Skip to main content
SpringerLink
Log in

Compressive Sensing Signal Reconstruction Using L0-Norm Normalized Least Mean Fourth Algorithms

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

Abstract

Stochastic gradient-based adaptive algorithm has recently attracted considerable attention as one of the best candidates for solving compressive sensing (CS) problems due to its two obvious advantages: low complexity and robust performance. In this paper, in order to further improve the reconstruction accuracy for CS problems under Gaussian background noise, two novel sparse fourth-order error criterion adaptive algorithms, i.e., the \({\ell }_{{0}}\)-norm normalized least mean fourth (\({\ell }_{{0}}\)-NLMF) algorithm and the \({\ell }_{{0}}\)-norm exponentially forgetting window NLMF (\({\ell }_{{0}}\)-EFWNLMF) algorithm, are proposed. In addition, to extend the obtained results to non-Gaussian noise environment, the variants of the above two algorithms, i.e., the sign \({\ell }_{{0}}\)-NLMF (\({\ell }_{{0}}\)-SNLMF) algorithm and the sign \({\ell }_{{0}}\)-EFWNLMF (\({\ell }_{{0}}\)-EFWSNLMF) algorithm, are presented which can effectively mitigate certain impulsive noises. Numerical simulations are also given to demonstrate the evident performance improvement and extensive stability of the proposed algorithms.

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
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  1. B. Aazhang, H.V. Poor, Performance of DS/SSMA communications in impulsive channels–part I: linear correlation receivers. IEEE Trans. Commun. 35(11), 1179–1187 (1987). doi:10.1109/TCOM.1987.1096707

    Article  Google Scholar 

  2. J. Benesty, S.L. Gay, An improved PNLMS algorithm, in Proc of the IEEE ICASSP (2002), pp. II–1881–II–1884. doi:10.1109/ICASSP.2002.5744994. ISSN: 1520-6149

  3. R.G. Baraniuk, More is less: signal processing and the data deluge. Science 331(6018), 717–719 (2011). doi:10.1126/science.1197448

    Article  Google Scholar 

  4. C.F.N. Cowan, P.M. Grant, Adaptive Filters (Prentice-Hall, Englewood Cliffs, 1985)

    MATH  Google Scholar 

  5. E.J. 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). doi:10.1109/TIT.2005.862083

    Article  MathSciNet  MATH  Google Scholar 

  6. E.J. Candes, T. Tao, Decoding by linear programming. IEEE Trans. Inf. Theory 51(12), 4203–4215 (2005). doi:10.1109/TIT.2005.858979

    Article  MathSciNet  MATH  Google Scholar 

  7. M.A. Chitre, J.R. Potter, S.H. Ong, Optimal and near-optimal signal detection in snapping shrimp dominated ambient noise. IEEE J. Ocean. Eng. 31(2), 497–503 (2006). doi:10.1109/JOE.2006.875272

    Article  Google Scholar 

  8. S.S. Chen, D.L. Donoho, M.A. Saunders, Atomic decomposition by basis pursuit. SIAM J. Sci. Comput. 20(1), 33–61 (1998). doi:10.1137/S1064827596304010

    Article  MathSciNet  MATH  Google Scholar 

  9. S.S. Chen, D.L. Donoho, M.A. Saunders, Atomic decomposition by basis pursuit. SIAM Rev. 43(1), 129–159 (2001). doi:10.1137/S003614450037906X

    Article  MathSciNet  MATH  Google Scholar 

  10. Y. Chen, Y. Gu, A.O. Hero, Sparse LMS for system identification, in Proceedings of the IEEE ICASSP (2009), pp. 3125–3128. doi:10.1109/ICASSP.2009.4960286

  11. D.L. Donoho, Compressed sensing. IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006). doi:10.1109/TIT.2006.871582

    Article  MathSciNet  MATH  Google Scholar 

  12. I. Daubechies, R. DeVore, M. Fornasier, C.S. Gunturk, Iteratively re-weighted least squares minimization for sparse recovery. Commun. Pure Appl. Math. 63(1), 1–38 (2010). doi:10.1002/cpa.20303

    Article  MATH  Google Scholar 

  13. L. Dai, Z. Wang, Z. Yang, Spectrally efficient time-frequency training OFDM for mobile large-scale MIMO systems. IEEE J. Sel. Areas Commun. 31(2), 251–263 (2013). doi:10.1109/JSAC.2013.130213

    Article  Google Scholar 

  14. L. Dai, Z. Wang, Z. Yang, Compressive sensing based time domain synchronous OFDM transmission for vehicular communications. IEEE J. Sel. Areas Commun. 31(9), 460–469 (2013). doi:10.1109/JSAC.2013.SUP.0513041

