Adaptive Volterra Filter Research Articles

Pipelined set-membership approach to adaptive Volterra filtering

Due to the high computational complexity required by Volterra filter, some of its practical implementations consider pipelined adaptive Volterra filter architecture with two layers structure. However, its main challenges are the poor robustness against impulsive noise, slow convergence and high computational complexity for long memory high order expansion on each module. In this paper, we firstly extend the pipelined second-order adaptive Volterra filter to its high order version. Then, to reduce the computational complexity and improve the robust performance of pipelined adaptive Volterra filter architecture, the pipelined adaptive Volterra set-membership (PAVF-SM) algorithm and its robust version (PAVF-RSM) are proposed, which are derived from the least-perturbation property and adaptive approximation principle. Due to the inherent variable step size and nonlinear selective update mechanisms, the proposed PAVF-SM and PAVF-RSM algorithms achieve lower complexity and improved convergence performance. Simulations also verify the improved performance of the PAVF-SM and PAVF-RSM algorithms under Gaussian noise and impulsive noise environments.

A hierarchical alternative updated adaptive Volterra filter with pipelined architecture

The pipelined adaptive Volterra filters (PAVFs) with a two-layer structure constitute a class of good low-complexity filters. They can efficiently reduce the computational complexity of the conventional adaptive Volterra filter. Their major drawbacks are low convergence rate and high steady-state error caused by the coupling effect between the two layers. In order to remove the coupling effect and improve the performance of PAVFs, we present a novel hierarchical pipelined adaptive Volterra filter (HPAVF)-based alternative update mechanism. The HPAVFs with hierarchical decoupled normalized least mean square (HDNLMS) algorithms are derived to adaptively update weights of its nonlinear and linear subsections. The computational complexity of HPAVF is also analyzed. Simulations of nonlinear system adaptive identification, nonlinear channel equalization, and speech prediction show that the proposed HPAVF with different independent weight vectors in nonlinear subsection has superior performance to conventional Volterra filters, diagonally truncated Volterra filters, and PAVFs in terms of initial convergence, steady-state error, and computational complexity.

The Volterra adaptive prediction method based on matrix decomposition

This paper comes up with the adaptive prediction method of Volterra series method in chaotic time series based on matrix factorization method. Taking the monthly runoff of the Huaxian Hydrological Station as example, based on phase-space reconstruction, it identifies the chaotic characteristic through correlation dimension and Lyapunov index. Based on Volterra adaptive filter model, use matrix factorization to solve the equation, which avoid the local optimum problem caused by selecting initial value in Normalized Least Mean Square (NLMS), and at the same time, obtain the global optimal Volterra filter coefficient. According to the Numerical experiments, the prediction performance based on matrix factorization method has no problems in selecting the initial value and can achieve highly-accurate prediction. The conclusions are as follows: (1) The monthly runoff in Huaxian has chaotic characteristic. It’s embedding dimension and optimal embedding dimension is 3; (2) Matrix factorization avoids square root operation and improves the speed; (3) Avoiding selecting the initial value improves the prediction accuracy of the optimal filter coefficient.

Adaptive Second-Order Volterra Filtering Based on an H∞ Criterion

This paper presents an adaptive Volterra filter implementation based on an exponentially-weighted a posteriori H∞ filtering algorithm. Its fast array form is immediately obtained by following the derivation of the fast multichannel RLS filter. Also the steady-state performance of the H∞ Volterra filter is analyzed by using an approximate expression for its excess mean-square errors. Several numerical examples show that the H∞ Volterra filter could achieve a balance between convergence and tracking capability when its attenuation parameter is chosen adequately.

