Abstract
The use of Volterra filters in practical applications is often limited by their high computational burden. To cope with this problem, many strategies for implementing Volterra filters with reduced complexity have been proposed in the open literature. Some of these strategies are based on reduced-rank approaches obtained by defining a matrix of filter coefficients and applying the singular value decomposition to such a matrix. Then, discarding the smaller singular values, effective reduced-complexity Volterra implementations can be obtained. The application of this type of approach to higher-order Volterra filters (considering orders greater than 2) is however not straightforward, which is especially due to some difficulties encountered in the definition of higher-order coefficient matrices. In this context, the present paper is devoted to the development of a novel reduced-rank approach for implementing higher-order Volterra filters. Such an approach is based on a new form of Volterra kernel implementation that allows decomposing higher-order kernels into structures composed only of second-order kernels. Then, applying the singular value decomposition to the coefficient matrices of these second-order kernels, effective implementations for higher-order Volterra filters can be obtained. Simulation results are presented aiming to assess the effectiveness of the proposed approach.
Highlights
The first challenge in filtering applications involving nonlinear systems is to choose an adequate model of the nonlinear filter [1]
The well-known Volterra filter [1] represents one extreme of this trade-off, since its universal approximation capability [2,3,4] comes at the cost of a high computational complexity [1, 5,6,7,8,9]
For higher-order filters, matrix-based reduced-rank approaches are usually obtained by considering non-trivial definitions of coefficient matrices [24], which occasionally lead to ineffective reduced-rank implementations
Summary
The first challenge in filtering applications involving nonlinear systems is to choose an adequate model of the nonlinear filter [1]. For higher-order filters, matrix-based reduced-rank approaches are usually obtained by considering non-trivial definitions of (often rectangular) coefficient matrices [24], which occasionally lead to ineffective reduced-rank implementations. In this context, the present paper is focused on the development of a novel reduced-rank approach for implementing higher-order Volterra filters. The present paper is focused on the development of a novel reduced-rank approach for implementing higher-order Volterra filters This approach is based on a new form of Volterra kernel implementation that allows converting a higher-order Volterra kernel into a structure composed of second-order kernels. Underbars specify variables related to the redundancy-removed Volterra representation and overbars indicate variables related to the proposed approach
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