    Article  Google Scholar 

  15. E. Eweda, Global stabilization of the least mean fourth algorithm. IEEE Trans. Signal Process. 60(3), 1473–1477 (2012). doi:10.1109/TSP.2011.2177976

    Article  MathSciNet  MATH  Google Scholar 

  16. E. Eweda, N.J. Bershad, Stochastic analysis of a stable normalized least mean fourth algorithm for adaptive noise canceling with a white Gaussian reference. IEEE Trans. Signal Process. 60(12), 6235–6244 (2012). doi:10.1109/TSP.2012.2215607

    Article  MathSciNet  Google Scholar 

  17. E. Eweda, A. Zerguine, New insights into the normalization of the least mean fourth algorithm. Signal Image Video Process 7(2), 255–262 (2013). doi:10.1007/s11760-011-0231-y

    Article  Google Scholar 

  18. G. Gui, F. Adachi, Sparse least mean fourth filter with zero-attracting \({\ell }_{1}\)-norm constraint. Inf. Commun. Signal Process. (ICICS) (2013). doi:10.1109/ICICS.2013.6782810

  19. G. Gui, F. Adachi, Adaptive sparse system identification using normalized least-mean fourth algorithm. Int. J. Commun. Syst. 28(1), 38–48 (2015). doi:10.1002/dac.2637

    Article  Google Scholar 

  20. G. Glentis, K. Berberidis, S. Theodoridis, Efficient least squares adaptive algorithms for FIR transversal filtering. IEEE Signal Process. Mag. 16(4), 13–14 (1999). doi:10.1109/79.774932

    Article  Google Scholar 

  21. G. Gui, A. Kuang, L. Wang, Low-speed ADC sampling based high-resolution compressive channel estimation, in 15th International Symposium on WPMC (2012). ISSN: 1347-6890

  22. G. Gui, A. Mehbodniya, F. Adachi, Adaptive sparse channel estimation using re-weighted zero-attracting normalized least mean fourth, in IEEE/CIC International Conference on Communications in China (ICCC) (2013), pp. 1439–1461. doi:10.1109/ICCChina.2013.6671144

  23. G. Gui, L. Xu, F. Adachi, RZA-NLMF algorithm-based adaptive sparse sensing for realizing compressive sensing. EURASIP J. Adv. Signal Process. 2014, 125 (2014). doi:10.1186/1687-6180-2014-125

    Article  Google Scholar 

  24. G. Gui, L. Xu, L. Dai, J. Dan, F. Adachi, ROSA: Robust sparse adaptive channel estimation in the presence of impulsive interferences, in IET Communications (2015), pp. 628–632. doi:10.1109/ICDSP.2015.7251950

  25. Y. Gu, J. Jin, S. Mei, l0 norm constraint LMS algorithm for sparse system identification. IEEE Signal Process. Lett. 16(9), 774–777 (2009)

    Article  Google Scholar 

  26. Z. Gao, L. Dai, Z. Lu, C. Yuen, Z. Wang, Super-resolution sparse MIMO-OFDM channel estimation based on spatial and temporal correlations. IEEE Commun. Lett. 18(7), 1266–1269 (2014). doi:10.1109/LCOMM.2014.2325027

    Article  Google Scholar 

  27. J. Hong, T. Ohtsuki, Signal eigenvector-based device-free passive localization using array sensor. IEEE Trans. Veh.Technol. 64(4), 1354–1363 (2015). doi:10.1109/TVT.2015.2397436

    Article  Google Scholar 

  28. S. Haykin, Adaptive Filter Theory (Prentice-Hall, Englewood Cliffs, 1986)

    MATH  Google Scholar 

  29. J. Jin, Y. Gu, S. Mei, A stochastic gradient approach on compressive sensing signal reconstruction based on adaptive filtering framework. IEEE J. Sel. Top. Signal Process. 4(2), 409–420 (2010). doi:10.1109/JSTSP.2009.2039173

    Article  Google Scholar 

  30. Y.P. Li, T.S. Lee, B.F. Wu, A variable step-size sign algorithm for channel estimation. Signal Process. 102, 304–312 (2014). doi:10.1016/j.sigpro.2014.03.030

    Article  Google Scholar 

  31. D. Middleton, Non-Gaussian noise models in signal processing for telecommunications: new methods and results for class A and class B noise models. IEEE Trans. Inf. Theory 45(4), 1129–1149 (1999). doi:10.1109/18.761256

