Symplectic analysis of time-frequency spaces

IF 2.1 1区 数学 Q1 MATHEMATICS
Elena Cordero , Gianluca Giacchi
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引用次数: 0

Abstract

We present a different symplectic point of view in the definition of weighted modulation spaces Mmp,q(Rd) and weighted Wiener amalgam spaces W(FLm1p,Lm2q)(Rd). All the classical time-frequency representations, such as the short-time Fourier transform (STFT), the τ-Wigner distributions and the ambiguity function, can be written as metaplectic Wigner distributions μ(A)(fg¯), where μ(A) is the metaplectic operator and A is the associated symplectic matrix. Namely, time-frequency representations can be represented as images of metaplectic operators, which become the real protagonists of time-frequency analysis. In [13], the authors suggest that any metaplectic Wigner distribution that satisfies the so-called shift-invertibility condition can replace the STFT in the definition of modulation spaces. In this work, we prove that shift-invertibility alone is not sufficient, but it has to be complemented by an upper-triangularity condition for this characterization to hold, whereas a lower-triangularity property comes into play for Wiener amalgam spaces. The shift-invertibility property is necessary: Rihaczek and conjugate Rihaczek distributions are not shift-invertible and they fail the characterization of the above spaces. We also exhibit examples of shift-invertible distributions without upper-triangularity condition which do not define modulation spaces. Finally, we provide new families of time-frequency representations that characterize modulation spaces, with the purpose of replacing the time-frequency shifts with other atoms that allow to decompose signals differently, with possible new outcomes in applications.

时频空间的辛分析
在定义加权调制空间Mmp,q(Rd)和加权Wiener混合空间W(FLm1p,Lm2q)(Rd。所有经典的时频表示,如短时傅立叶变换(STFT)、τ-Wigner分布和模糊函数,都可以写成元辛Wigner分布μ(A)(f⊗g’),其中μ(A)是元算子,A是相关的辛矩阵。也就是说,时间-频率表示可以表示为元算子的图像,它们成为时间-频率分析的真正主角。在[13]中,作者提出,在调制空间的定义中,任何满足所谓的移位可逆条件的元辛Wigner分布都可以取代STFT。在这项工作中,我们证明了仅移位可逆性是不够的,但它必须由上三角性条件来补充,才能保持这种刻画,而下三角性性质在维纳汞齐空间中发挥作用。移位可逆性是必要的:Rihaczek和共轭Rihaczek分布不是移位可逆的,并且它们不能表征上述空间。我们还展示了没有上三角性条件的移位可逆分布的例子,这些条件不定义调制空间。最后,我们提供了表征调制空间的新的时频表示族,目的是用允许以不同方式分解信号的其他原子取代时频偏移,从而在应用中获得可能的新结果。
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来源期刊
CiteScore
4.30
自引率
0.00%
发文量
84
审稿时长
6 months
期刊介绍: Published from 1836 by the leading French mathematicians, the Journal des Mathématiques Pures et Appliquées is the second oldest international mathematical journal in the world. It was founded by Joseph Liouville and published continuously by leading French Mathematicians - among the latest: Jean Leray, Jacques-Louis Lions, Paul Malliavin and presently Pierre-Louis Lions.
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