论文标题
时频空间的符号分析
Symplectic Analysis of Time-Frequency Spaces
论文作者
论文摘要
在加权调制空间的定义中，我们介绍了不同的符号观点，$ m^{p，q} _m（\ Mathbb {r}^d）$和加权的Wiener amalgam spaces $ w（\ Mathcal {f}所有经典的时频表示，例如短时傅立叶变换（STFT），$τ$ - 符号分布和歧义功能，都可以写成元素wigner wigner分布$μ（\ nathcal {a}）（f \ otimes \ otimes \ bar {g}）$（\ bar {g}）$，$ usp $μ（\ math）and a calcal和a calcal calcal and a a（a） $ \ Mathcal {a} $是关联的符号矩阵。也就是说，可以将时间频表示为元容器的图像，这成为时频分析的真正主角。在[E. Cordero和L. Rodino（2022）“通过互合式表示和对Schrödinger方程的应用对调制空间的表征”，Arxiv：2204.14124]，作者认为，任何满足所谓的“可变性”条件可以替换调制空间中的STFT的任何转移WIGNER分布都可以替换STFT。在这项工作中，我们证明单独的可转换性是不够的，但是必须通过三角形条件来补充这种表征，而较低三角形的特性则可以为维也纳amalgam空间发挥作用。偏移性属性是必要的：ryhaczek和和共轭ryhaczek分布不可移动，并且未能使上述空间的表征失败。我们还展示了没有上跨性别条件的可移动分布的示例，这些分布没有定义调制空间。最后，我们提供了特征在于调制空间的时频表示的新家族，目的是用其他原子代替允许以不同方式分解信号的时间频率转移，并且在应用程序中可能会带来新的结果。
We present a different symplectic point of view in the definition of weighted modulation spaces $M^{p,q}_m(\mathbb{R}^d)$ and weighted Wiener amalgam spaces $W(\mathcal{F} L^p_{m_1},L^q_{m_2})(\mathbb{R}^d)$. All of 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 $μ(\mathcal{A})(f\otimes \bar{g})$, where $μ(\mathcal{A})$ is the metaplectic operator and $\mathcal{A}$ 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 [E. Cordero and L. Rodino (2022) "Characterization of Modulation Spaces by symplectic representations and applications to Schrödinger equations", arXiv:2204.14124], 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 in to play for Wiener amalgam spaces. The shift-invertibility property is necessary: Ryhaczek and and conjugate Ryhaczek distributions are not shift-invertible and they fail the characterization of the above spaces. We also exhibit examples of shift-invertible distributions without upper-tryangularity 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.
