Skip to main content
SpringerLink
Log in

Performance Analysis of Toeplitz Block Sampling Matrix in Compressed Sensing

  • Published:
Wireless Personal Communications Aims and scope Submit manuscript

Abstract

This paper analyzes the recovery performance when a Toeplitz block matrix is used for sampling in compressed sensing. Different from current work that mainly discusses the restricted isometry property and the applications of such matrix, we provide an upper bound for the mean squared reconstruction error. By comparing with the random matrix in which its entries are drawn independently from certain probability distributions, the results show that the Toeplitz block matrix is efficient for compressed sampling. Simulation results validate that the sampling performance of Toeplitz block matrix can approach that of random matrix by choosing its parameters properly.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Candes, E. J. (2008). The restricted isometry property and its implications for compressed sensing. Computes Rendus de l’Acadmie des sciences, 346(9), 589–592.

    MATH  MathSciNet  Google Scholar 

  2. Li, G., Zhu, Z., Yang, D., & Chang, L. (2013). On projection matrix optimization for compressive sensing systems. IEEE Transactions on Signal Processing, 61(11), 2887–2898.

    Article  MathSciNet  Google Scholar 

  3. Candes, E. J., Romberg, J., & Tao, T. (2006). Stable signal recovery from incomplete and inaccurate measurements. Communications on Pure and Applied Mathematics, 59(8), 1207–1223.

    Article  MATH  MathSciNet  Google Scholar 

  4. Tibshirani, R. (1996). Regression shrinkage and selection via the LASSO. Journal of Royal Statistical Society, 58(1), 267–288.

    MATH  MathSciNet  Google Scholar 

  5. Li, K., & Cong, S. (2015). State of the art and prospects of structured sensing matrices in compressed sensing. Frontiers of Computer Science, 9(5), 665–677.

    Article  Google Scholar 

  6. Baraniuk, R., Davenport, M., DeVore, R., et al. (2008). A simple proof of the restricted isometry property for random matrices. Constructive Approximation, 28(3), 256–263.

    Article  MATH  MathSciNet  Google Scholar 

  7. Xu, S., Yin, H., Chai, Y., et al. (2014). An improved Toeplitz measurement matrix for compressive sensing. International Journal of Distributed Sensor Networks,. doi:10.1155/2014/846757.

    Google Scholar 

  8. Bajwa, W. U., Haupt, J. D., Raz, G. M., et al. (2007). Toeplitz-structured compressed sensing matrices. In IEEE Workshop on Statistical Signal Processing, (pp. 294–298).

  9. Bajwa, W. U., Haupt, J. D., Raz G. M., & Nowak, R. (2008). Compressed channel sensing. In 42nd Annual Conference on Information Sciences and Systems (pp. 5–10).

  10. Rauhut, H. (2009) Circulant and Toeplitz matrices in compressed sensing. In Signal Processing with Adaptive Sparse Structured Representations. arXiv:0902.4394.

  11. Yin, W., Morgan, S., Yang, J., & Zhang, Y. (2010). Practical compressive sensing with toeplitz and circulant matrices. In Visual Communications and Image Processing.

  12. Sebert, F., Ying, L., & Zou, Y. M. Toeplitz block matrices in compressed sensing. arXiv:0803.0755.pdf.

  13. Sebert, F., Zou, Y. M., & Ying, L. (2008). Toeplitz block matrices in compressed sensing and their applications in imaging. In International Conference on Information Technology and Applications in Biomedicine (pp. 47–50).

  14. Yap, H. L., Eftekhari, A., Wakin, M. B., & Rozell, C. J. (2011). The restricted isometry property for block diagonal matrices. In Conference on Information Sciences and Systems (pp. 1–6).

  15. Eftekhari, A., Yap, H. L., Rozell, C. J., & Wakin, M. B. (2015). The restricted isometry property for random block diagonal matrices. Applied and Computational Harmonic Analysis, 38(1), 1–31.

    Article  MATH  MathSciNet  Google Scholar 

  16. Berinde, R., & Indyk, P. (2008). Sparse recovery using sparse random matrices. Lecture Notes in Computer Science, 6034, 157.

  17. Xia, S., Liu, X., Jiang, Y., & Zheng, H. (2015). Deterministic constructions of binary measurement matrices from finite geometry. IEEE Transactions on Signal Processing, 63(4), 1017–1029.

    Article  MathSciNet  Google Scholar 

  18. Li, Y., Liu, D., & Wang, Z. (2010). Limit distributions of eigenvalues for random block Toeplitz and Hankel matrices. Journal of Theoretical Probability, 24(4), 1063–1086.

    Article  MATH  MathSciNet  Google Scholar 

  19. Dehghan, H., Dansereau, R. M., & Chan, A. D. C. (2015). Restricted isometry property on banded block Toeplitz matrices with application to multi-channel convolutive source separation. IEEE Transactions on Signal Processing, 63(21), 5665–5676.

    Article  MathSciNet  Google Scholar 

  20. Harmany, Z. T., Marcia, R. F., & Willett, R. M. (2012). Spatio-temporal compressed sensing with coded apertures and keyed exposures. arXiv:1111.7247v2.

  21. Romberg, J., & Neelamani, R. (2010). Sparse channel separation using random probes. Inverse Problems, 26(11), 115.

    Article  MATH  MathSciNet  Google Scholar 

  22. Bajwa, W. U. (2012). Geometry of random Toeplitz-block sensing matrices bounds and implications for sparse signal processing. SPIE,. doi:10.1117/12.919475.

    Google Scholar 

  23. Haupt, J., & Nowak, R. (2006). Signal reconstruction from noisy random projections. IEEE Transactions on Information Theory, 52(9), 4036–4048.

    Article  MATH  MathSciNet  Google Scholar 

  24. Taheri, O., & Vorobyov, S. A. (2011). Segmented compressed sampling for analog-to-information conversion: Method and performance analysis. IEEE Transactions on Signal Processing, 59(2), 554–572.

    Article  MathSciNet  Google Scholar 

  25. Xu, W., Tian, Y., & Lin, J. (2013). Performance analysis of partial segmented compressed sampling. Signal Processing, 93(9), 2653–2663.

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (61302084).

Author information

Authors and Affiliations

  1. Key Lab of Universal Wireless Communications, Ministry of Education, Beijing University of Posts and Telecommunications, Beijing, China

    Wenbo Xu, Yifan Wang, Yupeng Cui & Jiaru Lin

  2. Beijing Institute of Micro-Tech, No. 16, Futong East Street, Beijing, China

    Yibing Gai

Authors
  1. Wenbo Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yifan Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yupeng Cui
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yibing Gai
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Jiaru Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wenbo Xu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xu, W., Wang, Y., Cui, Y. et al. Performance Analysis of Toeplitz Block Sampling Matrix in Compressed Sensing. Wireless Pers Commun 97, 1141–1154 (2017). https://doi.org/10.1007/s11277-017-4558-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11277-017-4558-8

Keywords

Search

Navigation