A multichannel hierarchical approach to adaptive Volterra filters employing filtered-X affine projection algorithms

It is shown in this paper how the use of a recently introduced algebra, called V-vector algebra, can directly lead to the implementation of Volterra filters of any order P in the form of a multichannel filterbank. Each channel in this approach is modeled as a finite impulse response (FIR) filter, and the channels are hierarchically arranged according to the number of the filter coefficients. 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 P. The derivation of the so-called Filtered-X AP algorithms for nonlinear active noise controllers is easily obtained using the elements of the V-vector algebra. These algorithms can efficiently replace the standard LMS and NLMS algorithms usually applied in this field, especially when, in practical applications, a reduced-complexity multichannel structure can be exploited.