IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 4, APRIL 2005 1463 A Multichannel Hierarchical Approach to Adaptive Volterra Filters Employing Filtered-X Affine Projection Algorithms Giovanni L. Sicuranza, Senior Member, IEEE, and Alberto Carini, Member, IEEE Abstract—It is shown in this paper how the use of a recently in- troduced algebra, called V-vector algebra, can directly lead to the implementation of Volterra filters of any order in the form of a multichannel filterbank. Each channel in this approach is mod- eled as a finite impulse response (FIR) filter, and the channels are hierarchically arranged according to the number of the filter coef- ficients. In such a way, it is also possible to devise models of reduced complexity by cutting the less relevant channels. This model is then used to derive efficient adaptation algorithms in the context of nonlinear active noise control. In particular, it is shown how the affine projection (AP) algorithms used in the linear case can be extended to a Volterra filter of any order . The deriva- tion of the so-called Filtered-X AP algorithms for nonlinear ac- tive noise controllers is easily obtained using the elements of the V-vector algebra. These algorithms can efficiently replace the stan- dard LMS and NLMS algorithms usually applied in this field, espe- cially when, in practical applications, a reduced-complexity multi- channel structure can be exploited. Index Terms—Adaptive Volterra filters, affine-projection algo- rithms, multichannel filterbank, nonlinear active noise control, V-vector algebra. I. INTRODUCTION V OLTERRA filters [1] have been thoroughly studied in the context of nonlinear filters because of their nice properties that permit the derivation of a theoretical framework, including the well-known linear filters as a particular case. According to these properties and, in particular, to the linearity of the filter output with respect to the filter coefficients, adaptation rules for Volterra filters can be obtained by extending classical algorithms exploited for linear filters. This extension is often based on a multichannel approach [2]–[4] in which the Volterra filter is re- alized by means of a linear filterbank where each filter processes a product of samples of the input signal. The multichannel structure has been more recently exploited in [5] to implement the so-called simplified Volterra filters (SVFs). Even though this model can be in principle applied to kernels of arbitrary order, the simplest example of a Volterra filter, i.e., the homogeneous quadratic filter, has been specif- ically considered in [5] with reference to the acoustic echo Manuscript received October 15, 2003; revised April 13, 2004. This work was suppoted in part by Grant MIUR PRIN 2004092314. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Kenneth E. Barner. G. L. Sicuranza is with the Department of Electrical Electronic and Computer Engineering, University of Trieste, 34127 Trieste, Italy (email: sicuranza@univ.trieste.it). A. Carini is with the Information Science and Technology Institute, Univer- sity of Urbino, 61029 Urbino, Italy (e-mail: carini@sti.uniurb.it). Digital Object Identifier 10.1109/TSP.2005.843705 cancellation problem. The stress in [5] is on the fact that, according to the characteristics of the measured second-order kernel, it is possible to use a reduced number of active channels. In fact, it has been noted that the relevance of the quadratic kernel elements is strongly decreasing moving far from the main diagonal. Therefore, remarkable savings in implementation complexity can be achieved. This is an important aspect since one of the main drawbacks of Volterra filters is their implemen- tation complexity. It is worth noting that similar behaviors have been observed in other real-world nonlinear systems, as well. The argument of reduced number of active channels has been also presented in [6] with reference to the so-called banded regular Volterra kernels. As far as the adaptation algorithm is concerned, in [5], an updating rule based on the so-called affine projection (AP) algorithm has been devised extending it from the linear case, as originally proposed by Ozeki and Umeda [7], to quadratic filters. The significant aspect of this class of algorithms is the ability to offer, in the presence of correlated signals, convergence rates higher than those of LMS algorithms and tracking capabilities better than those of RLS algorithms [8]. In fact, while least mean squares (LMS) and normalized LMS (NLMS) adaptive algorithms minimize, according to the stochastic approximation, the error between the conditioning signal and the actual output signal at time , AP algorithms try to minimize the last a posteriori errors at time , . As a consequence, AP algorithms behave as the LMS algorithms for while tending to behave as the RLS algorithms for an increasing number of errors considered. On the other hand, even when the input signal is white, the presence of products of samples in the input signal to a Volterra filter introduces correlation among its entries. This is the reason why AP techniques can offer, for this class of filters, better convergence and tracking capabilities than the classical LMS and NLMS algorithms. According to the structure of SVFs, these performances are obtained with a limited increment of the computational complexity. A further example of application of the multichannel struc- ture can be found in [9], where adaptive Volterra filters are used to implement nonlinear active noise controllers. In recent years, intensive studies have been carried out in the field of vibration and acoustic noise control with promising results [10], [11]. The technique used in a single-channel active noise controller is based on the destructive interference in a given location of the noise produced by a primary source and the interefering signal generated by a secondary source. However, most of the studies presented in the literature refer to linear models, whereas it is 1053-587X/$20.00 © 2005 IEEE