Uncertainty inequalities for linear canonical transform

Abstract

The novel Hausdorff–Young inequalities associated with the linear canonical transform (LCT) are derived based on the relation between the Fourier transform and the LCT in p-norm space (0<p<∞). Uncertainty relations for Shannon entropy and Rényi entropy based on the derived Hausdorff–Young inequality are yielded. It shows that these relations are functions of the transform parameters (a, b, c, d). Meanwhile, from the uncertainty relation for Shannon entropy the Heisenberg's uncertainty relation in LCT domains is derived, which holds for both real and complex signals. Moreover, the Heisenberg's uncertainty principle for the windowed fractional Fourier transform is obtained. Finally, one review of the uncertainty relations for the LCT and other transforms is listed in tables systematically for the first time.