Cramér-Rao Bound Conditioned by the Energy Detector

A wide variety of processing incorporates a binary detection test that restricts the set of observations available for parameter estimation and requires to take this statistical conditioning into account to compute the Cramer-Rao bound (CRB). Therefore, we propose a derivation of the CRB for the deterministic signal model conditioned by the energy detector widely used in signal processing applications. This derivation has lead us to introduce novel identities on some conditional expectations of complex circular Gaussian random vectors that may be useful for other derivations.