Standard Heisenberg's uncertainty principles of Cohen's class time-frequency distribution with specific kernels
Zhichao Zhang
TL;DR
The paper tackles the problem of establishing a standard Heisenberg-type uncertainty principle for Cohen's class time-frequency distributions under a restricted kernel class called UMITRSK-CCTFD, where $|\phi(\mathbf{v},\mathbf{y})|=1$ and $\phi(\mathbf{v},2(\mathbf{z}-\mathbf{t}))=\phi(\mathbf{t})$. By leveraging Parseval's relation and a concise frequency-domain formulation, it extends known Wigner-distribution results to this CCTFD setting and derives attainable lower bounds on the time-frequency spread product in the UMITRSK-CCTFD domain. The core contributions are four variant Heisenberg-type inequalities corresponding to real/complex $f$ and real/complex kernels, with equalities achieved by optimal Gaussian enveloped chirp functions under suitable conditions; these results strengthen the weak Heisenberg principle and clarify kernel-induced limitations on concentration and resolution. Overall, the work provides a rigorous, kernel-aware framework for time-frequency concentration in Cohen's class distributions, with implications for kernel design and signal analysis in the time-frequency domain.
Abstract
Time-frequency concentration and resolution of the Cohen's class time-frequency distribution (CCTFD) has attracted much attention in time-frequency analysis. A variety of uncertainty principles of the CCTFD is therefore derived, including the weak Heisenberg type, the Hardy type, the Nazarov type, and the local type. However, the standard Heisenberg type still remains unresolved. In this study, we address the question of how the standard Heisenberg's uncertainty principle of the CCTFD is affected by fundamental properties. The investigated distribution properties are Parseval's relation and the concise frequency domain definition (i.e., only frequency variables are explicitly found in the tensor product), based on which we confine our attention to the CCTFD with some specific kernels. That is the unit modulus and v-independent time translation, reversal and scaling invariant kernel CCTFD (UMITRSK-CCTFD). We then extend the standard Heisenberg's uncertainty principles of the Wigner distribution to those of the UMITRSK-CCTFD, giving birth to various types of attainable lower bounds on the uncertainty product in the UMITRSK-CCTFD domain. The derived results strengthen the existing weak Heisenberg type and fill gaps in the standard Heisenberg type.