Filtering Out Time-Frequency Areas Using Gabor Multipliers

Kreme, Ama Marina; Emiya, Valentin; Chaux, Caroline; Torrésani, Bruno

doi:10.1109/icassp40776.2020.9053482

Cited by 4 publications

(2 citation statements)

References 14 publications

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“…) (or vice versa) then the window correlation function satisfies condition (45). In fact, by Proposition 2.6 it follows that…”

Section: Smoothing Effects Of Stft Multipliersmentioning

confidence: 86%

“…In many chosen settings the results also closely match the expectations (see Example 6.4). The problem of representation and approximation of linear operators by means of Gabor multipliers (and suitable modifications) was studied by Dörfler and Torrésani in [20], further investigations are contained in [33,45]. More generally, approximating problems for pseudodifferential operators via STFT multipliers ("wave packets" were exhibited in the work by Cordoba and Fefferman [18], see also Folland [32] and the PhD thesis [21]).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Comparisons between Fourier and STFT multipliers: the smoothing effect of the Short-time Fourier Transform

Balázs¹,

Bastianoni²,

Cordero³

et al. 2022

Preprint

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We study the connection between STFT multipliers A g1,g2 1⊗m having windows g 1 , g 2 , symbols a(x, ω) = (1 ⊗ m)(x, ω) = m(ω), (x, ω) ∈ R 2d , and the Fourier multipliers T m2 with symbol m 2 on R d . We find sufficient and necessary conditions on symbols m, m 2 and windows g 1 , g 2 for the equality1⊗m . For m = m 2 the former equality holds only for particular choices of window functions in modulation spaces, whereas it never occurs in the realm of Lebesgue spaces. In general, the STFT multiplier A g1,g2 1⊗m , also called localization operator, presents a smoothing effect due to the so-called two-window short-time Fourier transform which enters in the definition of A g1,g21⊗m . As a by-product we prove necessary conditions for the continuity of anti-Wick operators A g,g 1⊗m : L p → L q having multiplier m in weak L r spaces. Finally, we exhibit the related results for their discrete counterpart: in this setting STFT multipliers are called Gabor multipliers whereas Fourier multiplier are better known as linear time invariant (LTI) filters.

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“…) (or vice versa) then the window correlation function satisfies condition (45). In fact, by Proposition 2.6 it follows that…”

Section: Smoothing Effects Of Stft Multipliersmentioning

confidence: 86%

Section: Introductionmentioning

confidence: 99%

Comparisons between Fourier and STFT multipliers: the smoothing effect of the Short-time Fourier Transform

Balázs¹,

Bastianoni²,

Cordero³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

Comparisons between Fourier and STFT multipliers: The smoothing effect of the short-time Fourier transform

Balázs

Bastianoni

Cordero

et al. 2024

Journal of Mathematical Analysis and Applications

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Time-Frequency Fading Algorithms Based on Gabor Multipliers

Kreme

Emiya

Chaux

et al. 2021

IEEE J. Sel. Top. Signal Process.

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In this paper, we address a particular instance of time-frequency filter design, which we call Time-Frequency Fading (TFF). In TFF the only available information concerns the time-frequency localization of the component to be filtered out or attenuated: the signal of interest is supposed to be spread out in the time-frequency plane, whereas the perturbation signal is concentrated within a specified time-frequency region Ω.The problem is formulated as an optimization problem designed to fade out the perturbation with accurate control on the fading level. The corresponding objective function involves a data fidelity term that aims to match the TF coefficients of the estimated signal to those of the observed signal outside the perturbation support. It also involves a penalty term that controls the energy of the reconstructed signal, within that region.We obtain the closed-form solution of the problem which involves Gabor multipliers, i.e. linear operators of the pointwise product by a time-frequency transfer function called a mask. We study the TF localization properties of dominant eigenvectors of these Gabor multipliers, with particular attention to the case where the region Ω is a disjoint union of several sub-regions. The decay properties of eigenvalues naturally lead to reducedrank approximations, and further approximations are obtained in the multiply connected region case. Also, we exploit random projection methods to speed up eigenvalue decompositions and rank reduction. This is implemented in two TFF algorithms, that cover the cases of single or multiple regions.The efficiency of the proposed approach is demonstrated on several audio signals where the perturbations are filtered while leading to a good quality of signal reconstruction.

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Filtering Out Time-Frequency Areas Using Gabor Multipliers

Cited by 4 publications

References 14 publications

Comparisons between Fourier and STFT multipliers: the smoothing effect of the Short-time Fourier Transform

Comparisons between Fourier and STFT multipliers: the smoothing effect of the Short-time Fourier Transform

Comparisons between Fourier and STFT multipliers: The smoothing effect of the short-time Fourier transform

Time-Frequency Fading Algorithms Based on Gabor Multipliers

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