General framework for the higher-order correlation domain

A general framework that theoretically links the higher-order correlation (HOC) domain with statistical decision theory is explored. This is achieved by showing that the HOC domain provides an equivalent implementation of an average likelihood-ratio function (ALF) under certain conditions. Within this framework, statistics such as the energy of the first- and second-order correlation functions, among others, arise as terms in a power series expansion of an ALF. This ALF is associated with an AWGN-mutilated sinusoidal signal whose frequency is continuous and uniform. General analytical results on detection are produced. Special emphasis is placed on comparing the performance associated with the HOC with those achieved by the standard channelized and radiometric approaches. Receiver operating characteristic expression is also given to determine the effect on the overall performance of truncating the optimal non-linearity at different orders.