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Star DGT: a robust Gabor transform for speech denoising

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Abstract

In this paper, we address the speech denoising problem, where Gaussian and coloured additive noises are to be removed from a given speech signal. Our approach is based on a redundant, analysis-sparse representation of the original speech signal. We pick an eigenvector of the Zauner unitary matrix and—under certain assumptions on the ambient dimension—we use it as window vector to generate a spark deficient Gabor frame. The analysis operator associated with such a frame, is a (highly) redundant Gabor transform, which we use as a sparsifying transform in the denoising procedure. We conduct computational experiments on real-world speech data, using as baseline three Gabor transforms generated by state-of-the-art window vectors in time-frequency analysis and compare their performance to the proposed Gabor transform. The results show that the proposed redundant Gabor transform outperforms previous ones consistently for all types of examined signals of noise.

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Data Availability

The Librispeech dataset we utilized is publicly available at http://www.openslr.org/12.

Notes

  1. Equivalently, we call such frames full spark frames.

  2. That is, the sparsity and cosparsity of a signal with respect to a sparsifying operator.

  3. Such a frame can also be generated by the eigenvectors of certain unitaries belonging to the Clifford group, under certain assumptions. However, since the algebraic nature of these assumptions goes beyond the scope of the present paper, we preferred to employ only the Zauner unitary matrix.

  4. The autocorrelation of a signal is defined as the inner product between the signal and its time-translated version.

  5. Since we target at speech denoising, a tractable alternative could be some perceptual variant of the matching pursuit (MP) method, e.g. [45]; however—to our knowledge—perceptual variants of MP account for synthesis sparsity, while we are interested in analysis-sparsity-based denoising.

  6. In terms of optimization, it is preferred to solve (4) instead of (3).

  7. We will interchangeably use both terms in the sequel.

  8. \(\beta \beta ^{-1}\equiv 1\textrm{mod}\,L\).

  9. In the rest of the paper, when we speak of coloured noises, we mean the examined cases of pink and blue noise.

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Acknowledgements

V. Kouni was financially supported for this work by the German Academic Exchange Service (DAAD) through the program Research Grants - One-Year Grants for Doctoral Candidates, 2020–2021. V. Kouni would also like to thank G. Paraskevopoulos for his valuable advice and insightful discussions around the framework presented in this paper.

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Authors and Affiliations

  1. Department of Informatics and Telecommunications, National and Kapodistrian University of Athens, Athens, Greece

    Vicky Kouni & Theoharis Theoharis

  2. Chair for Mathematics of Information Processing, RWTH Aachen University, Aachen, Germany

    Holger Rauhut

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  1. Vicky Kouni
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Correspondence to Vicky Kouni.

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Communicated by Ron Levie.

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Kouni, V., Rauhut, H. & Theoharis, T. Star DGT: a robust Gabor transform for speech denoising. Sampl. Theory Signal Process. Data Anal. 21, 14 (2023). https://doi.org/10.1007/s43670-023-00053-x

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