Adaptive polynomial filters

Journal ArticleWhile linear filter are useful in a large number of applications and relatively simple from conceptual and implementational view points. there are many practical situations that require nonlinear processing of the signals involved. This article explains adaptive nonlinear filters equipped with polynomial models of nonlinearity. 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 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. followed by adaptive algorithms using system models involving recursive nonlinear difference equations. Such systems are attractive because they may be able to approximate many nonlinear systems with great parsimony in the use pf 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 also described

A fast recursive least-squares second order volterra filter

Journal ArticleABSTRACT This paper presents a fast, recursive least-squares (RLS) adaptive nonlinear filter. The nonlinearity is modelled using a second order Volterra series expansion. The structure presented in the paper makes use of the ideas of fast RLS multichannel filters and has a computational complexity of OW3) multiplications. This compares with OW6) multiplications required for direct implementation. Simulation examples in which the filter is employed to identify nonlinear systems using noisy output observations are also presented. Further simplification to the structure through a simplified model is discussed very briefly in the paper

Sparse Volterra and Polynomial Regression Models: Recoverability and Estimation

Volterra and polynomial regression models play a major role in nonlinear system identification and inference tasks. Exciting applications ranging from neuroscience to genome-wide association analysis build on these models with the additional requirement of parsimony. This requirement has high interpretative value, but unfortunately cannot be met by least-squares based or kernel regression methods. To this end, compressed sampling (CS) approaches, already successful in linear regression settings, can offer a viable alternative. The viability of CS for sparse Volterra and polynomial models is the core theme of this work. A common sparse regression task is initially posed for the two models. Building on (weighted) Lasso-based schemes, an adaptive RLS-type algorithm is developed for sparse polynomial regressions. The identifiability of polynomial models is critically challenged by dimensionality. However, following the CS principle, when these models are sparse, they could be recovered by far fewer measurements. To quantify the sufficient number of measurements for a given level of sparsity, restricted isometry properties (RIP) are investigated in commonly met polynomial regression settings, generalizing known results for their linear counterparts. The merits of the novel (weighted) adaptive CS algorithms to sparse polynomial modeling are verified through synthetic as well as real data tests for genotype-phenotype analysis.Comment: 20 pages, to appear in IEEE Trans. on Signal Processin

Volterra and general polynomial related filtering

Journal ArticleThis paper presents a review of polynomial filtering and, in particular, of tlie truncated Volterra filters. Following the introduction of the general properties of such filters, issues such as eficieiit realizations, design, adaptive algoritlims and stability are discussed

Adaptive weighted least squares algorithm for Volterra signal modeling

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Fast recursive least squares adaptive second-order volterra filter and its performance analysis

Journal ArticleAbstract-This paper presents a fast, recursive least squares (RLS) adaptive nonlinear filter. The nonlinearity is modeled using a second-order Volterra series expansion. The structure presented in the paper makes use of the ideas of fast RLS multichannel filters and has a computational complexity of 0 (N') multiplications per time instant where N - 1 represents the memory span in number of samples of the nonlinear system model. This compares with 0 (N 6) multiplications required for direct implementation. A theoretical performance analysis of the steady-state behavior of the adaptive filter operating in both stationary and nonstationary environments is presented in the paper. The analysis shows that, when the input is zero mean, 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 are included in the paper. These results show that the adaptive Volterra filter performs well in a variety of situations. Furthermore, the steady-state behavior predicted by the analysis is in very good agreement with the experimental results

Lattice algorithms for recursive least squares adaptive second-order volterra filtering

Journal ArticleThis paper presents two computationally efficient recursive least-square (RLS) lattice algorithms for adaptive nonlinear filtering based on a truncated second-order Volterra system model. The lattice formulation transforms the nonlinear filtering problem into an equivalent multichannel, linear filtering problem and then generalizes the lattice solution to the nonlinear filtering problem. One of the algorithms is a direct extension of the conventional RLS lattice adaptive linear filtering algorithm to the nonlinear case. The other algorithms is based on the QR decomposition of the prediction error covariance matrices using orthogonal transformations. Several experiments demonstrating and comparing the properties of the two algorithms in finite and "infinite" precision environments are included in the paper. The results indicate that both the algorithms retain the fast convergence behavior of the RLS Volterra filters and are numerically stable

An improved approximate QR-LS based second-order Volterra filter

This paper proposes a new transform-domain approximate QR least-squares-based (TA-QR-LS) algorithm for adaptive Volterra filtering (AVF). It improves the approximate QR least-squares (A-QR-LS) algorithm for multichannel adaptive filtering by introducing a unitary transformation to decorrelate the input signal vector so as to achieve better convergence and tracking performances. Further, the Givens rotation is used instead of the Householder transformation to reduce the arithmetic complexity. Simulation results show that the proposed algorithm has much better initial convergence and steady state performance than the LMS-based algorithm. The fast RLS AVF algorithm [J. Lee and V. J. Mathews, Mar 1993] was found to exhibit superior steady state performance when the forgetting factor is chosen to be 0.995, but the tracking performance of the TA-QR-LS algorithm was found to be considerably better.published_or_final_versio

A fast recursive least-squares adaptive nonlinear filter

Journal ArticleThis paper presents a fast, recursive least-squares (RLS) adaptive nonlinear filter. The nonlinearity in the system is modeled using the Hammerstein model, which consists of a memoryless polynomial nonlinearity followed by a finite impulse response linear system. The complexity of our method is about 3NP2+7NP+N+10P2+6P multiplications per iteration and is substantially lower than the computational complexities of fast RLS algorithms that are direct extensions of RLS adaptive linear filters to the nonlinear case