Adaptive Volterra filter with continuous lp-norm using a logarithmic cost for nonlinear active noise control

The filtered-x least mean lp-norm (FxLMP) algorithm is proven to be useful for nonlinear active noise control (NANC) systems. However, its performance deteriorates when the impulsive noises are presented in NANC systems. To surmount this shortcoming, a new nonlinear adaptive algorithm based on Volterra expansion model (VFxlogLMP) is developed in this paper, which is derived by minimizing the lp-norm of logarithmic cost. It is found that the FxLMP and VFxlogLMP require to select an appropriate value of p according to the prior information on noise characteristics, which prohibit their practical applications. Based on VFxlogLMP algorithm, we proposed a continuous lp-norm algorithm with logarithmic cost (VFxlogCLMP), which does not need the parameter selection and thresholds estimation. Benefiting from the various error norms for 1≤p≤2, it remains the robustness of VFxlogLMP. Moreover, the convergence behavior of VFxlogCLMP for moving average secondary paths and stochastic input signals is performed. Compared to the existing algorithms, two versions of the proposed algorithms have much better convergence and stability in impulsive noise environments.

Adaptive recursive algorithm with logarithmic transformation for nonlinear system identification in α-stable noise

Although the least mean pth power (LMP) and normalized LMP (NLMP) algorithms of adaptive Volterra filters outperform the conventional least mean square (LMS) algorithm in the presence of α-stable noise, they still exhibit slow convergence and high steady-state kernel error in nonlinear system identification. To overcome these limitations, an enhanced recursive least mean pth power algorithm with logarithmic transformation (RLogLMP) is proposed in this paper. The proposed algorithm is adjusted to minimize the new cost function with the p-norm logarithmic transformation of the error signal. The logarithmic transformation, which can diminish the significance of outliers under α-stable noise environment, increases the robustness of the proposed algorithm and reduces the steady-state kernel error. Moreover, the proposed method improves the convergence rate by the enhanced recursive scheme. Finally, simulation results demonstrate that the proposed algorithm is superior to the LMP, NLMP, normalized least mean absolute deviation (NLMAD), recursive least squares (RLS) and nonlinear iteratively reweighted least squares (NIRLS) algorithms in terms of convergence rate and steady-state kernel error.

Gas load forecasting based on chaotic theory and Volterra adaptive filter

Gas is an important source of clean energy, so in the future, gas load forecasting will have a very important meaning for the gas company to develop an appropriate business plan between‘upstream’gas supplier and‘downstream’users. Aiming at the gas short-term load forecasting problem, a prediction model has been put forward based on chaotic theory and Volterra adaptive filter. Firstly, conduct day-to-day correlation analysis for collected gas hour load data, and then calculate delay times and find out the optimal embedding dimension by the approach of mutual information and pseudo-nearest-neighbor. Secondly, on the basis of phase-space reconstruction, carry out chaotic characteristic analysis for collected gas hour load data. Once again, aiming at the current situation of most existing prediction models being subjective, reduce the subjectivity in the process of gas load forecasting by introduction of Volterra adaptive filter prediction model. Finally, the influence of the different order of Volterra adaptive filter on prediction results, and the comparison of accuracy among the Volterra adaptive filter prediction model, ANN (artificial neural network, ANN) prediction model and Fourier series prediction model are discussed through gas load forecasting example. Additionally, the forecasting results of the second order Volterra adaptive filter prediction model showed that: compared with ANN prediction model and fourier series prediction model, the second order Volterra adaptive filter prediction model has higher accuracy, and may provide a useful reference for practical engineering applications of short-term gas load forecasting.

Improvement of Chaotic Signals De-Noising with the Self-Optimizing Method of Wavelet Threshold

For the purpose of improving adaptive performance of chaotic signals de-noising with wavelet transform, a method of Memetic-algorithm-based adaptive wavelet de-noising (MAWD) is presented. The MAWD based on generalized cross validation (GCV) is competent to obtain the global optimum thresholds and to raise the efficiency of adaptive searching computation. The de-noising results of simulative Lorenz time series are presented. The results show that the chaotic signals de-noised by MAWD can remove the white noise more effectively than the signals de-noised by using standard soft threshoding method (STM) and genetic-algorithm-based adaptive wavelet de-noising (GAWD), and the advantages are more apparent under the condition of lower SNR. The Lorenz time series with lower SNR de-noised by MAWD and GAWD respectively are predicted by Volterra adaptive filters, and the results show that the prediction absolute error of Lorenz time series de-noised by MAWD is nearly nine times smaller than that by GAWD. This method has a promising prospect in practical Chaotic signals de-noising.

