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

Guanlei Xu, Xiaogang Xu, Xun Wang, Xiaotong Wang

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 Cramér–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 Cramér–Rao inequality and the uncertainty relation on Fisher information to the generalized cases. Compared with the traditional Cramér–Rao inequality, the generalized Cramér–Rao inequalities’ bounds are sharper and tighter. In addition, the generalized Cramér–Rao inequalities build the relation between the Cramér–Rao bounds and the FrFT transform angles, which seem to be quaint compared with the traditional counterparts. Furthermore, the generalized Cramér–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.