Adaptive polynomial filters

Adaptive nonlinear filters equipped with polynomial models of nonlinearity are explained. The polynomial systems considered are those nonlinear systems whose output signals can be related to the input signals through a truncated Volterra series expansion or a recursive nonlinear difference equation. The Volterra series expansion can model a large class of nonlinear systems and is attractive in adaptive filtering applications because the expansion is a linear combination of nonlinear functions of the input signal. The basic ideas behind the development of gradient and recursive least-squares adaptive Volterra filters are first discussed. Adaptive algorithms using system models involving recursive nonlinear difference equations are then treated. Such systems may be able to approximate many nonlinear systems with great parsimony in the use of coefficients. Also discussed are current research trends and new results and problem areas associated with these nonlinear filters. A lattice structure for polynomial models is described     

Nonlinear echo cancelling using look-up tables and Volterra systems

Two classical approaches to nonlinear echo cancellation are to use look-up tables combined with FIR filters and Volterra systems; their practical interest is higher when adapted with LMS-type algorithms, because results are easily implementable and offer good tracking capabilities. A complete analysis of look-up-table-based schemes is presented, and a new one is introduced which offers near optimal characteristics but without requiring variable steps: in practice a piecewise-constant adaptation algorithm is enough. The theoretical equivalence of these schemes with those based on Volterra kernels is established; a comparison of operational characteristic is then made. The important option of applying growing Volterra structures is proposed for the first time, and its possibilities are verified (as well as the previous analysis) by means of simulation examples. The paper ends with suggestions to be explored to obtain more efficient nonlinear echo cancellers     

Efficient algorithms for Volterra system identification

In this paper, nonlinear filtering and identification based on finite-support Volterra models are considered. The Volterra kernels are estimated via input-output statistics or directly in terms of input-output data. It is shown that the normal equations for a finite-support Volterra system excited by zero mean Gaussian input have a unique solution if, and only if, the power spectral process of the input signal is nonzero at least at m distinct frequencies, where m is the memory of the system. A multichannel embedding approach is introduced. A set of primary signals defined in terms of the input signal serve to map efficiently the nonlinear process to an equivalent multichannel format. Efficient algorithms for the estimation of the Volterra parameters are derived for batch, as well as for adaptive processing. An efficient order-recursive method is presented for the determination of the Volterra model structure. The proposed methods are illustrated by simulations     

Subspace based blind identification of LTI FIR MIMO systems and equalization of finite memory SIMO Volterra systems

In this paper a novel algorithm based on subspace projections is developed for the blind kernel identification of LTI FIR multiple input multiple output (MIMO) systems, as well as blind equalization of finite memory SIMO Volterra systems. In addition, for Volterra systems, the algorithm computes the memory lengths of the nonlinearities involved. Simulations in the context of blind channel equalization and identification demonstrate good performance in comparison to existing schemes.     

Volterra series modeling of power conversion systems

The nonlinear control-to-output response of pulse width modulated (PWM) conversion systems is modeled via the Volterra functional series. A brief overview of the series is presented. It is seen that the Volterra series is a power series with memory. Each term in the series represents a convolution integral. The nonlinear response of the system, for any input, can thus be determined from a knowledge of the multidimensional Volterra kernels or impulse responses. The determination of the Volterra kernels in the transform domain is performed on a simplified state-space model of the converter. The dominant component of various harmonic and intermodulation distortion frequency products in the output spectrum is derived and is expressed in terms of these kernels. Experimental results are presented confirming the modeling procedure     

Volterra-mapping-based behavioral modeling of nonlinear circuits and systems for high frequencies

Presents and validates a discrete-time/frequency-domain approach to the problem of Volterra-series-based behavioral modeling for high-frequency systems. The proposed technique is based on the acquisition of samples of the input/output data, both of which are sampled at the Nyquist rate corresponding to the input signal. The method is capable of identifying the time-/frequency-domain Volterra kernels/transfer functions of arbitrary causal time-invariant weakly nonlinear circuits and systems operating at high frequencies subject to essentially a general random or multitone excitation. The validity and efficiency of the proposed modeling approach has been demonstrated by several examples in high-frequency applications and good agreement has been obtained between results calculated using the proposed model and results measured or simulated with commercial simulation tools.     

