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. >

Adaptive polynomial filters

https://www.dtc.umn.edu/s/resources/gg-se.pdf

A bibliography on nonlinear system identification

Polynomial (or Volterra) filters are introduced, and the quadratic filters are presented as the simplest example of such filters. The principle aspects and properties of quadratic filters are derived in the framework of the discrete Volterra expansion. Fixed as well as adaptive filters are considered in one-dimensional and multidimensional environments. Such issues as design and efficient realizations are thoroughly addressed, and standard and advanced adaptation algorithms are presented. Several examples of signal processing applications requiring quadratic filters are discussed. >

Quadratic filters for signal processing

An adaptive filter algorithm is developed for the class of stack filters, which is a class of nonlinear filters obeying a weak superposition property. The adaptation algorithm can be interpreted as a learning algorithm for a group of decision-making units, the decisions of which are subject to a set of constraints called the stacking constraints. Under a rather weak statistical assumption on the training inputs, the decision strategy adopted by the group, which evolves according to the proposed learning algorithm, is shown to converge to an optimal strategy in the sense that it corresponds to an optimal stack filter under the mean absolute error-criterion, this adaptive algorithm requires only increment, decrement, and comparison operations and only local interconnections between the learning units. Implementation of the algorithm in hardware is therefore very feasible. An example is provided to show how the adaptive stack filtering algorithm can be used in an application in image processing. >

Adaptive stack filtering under the mean absolute error criterion

An application for adaptive nonlinear filters, adaptive linearization of a loudspeaker, is presented. An adaptive predistortion linearization scheme is used for linearizing a loudspeaker. A model of a direct radiator loudspeaker is developed and studied which takes into account two principal sources of nonlinear distortions: nonlinearity in the suspension system and inhomogeneity in the flux density. Based on this model, simulations of the proposed method were performed. The results show that nonlinear distortions of a loudspeaker can be reduced significantly. >

/pdf/adaptive-linearization-of-a-loudspeaker-16s5pcrhmh.pdf

Adaptive linearization of a loudspeaker

An algorithm for updating the linear and quadratic weights of a second-order Volterra filter (SVF) is proposed. This algorithm uses a fast Kalman filter algorithm to calculate the Kalman gain vector used in updating the linear and quadratic weights of the SVF. The convergence of the algorithm for the quadratic weights is established. A simulation is then performed in which a fast Kalman filter implementation of an SVF is compared to an LMS implementation in a system identification problem. The fast Kalman filter implementation is shown to converge to the unknown system parameters considerably faster than the LMS implementation.

A second-order adaptive Volterra filter with rapid convergence

Adaptive estimation of single evoked potentials.

When the input to the adaptive line enhancer (ALE) contains colored noise, it is shown that implementations of adaptive line enhancers using recursive least squares (RLS) algorithms result in suboptimal estimates of the least squares weight vector Looking at RLS filtering as a Kalman filtering problem makes it possible to account for the presence of colored observation noise by implementing a model of the colored noise using an augmented state equation The RLS filter derived from this augmented state equation and the corresponding augmented observation equation is shown to have improved performance over the standard RLS filter for ALE prediction lengths greater than and significantly less than the decorrelation time of the colored observation noise

C. Davila

Papers

A second-order adaptive Volterra filter with rapid convergence

Adaptive estimation of single evoked potentials.

A Kalman filter algorithm for estimating sinusoids in colored noise