlinear satellite channels

Abstract: Digital satellite communication channels have a nonlinearity with memory due to saturation characteristics of the high power amplifier in the satellite and transmitter/receiver linear filters used in the overall system. In this paper, we propose a network structure and a learning algorithm for complex pi-sigma network (CPSN) and exploit CPSN in the problem of equalization of nonlinear satellite channels. The proposed CPSN is a complex-valued extension of real-valued pi-sigma network (PSN) that is a higher-order feedforward network with fast learning while greatly reducing network complexity by utilizing efficient form of polynomials for many input variables. The performance of the proposed CPSN is demonstrated by computer simulation on the equalization of complex-valued QPSK input symbols distorted by a nonlinear channel modeled as a Volterra series and additive noise. The results indicate that the CPSN shows good equalization performance, fast convergence, and a lot less computations as compared to conventional higher-order neural networks such as Volterra filters.

Truncation of nonlinear system expansions in the frequency domain

Abstract: The truncation of Volterra series expansions is studied using the output frequency characteristics of nonlinear systems to develop a new algorithm for determining the terms to include in a Volterra series expansion. The results show the influence of both the generalized frequency response functions and properties of the input spectra on the significance of individual terms in the series. The effectiveness of the proposed method is demonstrated using simulation studies including the analysis of a single degree of freedom mechanical oscillator. Nonlinear system analyses using Volterra series theory must always be based on a truncated Volterra series description. The present study provides an effective strategy for determining which terms to include in the analysis of practical nonlinear systems based on Volterra series models.

Maximum Length Sequences and Volterra Series in the Analysis of Transient Evoked Otoacoustic Emissions

Abstract: Previous work from this group (IHR, Southampton) has shown the feasibility of using maximum length sequence (MLS) stimulation to obtain evoked otoacoustic emissions (OAE). Because an MLS is one of a set of inputs that enables the Volterra series to be computed, we investigated its use with OAE. We wanted to see if the Volterra series could model the system and if we could extract the higher order kernels. In order to realise a practicable MLS system, a variant of the MLS has been used which permits real time recovery of the response and so enables the rejection of noisy epochs. This paper shows that the MLS used does enable the higher order kernels to be measured. The problems of selecting the correct sequences are discussed and the necessary system checks described.

Time varying Volterra series and its application to the distortion analysis of a sampling mixer

Abstract: We develop the time varying theory of Volterra series, and use the theory in the sampled data domain to solve for harmonic distortion of a passive MOS-based track-and-hold sampling mixer with a finite falling time LO voltage. We also quantify distortion due to sampling error. These results, when combined with the time invariant solution, quantify harmonic distortion and intermodulation of the mixer completely.

Design method of exact model matching control for finite volterra series systems

Abstract: This paper is concerned with exact model matching control (EMM) for finite Volterra series systems. First, we show a structure of a causal controller which can achieve the exact model matching to a reference model and clarify the relationship between the proposed method and the EMM for linear systems. Second, to analyse the stability of the proposed control system, we present an input dependent small gain theorem for the system with an external input, then extend it for the system with two external inputs. With the help of this theorem, we clarify the condition under which the control system is stable for the reference input magnitude within a certain range, and is also robust for small disturbances. Finally, the effectiveness of the proposed method is illustated through numerical simulations.

High Order Volterra Series Analysis using Parallel Computing

Abstract: This paper addresses the determination of high-order Volterra transfer functions of non-linear multiport networks containing multidimensional non-linear elements. A new method is developed for utilizing parallel computing which is very efficient for approximately 10-20 processors. The utilization of each processor may be as high as 80%-95%. The developed program is very flexible as it is ANSI C++ compatible and may run on both single- and multiprocessor computers. Using about eight processors it is possible to analyse rather complicated non-linear circuits up to ninth order in a few hours.

Volterra adaptive prediction of multipath fading channel

Abstract: The Volterra adaptive prediction of the multipath fading channel is studied. The performances of the Volterra filter and the conventional linear transverse filter are compared. Computer simulation results indicate that the Volterra filter behaves better than a linear transverse filter.

Efficient equalization of nonlinear communication channels

Abstract: Nonlinear intersymbol interference (ISI) often arises in voice-band communication channels at high transmission rates or in satellite channels due to nonlinearities in power amplifiers. The proposed equalizers for the cancellation of the nonlinear interference are mainly based on the Volterra series expansion, which is an elegant but very complex model. This paper presents a decision feedback equalizer (DFE) which is based on a new nonlinear filter structure. It is composed only of linear taped delay line filters and multipliers. Hence, the complexity of this still very general structure is comparable to linear filtering. Simulation of data transmission over a telephone channel show that the proposed DFE clearly outperforms the conventional DFE and is also superior to the Volterra DFE with a comparable complexity.