Journal of Fourier Analysis and Applications

, Volume 25, Issue 6, pp 2899–2922 | Cite as

Uncertainty Principle for Real Functions in Free Metaplectic Transformation Domains

  • Zhichao ZhangEmail author


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 \({\mathbf {A}}_j,{\mathbf {B}}_j\), \(j=1,2\) 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.


Heisenberg’s uncertainty principle Metaplectic transformation Free symplectic matrix Inequality Covariance matrix Trace Singular value 

Mathematics Subject Classification

15A42 42A38 42B10 70H15 



The research was supported by the Startup Foundation for Introducing Talent of NUIST (Grant 2019r024) and the China Scholarship Council (CSC) joint Ph.D. student scholarship (Grant 201706240025). The author would also like to thank the anonymous reviewers for making many useful suggestions (especially the suggested mathematical terminology) to the manuscript.


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Authors and Affiliations

  1. 1.School of Mathematics and StatisticsNanjing University of Information Science & TechnologyNanjingChina
  2. 2.Department of MathematicsSichuan UniversityChengduChina
  3. 3.Department of Electrical and Computer Engineering, Tandon School of EngineeringNew York UniversityBrooklynUSA

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