Weighted Heisenberg–Pauli–Weyl uncertainty principles for the linear canonical transform

The classical uncertainty principle plays an important role in quantum mechanics, signal processing and applied mathematics. With the development of novel signal processing methods, the research of the related uncertainty principles has gradually been one of the most hottest research topics in modern signal processing community. In this paper, the weighted Heisenberg–Pauli–Weyl uncertainty principles for the linear canonical transform (LCT) have been investigated in detail. Firstly, the Plancherel–Parseval–Rayleigh identities associated with the LCT are derived. Secondly, the weighted Heisenberg–Pauli–Weyl uncertainty principles in the LCT domain are investigated based on the derived identities. The signals that can achieve the lower bound of the uncertainty principle are also obtained. The classical Heisenberg uncertainty principles in the Fourier transform (FT) domain are shown to be special cases of our achieved results. Thirdly, examples are provided to show that our weighted Heisenberg–Pauli–Weyl uncertainty principles are sharper than those in the existing literature. Finally, applications of the derived results in time frequency resolution analysis and signal energy concentrations are also analyzed and discussed in detail.