New Adaptive Compensator Robust to Memoryless Nonlinear Distortion

In this paper, the new hybrid adaptive Volterra filter was proposed to be applied for compensating the nonlinear distortion of memoryless nonlinear systems with saturation characteristics. Through computer simulations as well as the analytical analysis, it could be shown that it is possible for both conventional Volterra filter and proposed Volterra filter, to be applied for linearizing the memoryless nonlinear system with nonlinear distortion. Also, the simulation results demonstrated that the proposed Volterra filter may have faster convergence speed and better capability of compensating the nonlinear distortion than the conventional Volterra filter.

Pipelined robust M-estimate adaptive second-order Volterra filter against impulsive noise

The main limits on adaptive Volterra filters are their computational complexity in practical implementation and significant performance degradation under the impulsive noise environment. In this paper, a low-complexity pipelined robust M-estimate second-order Volterra (PRMSOV) filter is proposed to reduce the computational burdens of the Volterra filter and enhance the robustness against impulsive noise. The PRMSOV filter consists of a number of extended second-order Volterra (SOV) modules without feedback input cascaded in a chained form. To apply to the modular architecture, the modified normalized least mean M-estimate (NLMM) algorithms are derived to suppress the effect of impulsive noise on the nonlinear and linear combiner subsections, respectively. Since the SOV-NLMM modules in the PRMSOV can operate simultaneously in a pipelined parallelism fashion, they can give a significant improvement of computational efficiency and robustness against impulsive noise. The stability and convergence on nonlinear and linear combiner subsections are also analyzed with the contaminated Gaussian (CG) noise model. Simulations on nonlinear system identification and speech prediction show the proposed PRMSOV filter has better performance than the conventional SOV filter and joint process pipelined SOV (JPPSOV) filter under impulsive noise environment. The initial convergence, steady-state error, robustness and computational complexity are also better than the SOV and JPPSOV filters.

An approach for parameter estimation of combined CPPM and LFM radar signal

In this paper, the problem of parameter estimation of the combined radar signal adopting chaotic pulse position modulation (CPPM) and linear frequency modulation (LFM), which can be widely used in electronic countermeasures, is addressed. An approach is proposed to estimate the initial frequency and chirp rate of the combined signal by exploiting the second-order cyclostationarity of the intra-pulse signal. In addition, under the condition of the equal pulse width, the pulse repetition interval (PRI) of the combined signal is predicted using the low-order Volterra adaptive filter. Simulations demonstrate that the proposed cyclic autocorrelation Hough transform (CHT) algorithm is theoretically tolerant to additive white Gaussian noise. When the value of signal noise to ratio (SNR) is less than −4dB, it can still estimate the intra-pulse parameters well. When SNR=−3dB, a good prediction of the PRI sequence can be achieved by the Volterra adaptive filter algorithm, even only 100 training samples.

부대역 비선형 Volterra 적응필터의 응용과 성능분석

본 논문에서는 부대역 신호를 사용한 병렬 적응 비선형 Volterra 필터를 소개하고, 이 시스템의 성능을 분석하는 것을 주요 내용으로 한다. 연구 결과 제시한 부대역 신호를 사용한 적응 Volterra 필터는 수렴성이 우수함을 입력신호의 상관함수의 eigenvalue 분포를 사용하여 해석적으로 도출되었으며, 이러한 응용이 최근에 보고된 적응 반향 제거기에서 유용하게 사용될 수 있음을 이론적으로 밝혔다. 또한 각 부대역 에서의 최적필터를 이론적으로 유도하였고 컴퓨터 모의실험으로 이를 검증하였다. In this paper, the subband nonlinear adaptive Volterra filters are introduced and its analysis are presented. From the eigenvalue analysis of the input correlation matrix, we show that the proposed subband adaptive Volterra filter has superior convergence performance as compared to the conventional one, which shows that the it can be useful for the recently proposed subband nonlinear adaptive echo canceller. Also, the optimum filter in each subband are introduced and verified from the computer simulations.

