[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Log in

Compressed Blind Signal Reconstruction Model and Algorithm

  • Published:
Circuits, Systems, and Signal Processing Aims and scope Submit manuscript

Abstract

The model of inherent connection between underdetermined blind signal separation and compressed sensing (CS) is analyzed first; then, the mathematical model of underdetermined blind signal reconstruction is built using CS. More specifically, the mixing matrix is estimated by exploiting the wavelet packet transform and k-means clustering methods up to permutation and scaling indeterminacy, and then, the measurement matrix and the measurement equation are obtained. To reconstruct the underdetermined sparse source signals, the proposed semi-blind compressed reconstruction algorithm is derived based on the blind signal reconstruction model and compressive sampling matching pursuit (CoSaMP) method. Our simulation results demonstrate that the proposed scheme is effective, irrespective of artificial data or real data. Moreover, the proposed scheme can be adjusted for different applications by modifying the mixing matrix estimation method and CoSaMP method with respect to the correspondence conditions.

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

Access this article

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Notes

  1. According to the number of source signals and mixed signals, there are three basic types of BSS models: overdetermined, determined and underdetermined. When the number of the source signals is less than or equal to that of the mixed signals, the BSS model is called overdetermined or determined; otherwise, it is called underdetermined.

  2. Here, \(\tilde{\beta }\) is a book keeping shift index selected from the original shift index \(\beta \). \(\tilde{\beta }\) with nonzero elements represent the corresponding rows of the decomposition coefficients of mixed signals, which are used for source signals reconstruction.

  3. To use complex-valued data with k-means clustering, one method is to split the real and complex parts to obtain real data in twice as many dimensions. Alternatively, one could modify the calculation of the squared Euclidean distance from something like sum(\((\cdot )\cdot {}^{\wedge }2\)) to something like \(\hbox {sum}(\hbox {abs}(\cdot )\cdot {}^{\wedge }2\)).

  4. In the conventional BSS algorithms, a meaningful result is obtained when \(\hbox {S/R}>10\,\hbox {dB}\).

  5. The complex-valued noiseless mixed signals are the same as the first experiment in this subsection.

References

  1. F. Abrard, Y. Deville, A time-frequency blind signal separation method applicable to underdetermined mixtures of dependent sources. Signal Process. 85(7), 1389–1403 (2005)

    Article  MATH  Google Scholar 

  2. T. Adali, P. Schreier, Optimization and estimation of complex-valued signals: theory and applications in filtering and blind source separation. IEEE Signal Proc. Mag. 31(5), 112–128 (2014)

    Article  Google Scholar 

  3. A. Aissa-El-Bey, N. Linh-Trung, K. Abed-Meraim et al., Underdetermined blind separation of nondisjoint sources in the time-frequency domain. IEEE Trans. Signal Process. 55(3), 897–907 (2007)

    Article  MathSciNet  Google Scholar 

  4. R. Baraniuk, Compressive sensing. IEEE Signal Proc. Mag. 24(4), 118–120 (2007)

    Article  MathSciNet  Google Scholar 

  5. R. Baraniuk, M. Davenport, R. De Vore, M. Wakin, A simple proof of the restricted isometry property for random matrices. Constr. Approx. 28(3), 253–263 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. T. Blumensath, M. Davies, Compressed Sensing and Source Separation. Lecture Notes Computer Science, vol 4666, pp. 341–348 (2007)

  7. P. Bofill, M. Zibulevsky, Underdetermined blind source separation using sparse representations. Signal Process. 81(11), 2353–2362 (2001)

    Article  MATH  Google Scholar 

  8. E. Candes, J. Romberg, T. Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory 52(2), 489–509 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. E. Candes, T. Tao, Decoding by linear programming. IEEE Trans. Inf. Theory 51(12), 4203–4215 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Cichocki, S. Amari, Adaptive Blind Signal and Image Processing: Learning Algorithms and Applications (Wiley, New York, 2002)

    Book  Google Scholar 

  11. A. Cichocki, J. Karhunen, W. Kasprzak, R. Vigario, Neural networks for blind separation with unknown number of sources. Neurocomputing 24(1), 55–93 (1999)

