Uncertainty principles for short-time free metaplectic transformation

This study devotes to extend Heisenberg's uncertainty inequalities in free metaplectic transformation (FMT) domains into short-time free metaplectic transformation (STFMT) domains. We disclose an equivalence relation between spreads in time-STFMT and time domains, as well as FMT-STFMT and FMT domains. We use them to set up an inequality relation between the uncertainty product in time-STFMT and FMT-STFMT domains and that in time and FMT domains and an inequality relation between the uncertainty product in two FMT-STFMT domains and that in two FMT domains. We deduce uncertainty inequalities of real-valued functions and complex-valued window functions for the STFMT and uncertainty inequalities of complex-valued (window) functions for the orthogonal STFMT, the orthonormal STFMT, and the STFMT without the assumption of orthogonality, respectively. To formulate the attainable lower bounds, we also propose some novel uncertainty inequalities of complex-valued (window) functions for the orthogonal FMT and the FMT without the assumption of orthogonality, respectively.