K-Wigner Distribution: Definition, Uncertainty Principles and Time-Frequency Analysis

Abstract

To tackle a challenge in high-dimensional complex features information processing, this study extends the permanent scale Wigner distribution and the single scale <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> -Wigner distribution (kWD, formerly known as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\tau $ </tex-math></inline-formula> -Wigner distribution) to a novel multiscale parameterized Wigner distribution. That is the so-called <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {K}$ </tex-math></inline-formula> -Wigner distribution (KWD) which is able to use different scales to extract different types of features at different dimensions. Heisenberg-type uncertainty inequalities of the KWD are established, giving rise to the tightest universal attainable lower bound for all functions on the uncertainty product in time-KWD and Fouier transform-KWD domains, and two versions of attainable lower bounds for complex-valued functions. The obtained results solve an important concern regarding the limit of the KWD’s time-frequency resolution influenced by the parameter matrix. As an application, the derived uncertainty inequalities are applied to estimate the bandwidth in KWD domains. The time-frequency resolution performance of the multiscale KWD, as compared with that of the single scale kWD, is investigated in details. The optimal parameter matrix of the KWD achieving the best performance is then generated, which solves an important concern regarding the KWD’s parameter matrix selection. Examples are also carried out to demonstrate the usefulness and effectiveness of the proposed technique.