Generalized Cramér–Rao inequality and uncertainty relation for fisher information on FrFT

Abstract

Uncertainty principle plays an important role in signal processing, physics and mathematics and so on. In this paper, four novel uncertainty inequalities including the new generalized Cramer–Rao inequalities and the new uncertainty relations on Fisher information associated with fractional Fourier transform (FrFT) are deduced for the first time. These novel uncertainty inequalities extend the traditional Cramer–Rao inequality and the uncertainty relation on Fisher information to the generalized cases. Compared with the traditional Cramer–Rao inequality, the generalized Cramer–Rao inequalities’ bounds are sharper and tighter. In addition, the generalized Cramer–Rao inequalities build the relation between the Cramer–Rao bounds and the FrFT transform angles, which seem to be quaint compared with the traditional counterparts. Furthermore, the generalized Cramer–Rao inequalities give the relation between the FrFT’s variance and FrFT’s gradient’s integral in only one single transform domain, which is fully novel. On the other hand, compared with the traditional uncertainty relation on Fisher information, the newly deduced uncertainty relations on Fisher information yield the sharper and tighter bounds. These deduced inequalities are novel, and they will yield the potential advantage in the parameter estimation in the FrFT domain. Finally, examples are given to show the efficiency of these newly deduced inequalities.