Nonlinear System Identification Using Deterministic Multilevel Sequences

A new exact method of measuring the Volterra kernels of finite-order discrete nonlinear systems is presented. The kernels are rearranged in terms of multivariate crossproducts in vector form. The one-, two-, . . . , and ℓ-dimensional kernel vectors are determined using a deterministic multilevel sequence with ℓ distinct levels at the input of the system. It is shown that the defined multilevel sequence with ℓ distinct levels is persistently exciting for a truncated Volterra filter with nonlinearities of polynomial degree ℓ. Examples demonstrating the rearrangement of the Volterra kernels and a novel method for estimation of the kernels are presented. Simulation results are given to illustrate the effectiveness of the proposed method.     

Numeric Variable Forgetting Factor RLS Algorithm for Second-Order Volterra Filtering

Nonlinear adaptive filtering techniques for system identification (based on the Volterra model) are widely used for the identification of nonlinearities in many applications. In this correspondence, the improved tracking capability of a numeric variable forgetting factor recursive least squares (NVFF-RLS) algorithm is presented for first-order and second-order time-varying Volterra systems under a nonstationary environment. The nonlinear system tracking problem is converted into a state estimation problem of the time-variant system. The time-varying Volterra kernels are governed by the first-order Gauss–Markov stochastic difference equation, upon which the state-space representation of this system is built. In comparison to the conventional fixed forgetting factor recursive least squares algorithm, the NVFF-RLS algorithm provides better channel estimation as well as channel tracking performance in terms of the minimum mean square error (MMSE) for first-order and second-order Volterra systems. The NVFF-RLS algorithm is adapted to the time-varying signals by using the updating prediction error criterion, which accounts for the nonstationarity of the signal. The demonstrated simulation results manifest that the proposed method has good adaptability in the time-varying environment, and it also reduces the computational complexity.     

沿轴线方向测量高阶Volterra核

本文给出沿轴线方向测量非线性电路和系统Volterra前三阶核的快速有效方法。这是作者们以前工作(1990)的续篇。引用以前工作的结果,并在输入信号中加入直流项,可以得到非线性系统在轴线上的高阶核H2(j,0),H3(j,0,0),H3(j1,j2,0)等。     

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     

Nonlinear channel modeling and identification using baseband Volterra-Parafac models

Baseband Volterra models are very useful for representing nonlinear communication channels. These models present the specificity to include only odd-order nonlinear terms, with kernels characterized by a double symmetry. The main drawback is their parametric complexity. In this paper, we develop a new class of Volterra models, called baseband Volterra-Parafac models, with a reduced parametric complexity, by using a doubly symmetric Parafac decomposition of high order Volterra kernels viewed as tensors. Three adaptive algorithms are then proposed for estimating the parameters of these models. Some Monte Carlo simulation results are presented to compare the performance of the proposed estimation algorithms, in the case of third-order baseband Volterra systems excited by PSK and QAM inputs.     

Identification of a class of nonlinear systems under stationarynon-Gaussian excitation

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     

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.     

基于最低误码率准则的奇次三阶Volterra均衡器用于干扰抑制

该文针对通信系统中的干扰抑制问题,提出一种基于最低误码率准则的Volterra均衡器.区别于以往研究中采用最小均方误差准则估计Volterra核,本文采用最低误码率准则.仿真表明:对于扩展的二元相移键控信号,在相对强的窄带干扰下,匹配滤波器和基于最小均方误差准则的线性均衡器已失效,而基于最低误码率准则的Volterra均衡器仍能表现出良好的性能,也大大优于最小均方误差准则的Volterra均衡器;并且在计算复杂度与误码率性能的权衡中,奇次三阶Volterra均衡器更有实用价值.     

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     

一种确定弱非线性双端口网络模型参数的新方法

非线性传输函数模型和非线性电流源模型是利用Volterra级数对非线性动态电路建模的两种形式;本文提出一种确定一类弱非线性双端口网络非线性电流源模型参数的方法,它通过测量网络的端口特性来确定其模型参数;这种方法比测量Volterra核的方法简单方便。     

弱非线性动态系统Volterra核自动测试系统的研究

本文讨论弱非线性动态系统Ｖｏｌｔｅｒｒａ核自动测试系统的研制，应用Ｖｏｌｔｅｒｒａ泛函级数和计算机技术于非线性系统，本文结合频率域多点测量弱非线性系统前二阶核的算法ＱＭＭＰ２及测量前三阶核的算法ＱＭＭＰ３，设计了自动测试系统。本文给出应用该系统的两个测量实例。     

Time-domain analysis of nonlinear circuits performed on the basis of functional Volterra series

A method is proposed for determination of Volterra kernels from known images. It is shown that frequency-domain kernels should be considered at the points where their analyticity is violated. The timedomain structure of Volterra kernels is determined. A nonlinear microwave amplifier is simulated in the presence of a wideband input signal.     

MEASURING HIGH-ORDER VOLTERRA KERNELS ALONG AXES

A fast and efficient method for measuring high-order Volterra kernels along axes is given. This is a sequel of the authors' former paper(1990). Using the conclusions of those papers mentioned above, and combining the DC component with the input signals, the high-order kernels of nonlinear system H2(jω,0), H3(jω,0,0), H3(Jω,jω2,0) can be measured at many points.     

Equivalent Low-Pass Representations for Bandpass Volterra Systems

Relationships are derived for finding equivalent low-pass representations of the Volterra kernels corresponding to a widely used model for a nonlinear communication channel. The results are a logical extension of the usual complex envelope representation for bandpass signals. The equivalent low-pass representations are useful for simplifying calculations in cases where the input and output signals of the system are bandpass.     

Soft Fault Diagnosis in Analog Circuits Based on Bispectral Models

Aiming at the problem to locate soft faults in analog circuits, a new approach based on bispectral models is proposed. First, the Volterra kernels of the circuit under test (CUT) are calculated. Then, the Volterra kernels are used to construct bispectral models. By comparison with the fault features of the constructed models, soft faults of linear and weak nonlinear components in the analog circuit are identified and the faults are located. Simulations and experiments show the effectiveness of the proposed method in analog circuits.,     

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     

A digital technique to estimate second-order distortion usinghigher order coherence spectra

A digital spectral method for evaluating second-order distortion of a nonlinear system, which can be represented by Volterra kernels up to second order and which is subjected to a random noise input, is discussed. The importance of departures from the commonly assumed Gaussian excitation is investigated. The Hinich test is shown to be an appropriate test for orthogonality in the system identification. Tests for Gaussianity of two important sources, which are commonly used for Gaussian inputs in nonlinear system identification, are presented: (1) commercial software routines for simulation experiments, and (2) noise generators for practical experiments. The deleterious effects of assuming a Gaussian input when it is not are demonstrated. The random input method for evaluating the second-order distortion of a nonlinear system is compared with the sine-wave input method using both simulation and experimental data. The approach is applied to a loudspeaker in the low-frequency band     

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.     

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.