Showing papers on "Volterra series published in 1993"

A fast recursive least squares adaptive second order Volterra filter and its performance analysis

Abstract: A fast, recursive least squares (RLS) adaptive nonlinear filter modeled using a second-order Volterra series expansion is presented. The structure uses the ideas of fast RLS multichannel filters, and has a computational complexity of O(N/sup 3/) multiplications, where N-1 represents the memory span in number of samples of the nonlinear system model. A theoretical performance analysis of its steady-state behaviour in both stationary and nonstationary environments is presented. The analysis shows that, when the input is zero mean and Gaussian distributed, and the adaptive filter is operating in a stationary environment, the steady-state excess mean-squared error due to the coefficient noise vector is independent of the statistics of the input signal. The results of several simulation experiments show that the filter performs well in a variety of situations. The steady-state behaviour predicted by the analysis is in very good agreement with the experimental results. >

QR-decomposition based algorithms for adaptive Volterra filtering

Abstract: A QR-recursive-least squares (RLS) adaptive algorithm for non-linear filtering is presented. The algorithm is based solely on Givens rotation. Hence the algorithm is numerically stable and highly amenable to parallel implementations. The computational complexity of the algorithm is comparable to that of the fast transversal Volterra filters. The algorithm is based on a truncated second-order Volterra series model; however, it can be easily extended to other types of polynomial nonlinearities. The algorithm is derived by transforming the nonlinear filtering problem into an equivalent multichannel linear filtering problem with a different number of coefficients in each channel. The derivation of the algorithm is based on a channel-decomposition strategy which involves processing the channels in a sequential fashion during each iteration. This avoids matrix processing and leads to a scalar implementation. Results of extensive experimental studies demonstrating the properties of the algorithm in finite and 'infinite' precision environments are also presented. The results indicate that the algorithm retains the fast convergence behavior of the RLS Volterra filters and is numerically stable. >

Exploitation of cyclostationarity for identifying the Volterra kernels of nonlinear systems

Abstract: A class of random time-series inputs for nonlinear time-invariant systems that permit the analytical specification of a set of operators on the input that are orthonormal over all time to the Volterra operators for all orders and all lag sets is introduced. The time series in this class are cyclostationary and complex valued. The orthonormal operators are used to obtain an input-output type of cross-correlation formula for identifying the individual Volterra kernels of arbitrary order for a nonlinear system of possibly infinite order and possibly infinite memory. The real parts of the complex-valued inputs in this class comprise a class of real-valued inputs for which the same sets of specified operators apply. However, the orthogonality for different orders holds for these real inputs only for Volterra operators of order less than the order of the specified operator. Thus, these real inputs can be used to identify Volterra kernels only for finite-order systems. Frequency-domain counterparts of the time-domain methods that can utilize an FFT algorithm are developed. >

Nonlinear channel equalization in digital satellite systems

Abstract: A new nonlinear equalizer for digital transmission over a nonlinear satellite channel with a PSK modulation is presented in this paper. The nonlinear channel under consideration is assumed to have a Volterra series representation. By considering the channel equalization as a fixed point problem and taking advantage of the properties of a contraction mapping, we show that the transmitted data symbols can be recovered in an iterative manner. The equalizer has a modular structure for easy implementation. The computer simulation shows that the proposed equalizer has an excellent performance. >

Nonlinear channel models for digital magnetic recording

Abstract: Methods for identifying the digital magnetic recording channel in which data dependent nonlinear effects are present are described. The models used include the finite state machine (FSM), Volterra series, and channel state dependent transition shifting. Procedures to compute the model parameters are described, and the models are compared with a view to their accuracy and use for studying receiver performance. Results are included from prototype channels with various recording densities with significant nonlinear distortion. >

Identification of nonlinear channels in digital transmission systems

Abstract: Nonlinearity in digital transmission channels has long been an important problem in digital communications. Being able to identify the nonlinear characteristics of the channels can help in the design of the nonlinear equalizer. The authors consider identification of PSK and QAM nonlinear channels which can be expressed as a third-order complex-valued Volterra series. Based on higher-order moment analysis, they derive a simple algorithm to identify the Volterra kernels. In addition, it is shown that, for properly designed input sequences, the estimate obtained by the proposed method is equal to the optimum minimum mean square error solution. >