Second-order adaptive Volterra system identification based ondiscrete nonlinear Wiener model

The authors present the nonlinear LMS adaptive filtering algorithm based on the discrete nonlinear Wiener (1942) model for second-order Volterra system identification application. The main approach is to perform a complete orthogonalisation procedure on the truncated Volterra series. This allows the use of the LMS adaptive linear filtering algorithm for calculating all the coefficients with efficiency. This orthogonalisation method is based on the nonlinear discrete Wiener model. It contains three sections: a single-input multi-output linear with memory section, a multi-input, multi-output nonlinear no-memory section and a multi-input, single-output amplification and summary section. For a white Gaussian noise input signal, the autocorrelation matrix of the adaptive filter input vector can be diagonalised unlike when using the Volterra model. This dramatically reduces the eigenvalue spread and results in more rapid convergence. Also, the discrete nonlinear Wiener model adaptive system allows us to represent a complicated Volterra system with only few coefficient terms. In general, it can also identify the nonlinear system without over-parameterisation. A theoretical performance analysis of steady-state behaviour is presented. Computer simulations are also included to verify the theory     

Stochastic gradient based third-order Volterra system identification by using nonlinear Wiener adaptive algorithm

The nonlinear Wiener stochastic gradient adaptive algorithm for third-order Volterra system identification application with Gaussian input signals is presented. The complete self-orthogonalisation procedure is based on the delay-line structure of the nonlinear discrete Wiener model. The approach diagonalises the autocorrelation matrix of an adaptive filter input vector which dramatically reduces the eigenvalue spread and results in more rapid convergence speed. The relationship between the autocorrelation matrix and cross-correlation matrix of filter input vectors of both nonlinear Wiener and Volterra models is derived. The algorithm has a computational complexity of O(M/sup 3/) multiplications per sample input where M represents the length of memory for the system model, which is comparable to the existing algorithms. It is also worth noting that the proposed algorithm provides a general solution for the Volterra system identification application. Computer simulations are included to verify the theory.     

Nonlinear system identification using Gaussian inputs

The paper is concerned with the identification of nonlinear systems represented by Volterra expansions and driven by stationary, zero mean Gaussian inputs, with arbitrary spectra that are not necessarily white. Procedures for the computation of the Volterra kernels both in the time as well as in the frequency domain are developed based on cross-cumulant information. The derived kernels are optimal in the mean squared error sense for noncausal systems. Order recursive procedures based on minimum mean squared error reduction are derived. More general input output representations that result when the Volterra kernels are expanded in a given orthogonal base are also considered     

Exploitation of cyclostationarity for identifying the Volterrakernels of nonlinear systems

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     

Soft Fault Feature Extraction in Nonlinear Analog Circuit Fault Diagnosis

Aiming at the problem to diagnose soft faults in nonlinear analog circuits, a novel approach to extract fault features is proposed. The approach is based on the Wigner–Ville distribution (WVD) of the subband Volterra model. First, the subband Volterra kernels of the circuit under test are cleared. Then, the subband Volterra kernels are used to obtain the WVD functions. The fault features are extracted from the WVD functions and taken as input data into the hidden Markov model (HMM). Finally, with classification of features using HMMs, the soft fault diagnosis of the nonlinear analog circuit is achieved. The simulations and experiments show that the method proposed in this paper can extract the fault features effectively and improve the fault diagnosis.     

