Free Metaplectic K-Wigner Distribution: Uncertainty Principles and Applications

This study focuses on a novel parameterized Wigner distribution, which is an organic integration of the free metaplectic Wigner distribution and the K-Wigner distribution. We style this as the free metaplectic K-Wigner distribution (FMKWD) and investigate its uncertainty principles and related applications. We establish a crucial equivalence relation between the uncertainty product in time-FMKWD and free metaplectic transformation (FMT)-FMKWD domains and those in two FMT domains, from which we derive two types of orthogonality conditions: an orthonormality condition; and two sub-types of minimum or maximum eigenvalue commutativity conditions on the FMKWD. Finally we separately formulate an uncertainty inequality in FMKWD domains for real-valued functions, three kinds of uncertainty inequalities in orthogonal FMKWD domains, an uncertainty inequality in orthonormal FMKWD domains, and four kinds of uncertainty inequalities in the minimum or maximum eigenvalue commutative FMKWD domains for complex-valued functions. The time-frequency resolution of the FMKWD is compared with those of the free metaplectic Wigner distribution, K-Wigner distribution, and N-dimensional Wigner distribution to demonstrate its superiority in super-resolution analysis. For applications, the uncertainty inequalities derived are used to estimate the bandwidth in FMKWD domains, and the FMKWD is applied to detect noisy linear frequency-modulated signals.