Orthogonalised frequency domain Volterra model for non-Gaussian inputs

Abstract: An orthogonalised Volterra system model, valid for both non-Gaussian and Gaussian inputs, is presented. The approach is based on ordered sets of conditioned orthogonal higherorder input vectors in the frequency domain, and utilises co-ordinate transformation to relate the orthogonal and nonorthogonal system models. The orthogonal model exhibits no interference effects, thus facilitating physical interpretation of the nonlinear system model. The importance of non-Gaussian excitation in the nonlinear system identification procedure is discussed. The performance of the orthogonalised Volterra model is measured in terms of a generalised nonlinear system coherence function, and compared with the results of the Wiener (for Gaussian input) and Volterra models. The advantages of the orthogonalised Volterra model are illustrated by using it to model the linear and quadratic responses of a tension leg platform subject to random seas, given experimental input-output time series data.

Nonlinear system identification with pseudorandom multilevel excitation sequences

Abstract: The authors consider pseudorandom multilevel sequences (PRMS) for the identification of nonlinear systems modeled via a truncated Volterra series with a finite degree of nonlinearity and finite memory length. It is shown that PRMS are persistently exciting (PE) for a Volterra series model with nonlinearities of polynomial degree N if and only if the sequence takes on N+1 or more distinct levels. A computationally efficient least squares identification algorithm based on PRMS inputs is developed that avoids forming the inverse of the data matrix. Simulation results comparing identification accuracy using PRMS and Gaussian white noise are given. >

On reduced-complexity approximations of quadratic filters

Abstract: Among the various mathematical representations of nonlinear systems, a popular description is given by means of the discrete-time Volterra series. In this paper we first briefly review the relevant nonlinear system identification techniques, with reference in particular to nonlinear systems with lengthy memory. In such cases, a parallel-cascade structure has been proved to be very effective since it is able to provide an arbitrarily close approximation, in the mean square error (MSE) sense, for a broad class of systems to be modelled. The parallel-cascade structure is indeed an exact representation for a quadratic filter, obtained by means of a matrix decomposition technique: we recall such a technique and then we show how it is possible to apply it for the adaptive identification of quadratic systems with large time delays. >

Nonlinearity-induced distortion of the transfer function shape in high-order OTA-C filters

Abstract: Effects caused by weak nonlinearities of operational transconductance amplifiers (OTAs) on the performance of continuous-time OTA-capacitor (OTA-C) filters are investigated. Nonlinear phenomena, such as compression/expansion and desensitization, are examined and the resulting distortion of the transfer function shape of high-order OTA-C filters with sinusoidal inputs is discussed. The Volterra series method is used to represent filters by their nonlinear transfer function for computer analysis and simulation. Using this approach, distortion and interference criteria are established for predicting analytically the onset of nonlinear behavior in the frequency responses of OTA-C filters. Finally, representative calculation results are presented and verified via SPICE simulations.

Realtime Loudspeaker Linearization

Abstract: Electrodynamic loudspeakers exhibit a nonlinear behaviour especially at low frequencies which shows up in harmonic and intermodulation distortions. Their nonlinear transfer function can be modeled by a Volterra series expansion and prefiltering of the loud speaker input signal by an appropriate Volterra, inverse filter can reduce the distortions. Problems with real time application arise from the requirement of large computational power to realize Volterra filters of high orders and long memories. To over come these problems approximations to the loud speaker model must be made. This paper presents a new Volterra kernel decomposition with good performance and small computational load. The pre-distortion filters were implemented on a DSP system and tested with several electrodynamic loudspeakers. The results show that the nonlinear distortions can be reduced significantly.

