A unifying and general approach to adaptive linear-quadratic discrete time Volterra filtering

A general and unifying approach is presented that has different features, depending on whether the third-order moments vanish. All the well known procedures used in the linear context, such as the stochastic least mean squares (LMS), root mean square, and fast transversal filtering procedures, are extended. The role played by the third-order moments is pointed out. For any procedure if the third-order moments are vanishing, the optimum approach is parallel operation of two separate procedures. Otherwise, the optimum approach is a coupled algorithm. It is shown that the performance of the linear Wiener filter is always improved by adding a quadratic filter even if the third-order moments are vanishing, provided that the observation and the unknown processes are not jointly Gaussian. A general theory of the convergence of the linear-quadratic LMS method is established, assuming the m-ary independence of the underlying process. The effect of the mismatch in the algorithms is considered as the effect of a noise whose variance is derived. All these ideas are illustrated by simulation