Enhanced Adaptive Volterra Filtering by Automatic Attenuation of Memory Regions and Its Application to Acoustic Echo Cancellation

This paper presents a novel scheme for nonlinear acoustic echo cancellation based on adaptive Volterra Filters with linear and quadratic kernels, which automatically prefers those diagonals contributing most to the output of the quadratic kernel with the goal of minimizing the overall mean-square error. In typical echo cancellation scenarios, not all coefficients will be equally relevant for the modeling of the nonlinear echo, but coefficients close to the main diagonal of the second-order kernel will describe most of the nonlinear echo distortions, such that not all diagonals need to be implemented. However, it is difficult to decide the most appropriate number of diagonals a priori, since there are many factors that influence this decision, such as the energy of the nonlinear echo, the shape of the room impulse response, or the step size used for the adaptation of kernel coefficients. Our proposed scheme incorporates adaptive scaling factors that control the influence of each group of adjacent diagonals contributing to the quadratic kernel output. An appropriate selection of these factors serves to emphasize or neglect diagonals of the model as required by the present situation. We provide adaptation rules for these factors based on previous works on combination of adaptive filters, and comprehensive simulations showing the reduced gradient noise reached by the new echo canceller.

On the performance of adaptive pruned Volterra filters

Because of the high computational burden required by adaptive Volterra filters, several of their practical implementations consider some type of sparseness for complexity reduction. Such implementations are obtained using application-oriented strategies to prune a standard Volterra filter by zeroing some of its coefficients. In this context, the main challenge is to choose a pruning strategy that leads to minimum loss of performance. Meeting this challenge is not a trivial task because of the variety of strategies available for obtaining pruned Volterra filters as well as due to the lack of a theoretical framework describing these strategies in a general scenario. Thus, the primary objective of this research work is to establish a basis for assessing adaptive pruned Volterra filters. For such, a unifying scheme describing the input vectors of different pruned Volterra implementations is proposed along with an extended version of a constrained approach used to represent sparseness in adaptive filters. Based on this foundation, an analysis of the performance of adaptive pruned Volterra filters in terms of the minimum mean-square error is carried out. Simulation results are presented attesting the effectiveness of the proposed approach.

A Humanoid Method for Extracting Abnormal Engine Sounds from Engine Acoustics Based on Adaptive Volterra Filter

The improvement of SNR (Signal-to-Noise Ratio) of abnormal engine sounds is of great help in improving the accuracy of engine fault diagnosis. By imitating the way that human technicians use to distinguish abnormal engine sounds from engine acoustics, a humanoid abnormal sound extracting method is proposed. By implementing adaptive Volterra filter in the canonical Adaptive Noise Cancellation (ANC) system, the proposed method is capable of tracing the engine baseline sound which exhibits an intrinsic nonlinear dynamics. Besides, by introducing a template noise tailored from the records of engine baseline sound and taking it as virtual input of the adaptive Volterra filter, the priori knowledge of engine baseline sound, such as inherent correlation, periodicity or phase information, and stochastic factors, is taken into consideration. The hybrid simulations prove that the proposed method is functional. Since the method proposed is essentially a single-sensor based ANC, hopefully, it may become an effective way to extricate the dilemma that canonical dual-sensor based ANC encounters when it is used in extracting fault-featured signals from observed signals.

Identification of a parametric loudspeaker system using an adaptive Volterra filter

Due to the parametric acoustic array effect in air, the input audible signal of a parametric loudspeaker system can be reproduced with high directivity at the target region. However, the reproduced audible signal suffers from harmonic distortion, which is the by-product of nonlinear interaction between the primary waves. In order to investigate this inherent nonlinear phenomenon, a nonlinear system identification model is developed based on an adaptive Volterra filter. Unlike the conventional loudspeaker, the nonlinear characteristic of a parametric loudspeaker system is dependent on several primary parameters in nonlinear acoustics, which include the initial pressure of the primary waves, the observing distance and angle, as well as ambient temperature and relative humidity. By using a truncated Volterra series up to the 2nd-order kernel, numerical simulations are conducted to develop a system model with one group of parameters and examine the quadratic nonlinear intensity for different parameters’ settings. Experimental measurements, which take into account of emitter’s response, are carried out to verify the modeling result and evaluate the model performance. Based on the Volterra system model, the sound pressure level and the harmonic distortion can be accurately predicted.