    Article  MATH  Google Scholar 

  12. P. Comon, C. Jutten, Handbook of Blind Source Separation: Independent Component Analysis and Applications (Elsevier, Amsterdam, 2010)

    Google Scholar 

  13. W. Dai, O. Ilenkovic, Subspace pursuit for compressive sensing: closing the gap between performance and complexity. IEEE Trans. Inf. Theory 55(5), 2230–2249 (2009)

    Article  Google Scholar 

  14. D.L. Donoho, Denoising by soft-thresholding. IEEE Trans. Infm. Theory 41(3), 613–627 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  15. D.L. Donoho, Compressed sensing. IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. D.L. Donoho, Y. Tsaig, I. Drori, J.L. Starck, Sparse solution of underdetermined systems of linear equations by stagewise orthogonal matching pursuit. IEEE Trans. Inf. Theory 58(2), 1094–1121 (2012)

    Article  MathSciNet  Google Scholar 

  17. M.F. Duarte, Y.C. Eldar, Structured compressed sensing: from theory to applications. IEEE Trans. Signal Process. 59(9), 4053–4085 (2011)

    Article  MathSciNet  Google Scholar 

  18. Y.C. Eldar, G. Kutyniok, Compressed Sensing: Theory and Applications (Cambridge University Press, New York, 2012)

    Book  Google Scholar 

  19. D.-Z. Feng, H. Zhang, W.X. Zheng, Bi-iterative algorithm for extracting independent components from array signals. IEEE Trans. Signal Process. 59(8), 3636–3646 (2011)

    Article  MathSciNet  Google Scholar 

  20. P.G. Georgiev, F. Theis, A. Cichocki, Sparse component analysis and blind source separation of underdetermined mixtures. IEEE Trans. Neural Netw. 16(4), 992–996 (2005)

    Article  Google Scholar 

  21. F.L. Gu, H. Zhang, W.W. Wang, D.S. Zhu, PARAFAC-based blind identification of underdetermined mixtures using Gaussian mixture model. Circuits Syst. Signal Process 33(6), 1841–1857 (2014)

    Article  MathSciNet  Google Scholar 

  22. F.L. Gu, H. Zhang, D.S. Zhu, Blind separation of complex sources using generalized generating function. IEEE Signal Process. Lett. 20(1), 71–74 (2013)

    Article  Google Scholar 

  23. Z. He, S. Xie, Y. Fu, Sparse representation and blind source separation of ill-posed mixtures. Sci. China Ser. E. 36(8), 864–879 (2006)

    MathSciNet  Google Scholar 

  24. Z. He, S. Xie, L. Zhang et al., A note on Lewicki–Sejnowski gradient for learning overcomplete representation. Neural Comput. 20(3), 636–643 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. J. Jin, Y. Gu, S. Mei, An introduction to compressed sampling and its applications. J. Electron. Inf. Technol. 32(2), 470–475 (2010)

    Article  Google Scholar 

  26. Y. Li, A. Cichocki, S. Amari, Analysis of sparse representation and blind source separation. Neural Comput. 16(6), 1193–1234 (2004)

    Article  MATH  Google Scholar 

  27. R. Li, F. Wang, Blind dependent sources separation method using wavelet. Int. J. Comput. Appl. Technol. 41(3/4), 296–302 (2011)

    Article  Google Scholar 

  28. S. Li, D. Wei, A survey on compressive sensing. Acta Autom. Sin. 35(11), 1–7 (2009)

    Google Scholar 

  29. Q. Lv, X.D. Zhang, A unified method for blind separation of sparse sources with unknown source number. IEEE Signal Process. Lett. 13(1), 49–51 (2006)

    Article  Google Scholar 

  30. A. Maliki, D. Donoho, Optimally tuned iterative reconstruction algorithms for compressed sensing. IEEE J. Sel. Top. Signal Process. 4(2), 330–341 (2010)

