Generalized entropic uncertainty principles on the complex signals in terms of fractional Hilbert transform

Abstract

Uncertainty principle is of much signification to physics and so on. Hilbert transform (HT) plays the role of changing the real signal into the complex signal, which is related to the Fourier transform (FT) in traditional time-frequency domain. The fractional Hilbert transform (FRHT) plays the same role as the traditional counterpart in the fractional Fourier transform (FRFT) domain as well as in optics. Different from the general complex signals, the complex signals derived from HT and FRHT have the new properties such as the absence of negative frequency and so on. In this paper, the generalized entropic uncertainty relations in terms of Shannon entropy, Rényi entropy and Tsallis entropy with respect to the complex signals derived from FRHT are demonstrated for the first time based on the mathematical principles, including the special cases for the traditional HT. Interestingly, these novel derived uncertainty bounds proved to be different from and much lower (e.g., always less by ln2) in some cases than that of the traditional complex (and real) signals, which reveals the new properties of these complex signals derived from FRHT for the further potential valuable application. Moreover, the relationships between the uncertainty principles from the two types of complex signals are established. Finally, the numerical simulation is performed in this paper to show the efficiency of the proposed principles.