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Structured Measurement Matrices Based on Deterministic Fourier Matrices and Gram Matrices

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Abstract

The measurement matrices play a crucial role in compressed sensing, directly impacting the performance of signal sampling and reconstruction. As one of the primary construction methods for measurement matrices, designing structured measurement matrices is a challenging problem. In practical sampling, the measurement matrices often have strong coherence. Therefore, it is significant to design structured measurement matrices with superior reconstruction performance at low sampling, although the coherence is strong. In this paper, by introducing a special Gram matrix and merging it with the deterministic Fourier matrix, we construct a kind of measurement matrices with superior signal recovery performance under strong coherence. Furthermore, utilizing Katz’ character sum estimation allows us to establish an upper bound on the coherence of the constructed matrices. All experimental results demonstrate that the performance of the proposed matrices outperform that of Fourier matrices and Gaussian random matrices. Consequently, the proposed matrices hold significant application in sparse signal processing.

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Acknowledgements

This work is supported by the National Natural Science Foundation of China under Grants 12261002 and 11761003; the Natural Science Foundation of Ningxia Province under Grant 2020AAC03231; and, in part, by the 2023 Graduate Innovation Project of North Minzu University under Grants YCX23071

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  1. School of Mathematics and Information Science, North Minzu University, Yinchuan, 750021, Ningxia, China

    Guojun Zhang & Yi Gao

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  1. Guojun Zhang
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  2. Yi Gao
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Correspondence to Yi Gao.

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Zhang, G., Gao, Y. Structured Measurement Matrices Based on Deterministic Fourier Matrices and Gram Matrices. Circuits Syst Signal Process (2024). https://doi.org/10.1007/s00034-024-02692-4

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