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.     

Perturbation Analysis of Nonlinear Distortion in Analog Integrated Circuits

A method for predicting the distortion in weakly nonlinear analog circuits is presented, which relies on the classical theory of regular perturbation. Accordingly, a nonlinear circuit is described and analyzed as a perturbation of its linearized model, and the response to a periodic signal is analytically calculated through frequency-domain recurrent formulas. The method is simple and quite straightforward to apply, as it involves the calculation of frequency-domain transfer functions and of Fourier coefficients only, making it easily adaptable to any circuit topology. The method can be a valid alternative to the Volterra series method. A relationship between the proposed method and the Volterra series method is established, showing that they lead to very similar approximants to the solution. The method has been numerically tested in practical circuits wherein the devices are modeled by polynomial and exponential nonlinearities.     

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     

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.     

A Scattering Variable Approach to the Volterra Analysis of Nonlinear Systems

A new mathematical model is developed which extends Volterra series analysis of nonlinear systems with memory to high-frequency systems, including those containing linear distributed component devices. A generalized set of nonlinear scattering parameters is defined which can be used to describe power transfer and distortion in nonlinear multiports, and which reduce to the classical scattering parameters for linear networks. The methodology is based on Volterra functional series, and is most useful for the small-signal case where the response can be approximated by a finite number of terms of the series. Nonlinear scattering kernels, derived by extending the Volterra analysis, are simply related to previously developed nonlinear voltage and current Volterra kernels. For sinusoidal inputs nonlinear scattering parameters are defined which are shown to be particularly helpful when power relationships are studied. The principal applications are for microwave networks terminated in real-valued site reference impedances. To evaluate the average power dissipated in a load at some intermodulation frequency, the concept of nonlinear transducer gain is defined and shown to be proportional to the squared magnitude of a nonlinear scattering parameter. Examples are presented illustrating the analysis procedure for a tunnel diode reflection amplifier and for a linear lossless transmission line terminated by a non-linear network.     

Linear multichannel blind equalizers of nonlinear FIR Volterrachannels

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     

Nonlinear Transient and Distortion Analysis via Frequency Domain Volterra Series

This paper presents a novel approach for transient and distortion analyses for time-invariant and periodically time-varying mildly nonlinear analog circuits. Our method is based on a frequency domain Volterra series representation of nonlinear circuits. It computes the nonlinear responses using a nonlinear current method that recursively solves a series of linear Volterra circuits to obtain linear and higher-order responses of a nonlinear circuit. Unlike existing approaches, where Volterra circuits are solved mainly in the time domain, the new method solves the linear Volterra circuits directly in the frequency domain via an efficient graph-based technique, which can derive transfer functions for any large linear network efficiently. As a result, both frequency domain characteristics, like harmonic and intermodulation distortion, and time domain waveforms can be computed efficiently. The new algorithm takes advantage of identical Volterra circuits for second- and higher-order responses, which results in significant savings in driving the transfer functions. Experimental results for two circuits—a low-noise amplifier and a switching mixer—are obtained and compared with SPICE3 to validate the effectiveness of this method.     

A digital method of modeling two-input quadratic systems withgeneral random inputs

A digital method of estimating linear and quadratic transfer functions of a two-input/multiple-output quadratically nonlinear system with general (i.e., non-Gaussian as well as Gaussian) random inputs is described. The approach is based on a frequency domain second-order Volterra functional series representation for a two-input/multiple-output system. Practical applications where it is important to have the capability to model dual-input nonlinear systems include: ground vibration tests of aircraft where two random exciting forces are applied in order to inject sufficient energy into all modes of the system; and experimental studies of transition to turbulence in fluids and plasmas, where the two inputs consist of two different velocity components, and plasma density and potential fluctuations, respectively. The approach described is being utilized to investigate such applications     

Spectral modeling of switched-mode power converters

A new modeling approach for the spectral analysis of pulsewidth modulated (PWM) converters with independent inputs is developed. The key of this approach is to extend the Volterra functional series to nonlinear systems with multiple independent inputs. After formulating the state-space equations describing the dynamical behavior of PWM converters, the Volterra transfer function characterizing the output frequency response can be obtained, which is then symmetrised to form the spectral model. Since the model is developed in a closed form, it is suitable for computer analysis. The modeling approach has been applied to various PWM converters, and the results are verified. The spectral models of different power converters can readily be obtained by using this general approach     

Error probability for digital transmission over nonlinear channelswith application to TCM

The computation of upper and lower bounds to error probability in digital transmission over nonlinear channels with a finite memory is considered. By using orthogonal Volterra series, the authors derive a canonical representation for discrete nonlinear systems, based on a linear convolutional code and a memoryless mapper. This representation shows that finite-memory, discrete nonlinear systems can be analyzed in much the same way as TCM (trellis-coded modulation) schemes. In particular, TCM over nonlinear channels can be analyzed. A technique is derived that expresses an upper bound to error probability based on the computation of the transfer of a state diagram with N branches, and whose branch labels are matrices rather than scalars. Some examples of its application are given. In particular, error bounds are derived for nonlinear TCM schemes and for TCM schemes operating on nonlinear channels     

Relationship between Volterra series and generalized power series

The relationship between the Volterra nonlinear transfer functions of a system and the elements of its generalized power series is established. A formula is derived which enables the Volterra nonlinear transfer functions to be obtained from the power series expansion of the nonlinear system.     

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.     

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.     

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     

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     

基于子空间比较的宽带信号下辐射源 功放"指纹"分类方法

 本文提出了一种基于功放宽带Volterra级数模型的辐射源"指纹"分类方法,以用于电子侦察对宽带信号源的特定辐射源识别.针对传统Volterra系统MIMO建模方法的适应性不足,以及系统辨识过程存在维数爆炸问题,本文推导出一种对编码类和调频类宽带信号都适用的MIMO建模方法,然后直接基于参数子空间比较来实现功放的指纹分类.对本方法的"独立性"和"可测性"进行了分析.数值仿真实验对子空间比较方法进行了验证.     

求解非线性系统传递函数的递推组合法

本文给出了求解一类非线性系统传递函数的递推组合法,即根据系统Volterra响应的递推关系和组合系统传递函数的公式求解非线性系统传递函数的方法。     

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     

Inter modulation distortion analysis of cascaded MESFET amplifiers using Volterra series representation

The third-order intermodulation distortion generated in a two-stage cascaded amplifier is derived analytically by means of the Volterra series expansion, for non-linear systems with memory. A unilateral transistor model is used, which takes into account the three major non-linearities, the gate capacitance, the trans-conductance and the output conductance, in each stage of the amplifier. The Volterra transfer functions are determined for this transistor model and closed-form expressions for the intermodulation distortion ratio are given, where the terms with a contribution less than 1% have been neglected. The equations identify the principal sources of the distortion in the amplifier circuit and the influence of the transistor parameters and load impedance is investigated. The analysis of the two-stage cascaded amplifier gives a new insight into the intermodulation distortion behaviour and is discussed in detail at the end of the paper. The results obtained here are compared with those received for single-stage amplifiers.