Complex transversal Volterra filters optimal for detection and estimation

A class of nonlinear filter, the p-th order complex transversal Volterra filter (TVF), is proposed. Its output is represented by a specific inner product of two vectors, which allows a unique formulation of problems such as the nonlinear detection, using the deflection criterion, and the nonlinear estimation in a mean square (MS) sense, by p-th order complex TV filtering, is introduced. It is shown that the p-th order complex TVF which maximizes the deflection is also the one which minimizes the distance, in a MS sense, to the likelihood ratio. The calculation of the optimal p-th order complex TVF requires the knowledge of the statistical properties of the data up to the 2p-th order