I.I.R. Volterra Filtering with Application to Bilinear Systems

A new nonlinear filtering technique by means of infinite impulse response (IIR) Volterra functionals is developed. It yields the projection onto the closed class of finite Volterra series with separable kernels of arbitrary degree k. Such an optimal estimator is finitely realizable as a bilinear system with parameters that are computable off line. Moreover, if the original system model is itself bilinear, this computation is finitely recursive through higher moments of degree 2 k. Two simple illustrating examples are developed: (i) estimation of the covariance of the internal white noise driving a linear system and (ii) filtering of a non-Gaussian linear system (driven by a Poisson process). The robustness with respect to the observation noise distribution is finally examined