Random and pseudorandom excitation sequences for nonlinear system identification

Abstract: This paper considers random and pseudorandom excitation sequences for the identification of nonlinear systems modelled via a truncated Volterra series with a finite degree of nonlinearity and finite memory length. Random i.i.d. sequences are studied and necessary and sufficient conditions on the input that guarantee persistence of excitation are derived. The condition number of the correlation matrix corresponding to the Volterra series is mathematically characterized. A computationally efficient least squares identification algorithm based on i.i.d. excitation is developed. Simulations comparing identification accuracy using random and pseudorandom inputs are given.

Digital compensation of nonlinear distortion in loudspeakers

Abstract: Loudspeakers produce nonlinear distortion. The authors present a method to compensate for this distortion in real time by nonlinear digital signal processing implemented on a digital signal processor (i.e., the TMS320C30 DSP). Based on the literature, an electrical equivalent circuit of an electrodynamic loudspeaker is developed, resulting in a linear lumped parameter model. The parameters in this model are matched with the measurements of a selected test loudspeaker. The linear model is extended to include nonlinear effects by developing the parameters as a function of the voice coil excursion of the loudspeaker in a Taylor series expansion. The resulting nonlinear system is described by a Volterra series. On the basis of this description, an inverse circuit is designed for the second-order nonlinear distortion. This circuit was implemented in real time on the DSP, using a high-level design and code generation system. Simulations and experiments are presented. >

Frequency-domain Volterra kernel estimation via higher-order statistical signal processing

Abstract: We discuss the utilization of higher-order spectral moments to determine frequency-domain Volterra kernels, given time series records of the random excitation and response of a nonlinear physical system. In particular, we consider frequency domain third-order Volterra kernel identification for nonGaussian excitation. Next an orthogonal Volterra like model valid for nonGaussian excitation is described. This model eliminates the interference terms associated with the nonorthogonal Volterra model, and thus facilitates decomposition of an observed power spectrum into its constituent linear quadratic, and cubic components. >

Performance assessment of subcarrier multiplexed optical systems : implications of laser nonlinearities.

Abstract: This thesis is concerned with a detailed study of laser nonlineanties and the implications for multichannel subcamer multiplexed fibre optic systems In view of the limitations of previously reported analytic treatments of laser nonlinear distortion, a considerable part of the study is dedicated to the development of modelling and analytic tchniques suitable for practical system design and optimisation The single-mode rate equations provide an adequate basis for the analysis of laser intnnsic dynamic distortion and relative intensity noise and their dependence on device parameters and operating conditions The Volterra series method of nonlinear system theory is then applied and an analytical model is obtained that describes the nonlineanty in the frequency domain by a set of nonlinear transfer functions This method provides a ngorous analytic nonlinear model that takes into account all the intermodulation products up to third-order as determined by the rate equations Moreover, the laser response to the important case of a sum of narrow-band signals is considered and under certain conditions, which are valid for the majority of systems of interest, the intermodulation power spectral density of the distortion products is determined This enables an accurate evaluation of the impact on the overall system performance of laser intrinsic distortion and optimum overall performance is identified after including the noise introduced in the detection process The relative importance of laser intnnsic distortion and clipping-distortion is also examined Finally, the analytic model is used to investigate design constraints and the overall system performance of relevant SCM systems Case studies are considered that demonstrate the applicability of the method devised

Nonlinear model and measurement method for microwave mixers

Abstract: A novel approach to characterizing and measuring the static and dynamic nonlinear properties of a microwave mixer is described. It is based on the use of generalized Volterra series. Its modeling capabilities are confirmed by simulation results for nonlinear microwave devices, as well as by measurements. A new measurement setup has been developed. It uses a sampling oscilloscope and yields both the amplitude and phase of the different spectral components of the measured signal. >

A neural network model for nonlinear predictive coding in Fock space

Abstract: A generic nonlinear autoregressive (AR) model for a random time series is presented. The model is obtained by a nonlinear predictive coding (NLPC) approach which expresses the minimum mean square error estimate of the current value of the series as a Volterra series in terms of its immediate N preceding values. This Volterra series is assumed to belong to a generalized Fock Hilbert space F. In the second stage, which is parametric, the model parameters, which are coefficients of a linear combination of known nonlinear random functions of the data, are obtained by linear mean square estimation. The implementations of the model and of the estimator appear respectively as two layer recurrent and feedforward neural networks. >

