A Unified Approach to Time-Frequency Representations and Generalized Spectrogram
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2024
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| author | Cordero, Elena Giacchi, Gianluca Rodino, Luigi |
| author_facet | Cordero, Elena Giacchi, Gianluca Rodino, Luigi |
| contents | To overcome the impossibility of representing the energy of a signal simultaneously in time and frequency, many time-frequency representations have been introduced in the literature. Some of these are recalled in the Introduction. In this work we propose a unified approach of the previous theory by means of metaplectic Wigner distributions $W_{\mathcal{A}}$, with $\mathcal{A}$ symplectic matrix in $Sp(2d,\mathbb{R})$, which were introduced by Cordero, Rodino (2022) and then widely studied in subsequent papers. Namely, the short-time Fourier transform and the most popular members of the Cohen's class can be represented via metaplectic Wigner distributions. In particular, we introduce $\mathcal{A}$-metaplectic spectrograms which contain the classical ones and their variations arising from the $τ$-Wigner distributions of Boggiatto, De Donno, and Oliaro (2010). We provide a complete characterization of those $\mathcal{A}$-Wigner distributions which give rise to generalized spectrograms. This characterization is related to the block decomposition of the symplectic matrix $\mathcal{A}$. Moreover, a characterization of the $L^p$-boundedness of both $\mathcal{A}$-Wigner distributions and related metaplectic pseudodifferential operators is provided. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_03882 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Unified Approach to Time-Frequency Representations and Generalized Spectrogram Cordero, Elena Giacchi, Gianluca Rodino, Luigi Analysis of PDEs 42A38, 46F10, 46F12, 81S30 To overcome the impossibility of representing the energy of a signal simultaneously in time and frequency, many time-frequency representations have been introduced in the literature. Some of these are recalled in the Introduction. In this work we propose a unified approach of the previous theory by means of metaplectic Wigner distributions $W_{\mathcal{A}}$, with $\mathcal{A}$ symplectic matrix in $Sp(2d,\mathbb{R})$, which were introduced by Cordero, Rodino (2022) and then widely studied in subsequent papers. Namely, the short-time Fourier transform and the most popular members of the Cohen's class can be represented via metaplectic Wigner distributions. In particular, we introduce $\mathcal{A}$-metaplectic spectrograms which contain the classical ones and their variations arising from the $τ$-Wigner distributions of Boggiatto, De Donno, and Oliaro (2010). We provide a complete characterization of those $\mathcal{A}$-Wigner distributions which give rise to generalized spectrograms. This characterization is related to the block decomposition of the symplectic matrix $\mathcal{A}$. Moreover, a characterization of the $L^p$-boundedness of both $\mathcal{A}$-Wigner distributions and related metaplectic pseudodifferential operators is provided. |
| title | A Unified Approach to Time-Frequency Representations and Generalized Spectrogram |
| topic | Analysis of PDEs 42A38, 46F10, 46F12, 81S30 |
| url | https://arxiv.org/abs/2401.03882 |