Among the various mathematical representations of nonlinear systems, a popular description is given by means of the discrete-time Volterra series. In this paper we first briefly review the relevant nonlinear system identification techniques, with reference in particular to nonlinear systems with lengthy memory. In such cases, a parallel-cascade structure has been proved to be very effective since it is able to provide an arbitrarily close approximation, in the mean square error (MSE) sense, for a broad class of systems to be modelled. The parallel-cascade structure is indeed an exact representation for a quadratic filter, obtained by means of a matrix decomposition technique: we recall such a technique and then we show how it is possible to apply it for the adaptive identification of quadratic systems with large time delays