Analysis of noise in cascaded non‐linear stages

Abstract: This paper presents a method to analyse noise in cascaded non-linear stages based on Volterra series for multiport systems. the cascaded stages are partitioned such that internal noise sources from different stages are uncorrelated and linear loading between all stages can be assumed. This partitioning scheme implies that the ensemble average of the magnitude-squared output response from the last stage in the cascade can be determined as a sum of noise contributions from the individual stages. the method allows both unmodulated (independent) and modulated (dependent) noise sources. A special case of the theory and two related examples are presented to illustrate the method.

Modeling and electronic implementation of directional selectivity

Abstract: A shunting model of motion detection treatment is proposed and implemented. It is analyzed using a perturbational technique to model the behavior and extract the velocity information including both direction and speed. The model's properties are illustrated by constructing a simple hardware analogue circuit. Response of the circuit to a variety of stimulus distributions show that the output is sensitive to speed and also is markedly different for movement in opposite directions. In order to analyze the behavior of the model, a perturbational technique is developed which describes the input-output relation in the form of a Volterra integral series. It is shown that the velocity term can be identified with the second order kernel while the second order Volterra operator is a running autocorrelation operator. This is in agreement with alternative methods of motion information analysis. >

A rotation based multichannel least squares lattice algorithm for adaptive nonlinear filters

Abstract: An efficient rotation based adaptive multichannel least squares lattice algorithm for multichannel filtering with different filter orders for each input channel is presented. It is shown that this algorithm is suited for adaptive nonlinear filtering based on Volterra and recursive polynomial system models. The new algorithm requires only orthogonal transformations offering superior numerical behavior. Its highly modular and regular structure facilitates VLSI and parallel implementations. >

The Neurone as a Nonlinear System: A Single Compartment Study

Abstract: A neurone is described as a single-input single-output nonlinear system. The Volterra kernels of a single compartmental model are extracted using a feedforward network. We then demonstrate that the derived kernels can represent features of the compartmental model. This technique can also be applied to intracellular recordings to allow the construction of improved models.

Nonlinear Control Systems Analysis using Computer Algebra

Abstract: In this paper we discuss the use of computer algebra for the development of software tools for analysis and design of nonlinear control systems We illustrate with a number of examples how the analysis can be simplified using some suitable computer algebra functions, which we have implemented The examples concern several different areas of nonlinear control theory; exact linearization, Volterra series expansions, stability theory and optimal control

On Reduced-Complexity Approximations of Quadratic Filters

Abstract: Among the various mathematical representations of nonlinear systems, a popular description is given by means of the discrete-time Volterra series In this paper we first briefly review the mlevant nonlinear rystem identification techniques, with reference in particular to nonlinear syrkms with knghty memory In such wes, a padleLcascade structure has been proved to be very effective since it is able to provide an arbitmady close approzimatwn, in the Mean Square Emr (MSE) sense, for a broad class of systems to be md elkd The pamallel-cascade structure is indeed an ezact representation for a quadmatic filter, obtained by means of a mat& decomposition technique: we recall such a technique and then we show how it is possible to apply it for the adaptive identijkation of quadmatic systems with lave time delays

A new adaptive equalizer structure for nonlinear channels

Abstract: Based on the analysis of nonlinear channel models, a new connectionist model of adaptive equalizer is constructed. Comparing with the connectionist model using the Volterra series to extend the input vector space, the number of weights with the new structure is reduced significantly. It is shown by simulations that the weight values of the new scheme converge to the optimal values closely for non-minimum phase channels as well minimum phase channels, if the channel noise is small enough. Testing results of the BER (Bit Error Rate) tell us that the new adaptive equalizer for nonlinear channels is superior to the conventional linear equalizers in the equalization performances.