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<p>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 <Emphasis Type="Underline">u</Emphasis>nit modulus and <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11868_2025_725_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\(\textbf{v}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">v</mi> </math></EquationSource> </InlineEquation>-independent <Emphasis Type="Underline">t</Emphasis>ime <Emphasis Type="Underline">a</Emphasis>ffine <Emphasis Type="Underline">i</Emphasis>nvariant kernel <Emphasis Type="Underline">CC</Emphasis>TFD (UTAICC). We then extend the standard Heisenberg’s uncertainty principles of the Wigner distribution to those of the UTAICC, giving birth to various types of attainable lower bounds on the uncertainty product in the UTAICC domain. The derived results strengthen the existing weak Heisenberg type and fill gaps in the standard Heisenberg type.</p>

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Standard Heisenberg’s uncertainty principles of Cohen’s class time-frequency distribution with specific kernels

  • 期刊论文

  • 发表日期：

    2025年8月4日

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  • Zhichao Zhang

摘要

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 \(\textbf{v}\) v -independent time affine invariant kernel CCTFD (UTAICC). We then extend the standard Heisenberg’s uncertainty principles of the Wigner distribution to those of the UTAICC, giving birth to various types of attainable lower bounds on the uncertainty product in the UTAICC domain. The derived results strengthen the existing weak Heisenberg type and fill gaps in the standard Heisenberg type.