Application of Volterra series to the problem of self-oscillatingmixer

A new approach to the nonlinear problem of self-oscillating mixer has been investigated using Volterra series. The circuit under consideration is first converted into a one-port network. The input and coupling impedances of various ports are represented by Volterra kernels generated by nonlinear current method. Advantage of this approach is that the phase relationships among signals are not required for the analysis. Also, no stability criterion testing is needed to ensure convergence to the correct solution numerically. It is computationally efficient and mathematically simple, yet reasonably accurate. Measured results with respect to RF frequency and power show good agreement with that calculated     

Modeling and Identification of Nonlinear Systems in the Short-Time Fourier Transform Domain

In this paper, we introduce a novel approach for improved nonlinear system identification in the short-time Fourier transform (STFT) domain. We first derive explicit representations of discrete-time Volterra filters in the STFT domain. Based on these representations, approximate nonlinear STFT models, which consist of parallel combinations of linear and nonlinear components, are developed. The linear components are represented by cross-band filters between subbands, while the nonlinear components are modeled by multiplicative cross-terms. We consider the identification of quadratically nonlinear systems and show that a significant reduction in computational cost as well as substantial improvement in estimation accuracy can be achieved over a time-domain Volterra model, particularly when long-memory systems are considered. Experimental results validate the theoretical derivations and demonstrate the effectiveness of the proposed approach.     

Optimal Volterra kernel estimation algorithms for a nonlinearcommunication system for PSK and QAM inputs

A fifth-order Volterra kernel estimation algorithm, which is optimal in the least mean square error sense, for a bandpass nonlinear system is derived. The algorithm is based on some characteristics of i.i.d. circularly symmetric zero-mean complex-valued Gaussian random variables. The proposed algorithm can be used to identify a nonlinear system under uniformly i.i.d. rectangular M-QAM input and under uniformly i.i.d. M-PSK input (M⩾4) with modest modification. The same approach has been used to derive an optimal Volterra kernel estimation algorithm up to the third order. However, in some cases, a third-order model is not of “high enough order” to capture the nonlinear system characteristics. A simulation example is given to show the necessity of deriving a fifth-order Volterra kernel estimation algorithm and to test for the correctness of the algorithm     

Baseband Volterra filters for implementing carrier basednonlinearities

A diagonal coordinate representation for Volterra filters is developed and exploited to derive efficient Volterra filter implementations for processing carrier based input signals. In the diagonal coordinate representation, the output is expressed as a sum of linear filters applied to modified input signals. Hence, linear filtering methods are employed to implement the nonlinear filter on a baseband version of the input. Downsampling is then used to reduce computational complexity. The same approach is employed to develop efficient implementations for processing continuous-time carrier-based signals, pulse amplitude-modulated signals, and frequency division multiplexed input signals     

非线性动态网络的离散算法及仿真

在非线性网络响应分析中,采用Volterra级数法可以导出与线性系统传递函数相似的非线性传递函数,能使非线性系统用线性化和系统化方法达到精确分析.文中给出了非线性网络响应的Volterra级数解的连续算式,为解决连续算式计算麻烦的问题,提出用方波脉冲技术处理用Volterra级数表示法描述的非线性网络响应与激励之间关系的一组广义卷积积分的迭加计算,从而得到非线性网络响应求解的Volterra级数解的离散算式.仿真表明该算法求出的非线性网络响应与真实模型曲线十分逼近,证明了它的有效性.     

Volterra series transfer function of single-mode fibers

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     

Estimating second-order Volterra system parameters from noisy measurements based on an LMS variant or an errors-in-variables method

This paper deals with the identification of a nonlinear SISO system modelled by a second-order Volterra series expansion when both the input and the output are disturbed by additive white Gaussian noises. Two methods are proposed. Firstly, we present an unbiased on-line approach based on the LMS. It includes a bias correction scheme which requires the variance of the input additive noise. Secondly, we suggest solving the identification problem as an errors-in-variables issue, by means of the so-called Frisch scheme. Although its computational cost is high, this approach has the advantage of estimating the Volterra kernels and the variances of both the additive noises and the input signal, even if the signal-to-noise ratios at the input and the output are low.