A reduced-rank approach for implementing higher-order Volterra filters

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.

Implementação Eficiente de Filtros Volterra Explorando Abordagens de Posto Reduzido e Caracterı́sticas de Hardware

“…Como consequência, a aplicação de filtros Volterra muitas vezesé feita a partir de implementações com complexidade reduzida, como as implementações de posto reduzido consideradas neste trabalho. Além disso, conforme mostrado em [14], a representação triangular nãoé a mais interessante para obter implementações de kernels de segunda ordem com posto reduzido. Assim, como o foco deste trabalho está em implementações de segunda ordem, a representação triangular será considerada apenas na Seção IV para fins de comparação de desempenho.…”

“…Fazendo uma análise similar a (12)-(13) para obter P [|x k (n)| > α k ] e substituindo o valor resultante em (14), obtém-se…”

“…Considerando então (15) e ainda que a Função Qé uma função decrescente,é possível concluir que (14)é satisfeita somente se…”

“…1), ela pode ser facilmente aplicada a kernels com ordens superiores. Para tal, a decomposição em kernels de segunda ordem apresentada em [14]é utilizada e a abordagem aqui propostaé aplicada a cada um dos kernels de segunda ordem resultantes.…”

Effective hardware implementation of Volterra filters based on reduced‐rank approaches

“…As a consequence, practical applications of Volterra filters usually rely on reducedcomplexity implementations. Among these implementations, the so-called reduced-rank ones have attracted considerable interest over the last decades [2][3][4][5][6][7][8][9]. In general terms, such implementations are based on the application of matrix or tensor decompositions to structured representations of the kernels that compose the Volterra filter.…”