Generalized Heisenberg uncertainty relations for complex-valued signals out of Hilbert transform associated with LCT

Heisenberg uncertainty principle is of much signification to physics and signal processing, and the most classical representation of Heisenberg uncertainty principle represents the relationship between time duration and frequency duration in traditional time-frequency domain. Linear canonical transform (LCT) is the generalization of Fresnel transform and fractional Fourier transform (FRFT), and has been used in physical optics and information processing. In this paper, from various aspects six new results of Heisenberg uncertainty principles in LCT domain are obtained for the complex-valued signals out of traditional Hilbert transform (HT) and generalized Hilbert transform (GHT) in the first time to our best knowledge, among which two ones are related with parameters a and b and another two ones are related with c and d, and the last two ones are related with four transform parameters a, b, c and d simultaneously. Since the complex-valued signals out of GHT have the merit of the absence in negative generalized frequency domain, their physical meanings are shown as well, and these results discover the inequalities’ relationships for durations and phase differential for these complex-valued signals. In addition, it has also been shown that any one of these Heisenberg uncertainty principles can turn to classical Heisenberg uncertainty principle in traditional time-frequency domain for the complex-valued signals out of traditional HT.