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1-Bit Hilbert Transform for Signed Signals with Sparse Prior

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Abstract

Owing to the advantages of low cost and easy implementation, the 1-bit sampling technique attracts more and more attention recently. In many signal processing areas, such as in the direction finding problem, the complete analytical 1-bit signal is required to provide the instantaneous phase information, rather than the real-valued 1-bit signal. However, how to generate the imaginary component of the 1-bit signal continues to be an open problem in the literature. To this end, in this paper the concept of 1-bit Hilbert transform is introduced. Moreover, we propose an algorithm to construct the analytical signal for the real-valued 1-bit signal based on the 1-bit compressive sensing method. Simulations show that the proposed method can obtain the analytical 1-bit signal with a reasonable accuracy.

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Data availability

The datasets generated and analyzed during the current study are available from the corresponding author on reasonable request.

Notes

  1. It is worth noting that the signum function \(\mathcal {S}(\cdot )\) defined here is different from the sign function \(\text {sgn}(\cdot )\) defined for 1-bit signals. In (3), \(\mathcal {S}(0)=0\). However, since there are only two status in 1-bit signals (+1 or -1), it is defined that \(\text {sgn}(0)=-1\) in this paper.

  2. The Hadamard product of two matrices \(\mathbf{A}\) and \(\mathbf{B}\) of same dimension is defined as \(\left[ \mathbf{A}\odot \mathbf{B}\right] _{ij}=\left[ \mathbf{A}\right] _{ij}\left[ \mathbf{B}\right] _{ij}\), i.e., the elements of the resulting matrix are obtained by the scalar product of the corresponding elements of \(\mathbf{A}\) and \(\mathbf{B}\) matrices.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant 61901273, the Stable Supporting Fund of Acoustics Science and Technology Laboratory, the State Key Laboratory of Acoustics, Chinese Academy of Sciences, and Guangdong marine economic development project (GDNRC2021-31).

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Authors and Affiliations

  1. School of Ocean Engineering and Technology, Sun Yat-sen University, Zhuhai, 519000, Guangdong, China

    Peng Xiao, Yidi Li, Yinglin Chen & Lingji Xu

  2. Southern Marine Science and Engineering Guangdong Laboratory (Zhuhai), Zhuhai, 519000, Guangdong, China

    Peng Xiao, Yidi Li, Yinglin Chen & Lingji Xu

  3. State Key Laboratory of Acoustics, Institute of Acoustics, Chinese Academy of Sciences, Beijing, 100190, China

    Jixing Qin

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  1. Peng Xiao
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Correspondence to Lingji Xu.

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Xiao, P., Li, Y., Chen, Y. et al. 1-Bit Hilbert Transform for Signed Signals with Sparse Prior. Circuits Syst Signal Process 42, 1848–1859 (2023). https://doi.org/10.1007/s00034-022-02162-9

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  • DOI: https://doi.org/10.1007/s00034-022-02162-9

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