Uncertainty Principle for Real Functions in Free Metaplectic Transformation Domains

This study devotes to the uncertainty principle under the free metaplectic transformation (an abbreviation of the metaplectic operator with a free symplectic matrix) of a real function. Covariance matrices in time, frequency and time–frequency domains are defined, and a relationship between these matrices and the free metaplectic transformation domain covariance is proposed. We then obtain two versions of lower bounds on the uncertainty product of the covariances of a real function in two free metaplectic transformation domains. It is shown here that a multivariable square integrable real-valued function cannot be both two free metaplectic transformations band limited. It is also seen that these two lower bounds depend not only on the minimum singular value of the blocks Aj,Bj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {A}}_j,{\mathbf {B}}_j$$\end{document}, j=1,2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j=1,2$$\end{document} found in free symplectic matrices but also on the covariance in time domain or in frequency domain. We thus reduce them to a new one which does not contain the covariances in time and frequency domains. Sufficient conditions that reach the lower bounds are derived. Example and simulation results are provided to validate the theoretical analysis.