New perspectives, extensions and applications of de Bruijn identity

This paper focuses on de Bruijn identity, a fundamental result that relates two important concepts: Fisher information and differential entropy. A novel relationship with Stein identity and extensions of de Bruijn identity are first presented. Then several applications of de Bruijn identity in deriving the Bayesian Cramér-Rao lower bound (BCRLB), Cramér-Rao lower bound (CRLB), and a new lower bound, tighter than the BCRLB, are presented. The paper concludes with an application of de Bruijn identity in designing min-max optimal training sequences for channel estimation and synchronization in the presence of unknown noise distribution.