Free Metaplectic Wigner Distribution: Definition and Heisenberg’s Uncertainty Principles
Zhichao Zhang, Zhicheng Zhu, Dong Li, Yangfan He
Abstract
Inspired by a definition of the closed-form instantaneous cross-correlation Wigner distribution (Zhang, 2019), we generalize the notion of Wigner distribution to the so-called free metaplectic Wigner distribution (FMWD) through three free metaplectic transforms, in order to tackle a challenge in high-dimensional complex features information processing. We provide some representative special cases for this general form, including the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> -dimensional nonseparable affine characteristic Wigner distribution, kernel function Wigner distribution, convolution representation Wigner distribution and instantaneous cross-correlation Wigner distribution. We establish the standard Heisenberg’s uncertainty principles (HUPs) of the real-valued function for the FMWD. We also establish the standard HUPs of the complex-valued function for some specific (i.e., the orthogonal, the orthonormal, the minimum eigenvalue commutative and the maximum eigenvalue commutative) FMWDs. In view of the one-dimensional case of our results, we solve a burning question regarding the limit of time-frequency superresolution triggered by the linear canonical transform free parameters.