    Article  Google Scholar 

  31. S. Mallat, A Wavelet Tour of Signal Processing (Academic Press, New York, 1999)

    MATH  Google Scholar 

  32. D. Needell, J.A. Tropp, CoSaMP: iterative signal recovery from incomplete and inaccurate samples. Appl. Comput. Harmon. A. 26(3), 301–321 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  33. D. Needell, R. Vershynin, Signal recovery from inaccurate and incomplete measurements via regularized orthogonal matching pursuit. IEEE J. Sel. Top. Signal Process. 4(2), 310–316 (2010)

    Article  Google Scholar 

  34. P.D. O’Grady, B.A. Pearlmutter, S.T. Rickard, Survey of sparse and non-sparse methods in source separation. Int. J. Imaging Syst. Technol. 15(1), 18–33 (2005)

    Article  Google Scholar 

  35. M.D. Plumbley, T. Blumensath, L. Daudet, R. Gribonval, M.E. Davies, Sparse representations in audio music: from coding to source separation. Proc. IEEE 98(6), 995–1005 (2010)

    Article  Google Scholar 

  36. V.G. Reju, S.N. Koh, I.Y. Soon, An algorithm for mixing matrix estimation in instantaneous blind source separation. Signal Process. 89(9), 1762–1773 (2009)

    Article  MATH  Google Scholar 

  37. J.L. Starck, F. Murtagh, J.M. Fadili, Sparse Image and Signal Processing: Wavelets, Curvelets, Morphological Diversity (Cambridge University Press, New York, 2010)

    Book  MATH  Google Scholar 

  38. J. Tropp, A. Gilbert, Signal recovery from random measurements via orthogonal matching pursuit. IEEE Trans. Inf. Theory 53(12), 4655–4666 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  39. J.A. Tropp, S.J. Wright, Computational methods for sparse solution of linear inverse problems. Proc. IEEE 98(6), 948–958 (2010)

    Article  Google Scholar 

  40. R. Walden, Analog-to-digital converter survey and analysis. IEEE J. Sel Areas Commun. 17(4), 539–550 (1999)

    Article  Google Scholar 

  41. F. Wang, H. Li, R. Li, Unified nonparametric and parametric ICA algorithm for hybrid source signals and stability analysis. Int. J. Innov. Comput. Inf. 4(4), 933–942 (2008)

    Google Scholar 

  42. D. Wu, W.-P. Zhu, M.N.S. Swamy, The theory of compressive sensing matching pursuit considering time-domain noise with application to speech enhancement. IEEE/ACM Trans. Audio Speech Lang. Process. 22(3), 682–696 (2014)

    Article  Google Scholar 

  43. R. Xu, D. Wunsch II, Survey of clustering algorithms. IEEE Trans. Neural Netw. 16(3), 645–678 (2005)

    Article  Google Scholar 

  44. Z.Y. Yang, B.H. Tan, G.X. Zhou, J.L. Zhang, Source number estimation and separation algorithms of underdetermined blind separation. Sci. China Ser. F. 51(10), 1623–1632 (2008)

    Article  MATH  Google Scholar 

  45. J.M. Ye, X.L. Zhu, X.D. Zhang, Adaptive blind separation with an unknown number of sources. Neural Comput. 16(8), 1641–1660 (2004)

    Article  MATH  Google Scholar 

  46. O. Yilmaz, S. Rickard, Blind separation of speech mixtures via time-frequency masking. IEEE Trans. Signal Process. 52(7), 1830–1847 (2004)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the editor and anonymous referees for their insightful comments and suggestions, which have greatly improved the paper. This research is financially supported by the National Natural Science Foundation of China (Nos. 61401401, 61571401, 61301150, 61172086, U1204607), the China Postdoctoral Science Foundation (Nos. 2014M561998, 2015T80779) and the open research fund of the National Mobile Communications Research Laboratory, Southeast University (No. 2016D02).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Fasong Wang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, F., Li, R., Wang, Z. et al. Compressed Blind Signal Reconstruction Model and Algorithm. Circuits Syst Signal Process 35, 3192–3219 (2016). https://doi.org/10.1007/s00034-015-0189-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-015-0189-z

Keywords

Navigation