Tensor product basis approximations for Volterra filters

| This paper studies approximations for a class of nonlinear lters known as Volterra lters. Although the Volterra lter provides a relatively simple and general representation for nonlinear ltering, often it is highly over-parameterized. Due to the large number of parameters, the utility of the Volterra lter is limited. The over-parameterization problem is addressed in this paper using a tensor product basis approximation (TPBA). In many cases a Volterra lter may be well approximated using the TPBA with far fewer parameters. Hence, the TPBA ooers considerable advantages over the original Volterra lter in terms of both implementation and estimation complexity. Furthermore, the TPBA provides useful insight into the lter response. This paper studies the crucial issue of choosing the approximation basis. Several methods for designing an appropriate approximation basis and error bounds on the resulting mean-square output approximation error are derived. Certain methods are shown to be nearly optimal.