Wavelet domain implementation of the estimator-correlator and weighted wavelet transforms

It is well known that the estimator-correlator (EC) is a maximum likelihood detector for random signals in Gaussian noise. In this paper we derive a continuous wavelet domain EC processor for the detection of signals that have propagated over stochastic propagation and scattering channels. The derivation shows that the wavelet transforms that are used for the conditional mean estimator (CME) and for the computation of the detection statistic must be defined by using reproducing kernel Hilbert space (RKHS) inner products rather than ordinary Hilbert space inner products. This fact suggests new weighted wavelet (as well as other time-frequency and time-scale) transforms. These new transforms have many applications to optimum signal processing.