Evaluating the potential of Volterra-PARAFAC IIR models

Abstract

The Volterra-PARAFAC (VP) nonlinear system model, which consists of a FIR filterbank followed by a memoryless nonlinearity, aims at offering a good compromise between accuracy and parametric complexity. Here, for an even better compromise, we propose a generalization with IIR filters (VPI model) and evaluate both models. For the evaluation, we consider the concrete case of two audio loudspeakers and initially compute reference Volterra kernels from their known physical state-space models, using an efficient procedure. Then, VP and VPI models are derived and their accuracy is tested. As shown, the VPI models have in this case only 15 to 26 % of the parametric complexity of VP models with the same accuracy, which points to a great potential for accurate and efficient nonlinear system modeling.

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

“…In general, the first step for the development of a reduced‐rank implementation for a Volterra kernel is to represent its input–output relationship as a function of a matrix (or eventually a tensor) of coefficients [2–9]. For instance, in the case of the implementation introduced in [3], the input–output relationship of the second‐order kernel is written as where is the input vector and is a symmetric coefficient matrix.…”

“…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.…”

“…Reduced-rank Volterra implementations: In general, the first step for the development of a reduced-rank implementation for a Volterra kernel is to represent its input-output relationship as a function of a matrix (or eventually a tensor) of coefficients [2][3][4][5][6][7][8][9]. For instance, in the case of the implementation introduced in [3], the input-output relationship of the second-order kernel is written as…”

“…It is important to point out that the proposed approach can be easily generalised to higher-order kernels by using the strategy presented in [9]. Moreover, the idea behind the proposed approach can potentially be extended to other reduced-rank implementations, since most of them involve parallel branches having different levels of significance [2][3][4][5][6][7][8][9]. Results: In this section, the effectiveness of the proposed approach is assessed considering the fixed-point implementation (with 8, 12, and 16 bits) of the second-order kernel described in [2] (whose memory size is 21).…”

Estimação da causalidade de Granger no caso de interação não-linear.

“…Como visto no Apêndice C, a série de Volterra, por ser de natureza não-paramétrica, leva a modelos com um número muito elevado de coeficientes, mesmo para sistemas de ordem reduzida, como descreve [98,99]. A fim de se diminuir essa alta complexidade dos Sistemas de Volterra, foram propostos os Modelos Orientados à Blocos (MOB) [12], que são estruturas compostas pela interconexão, em cascata ou paralelo, de sistemas lineares contendo não-linearidades estáticas (sem memória) 1 .…”