Chaotic property analysis and prediction model study for heating load time series

In order to reveal the internal dynamics characteristics of heating load time series, the existing chaotic behavior is validated by use of nonlinear analysis method. The data sets taken from heat source and substation of district heat supply are studied by which phase spaces are reconstructed, and the correlation dimensions and the largest Lyapunov exponent are computed to identify the presence of chaos in heat load time series. By the analysis of the results, chaotic characteristics obviously exist in the heat load time series, which is a theoretical basis for the correlative investigation of heat load prediction. According to the existing heat load predictive method almostly based subjective models, a novel predictive approach based on Volterra adaptive filter, which avoids the subjective model assumptions, is presented for heat load prediction. Finally the predictive results are presented, and the simulation results illustrate that the second-order Volterra adaptive filter has high predictive accuracy which can meet the demands of heat energy-saving control and heat dispatching in practical applications.

Adaptive Combination of Volterra Kernels and Its Application to Nonlinear Acoustic Echo Cancellation

The combination of filters concept is a simple and flexible method to circumvent various compromises hampering the operation of adaptive linear filters. Recently, applications which require the identification of not only linear, but also nonlinear systems are widely studied. In this paper, we propose a combination of adaptive Volterra filters as the most versatile nonlinear models with memory. Moreover, we develop a novel approach that shows a similar behavior but significantly reduces the computational load by combining Volterra kernels rather than complete Volterra filters. Following an outline of the basic principles, the second part of the paper focuses on the application to nonlinear acoustic echo cancellation scenarios. As the ratio of the linear to nonlinear echo signal power is, in general, a priori unknown and time-variant, the performance of nonlinear echo cancellers may be inferior to a linear echo canceller if the nonlinear distortion is very low. Therefore, a modified version of the combination of kernels is developed obtaining a robust behavior regardless of the level of nonlinear distortion. Experiments with noise and speech signals demonstrate the desired behavior and the robustness of both the combination of Volterra filters and the combination of kernels approaches in different application scenarios.

A Sparse-Interpolated Scheme for Implementing Adaptive Volterra Filters

In most practical applications, the major drawback for using adaptive Volterra filters is the large number of coefficients to cope with. Several research works discussing strategies to reduce the computational burden of these structures have been presented in the open literature. For such, a common approach has been the use of some type of sparseness in Volterra filter kernels. In this work, a sparse-interpolated approach, with the interpolation having the purpose of recreating (in an approximate way) the elements disregarded for obtaining sparse kernels, is presented and discussed. Thus, for the adaptive sparse-interpolated Volterra filter, coefficient update expressions considering both least-mean-square (LMS) and normalized LMS (NLMS) algorithms are derived by using a constrained approach. In general, the proposed strategy outperforms other sparse schemes in terms of the tradeoff between computational complexity and mean-square error (MSE) performance, as shown through numerical simulations.

Adaptive non-linear filter using a modular polynomial perceptron

This study presents a joint adaptive non-linear filter with pipelined second-order polynomial perceptron (PSOVNN) to reduce the computational complexity and improve the non-linear processing capability of adaptive direct-form second-order Volterra (SOV) filter. The PSOVNN is a nesting modular structure comprising a number of modules that are interconnected in a chained form. Each module is implemented by a small-scale direct-form SOV neural network (SOVNN). These cascaded modules can perform a non-linear mapping from the input space to an intermediate space. In addition, the linear filter of the complete PSOVNN performs a linear mapping from the intermediate space to the output space. A modified real-time recurrent learning (RTRL) algorithm is developed, and its performance is evaluated by a series of simulation experiments. Computer simulations indicate that the proposed non-linear filter exhibits better performance over the direct-form SOV filter with less computational complexity.