Uncertainty principle for linear canonical transform using matrix decomposition of absolute spread matrix

Abstract

We define a matrix, the absolute spread matrix (ASM), whose positive definite property is independent of signals. Based on the relation between the ambiguity function and the linear canonical power spectra, we give a new perspective on a key relation associated with spreads in time, frequency, time-frequency and linear canonical transform (LCT) domains, from which a new LCT uncertainty principle for complex signals using the ASM's rotation orthogonal decomposition is derived. The proposed lower bound of uncertainty product in two LCT domains is tighter than the latest one given by the author. We then obtain the sufficient and necessary condition that gives rise to this stronger result truly, and define a quantitative index to analyse the difference with the existing bound. We also reduce the uncertainty inequality to time and LCT domains. Furthermore, we present the example and simulation results that validate the previous theoretical analyses, and finally we demonstrate the effectiveness of the new proposal through discussing its applications in signal processing and optics.