Volterra Series - Methods To Estimate The Kernel Coefficients

Methods To Estimate The Kernel Coefficients

Estimating the Volterra coefficients individually is complicated since the basis functionals of the Volterra series are correlated. This leads to the problem of simultaneously solving a set of integral-equations for the coefficients. Hence, estimation of Volterra coefficients is generally performed by estimating the coefficients of an orthogonalized series, e.g. the Wiener series, and then recomputing the coefficients of the original Volterra series. The Volterra series main appeal over the orthogonalized series lies in its intuitive, canonical structure, i.e. all interactions of the input have one fixed degree. The orthogonalized basis functionals will generally be quite complicated.

An important aspect, with respect to which the following methods differ is whether the orthogonalization of the basis functionals is to be performed over the idealized specification of the input signal (e.g. gaussian, white noise) or over the actual realization of the input (i.e. the pseudo-random, bounded, almost-white version of gaussian white noise, or any other stimulus). The latter methods, despite their lack of mathematical elegance, have been shown to be more flexible (as arbitrary inputs can be easily accommodated) and precise (due to the effect that the idealized version of the input signal is not always realizable).