    Article  MathSciNet  MATH  Google Scholar 

  32. D. Malioutov, M. Cetin, A.S. Willsky, A sparse signal reconstruction perspective for source localization with sensor arrays. IEEE Trans. Signal Process. 53(8), 3010–3022 (2005). doi:10.1109/TSP.2005.850882

    Article  MathSciNet  MATH  Google Scholar 

  33. M. Mishali, Y.C. Eldar, From theory to practice: sub-Nyquist sampling of sparse wideband analog signals. IEEE J. Sel. Top. Signal Process. 4(2), 375–391 (2010). doi:10.1109/JSTSP.2010.2042414

    Article  Google Scholar 

  34. R.K. Martin, W.A. Sethares, R.C. Williamson, C.R. Johnson, Exploiting sparsity in adaptive filters. IEEE Trans. Signal Process. 50(8), 1883–1894 (2002). doi:10.1109/TSP.2002.800414

    Article  Google Scholar 

  35. W. Ma, H. Qu, G. Gui, Li Xu, J. Zhao, B. Chen, Maximum correntropy criterion based sparse adaptive filtering algorithms for robust channel estimation under non-Gaussian environments. The. J. Frankl. Inst. 352(7), 2708–2727 (2015)

    Article  Google Scholar 

  36. D. Needell, J.A. Tropp, Cosamp: iterative signal recovery from incomplete and inaccurate samples. Appl. Comput. Harmonic Anal. 26(3), 301–321 (2009). doi:10.1016/j.acha.2008.07.002

    Article  MathSciNet  MATH  Google Scholar 

  37. K. Pelekanakis, M. Chitre, Adaptive sparse channel estimation under symmetric alpha-stable noise. IEEE Trans. Wirel. Commun. 13(6), 3183–3195 (2014). doi:10.1109/TWC.2014.042314.131432

    Article  Google Scholar 

  38. Y.C. Pati, R. Rezaiifar, P.S. Krishnaprasad, Orthogonal matching pursuit: Recursive function approximation with applications to wavelet decomposition, in Proceedings of the 27th Annual Asilomar Conference on Signals, Systems and Computers Pacific (1993), pp. 40–44. doi:10.1109/ACSSC.1993.342465. ISSN: 1058-6393

  39. B.D. Raychaudhuri, N.B. Mandayam, Frontiers of wireless and mobile communications. Proc. IEEE 100(4), 824–840 (2012). doi:10.1109/JPROC.2011.2182095

    Article  Google Scholar 

  40. C. Rich, W. Yin, Iteratively reweighted algorithms for compressive sensing, in Proceedings of the IEEE ICASSP (2006), pp. 3869–3872. doi:10.1109/ICASSP.2008.4518498. ISSN: 1520-6149

  41. M. Rudelson, R. Veshynin, Geometric approach to error correcting codes and reconstruction of signals. Int. Math. Res. Note 2005(64), 4019–4041 (2005). doi:10.1155/IMRN.2005.4019

    Article  MathSciNet  MATH  Google Scholar 

  42. D.W.J. Stein, Detection of random signals in Gaussian mixture noise. IEEE Trans. Inf. Theory 41(6), 1788–1801 (1995). doi:10.1109/18.476307

    Article  MATH  Google Scholar 

  43. T. Sakai, Online algorithms of low-rank and sparse structure pursuit in practice. IEICE Tech. Rep. 116(395), 285–288 (2017)

    Google Scholar 

  44. T. Shao, Y.R. Zheng, J. Benesty, An affine projection sign algorithm robust against impulsive interferences. IEEE Signal Process. Lett. 17(4), 327–330 (2010). doi:10.1109/LSP.2010.2040203

    Article  Google Scholar 

  45. J. Tropp, A. Gilbert, Signal recovery from random measurements via orthogonal matching pursuit. IEEE Trans. Inf. Theory 53(12), 4655–4666 (2007). doi:10.1109/TIT.2007.909108

    Article  MathSciNet  MATH  Google Scholar 

  46. R. Tibshirani, Regression shrinkage and selection via the Lasso. J. R. Stat. Soc. B 58(1), 267–288 (1996). doi:10.1111/j.1467-9868.2011.00771.x

    MathSciNet  MATH  Google Scholar 

  47. L.R. Vega, H. Rey, J. Benesty, S. Tressens, A new robust variable step-size NLMS algorithm. IEEE Trans. Signal Process. 56(5), 1878–1893 (2008). doi:10.1109/TSP.2007.913142

    Article  MathSciNet  Google Scholar 

  48. B. Widrow, J.R. Glover, J.M. McCool, J. Kaunitz, C.S. Williams, R.H. Hearn, J.R. Zeidler, J.E. Dong, R.C. Goodlin, Adaptive noise cancelling: principles and applications. Proc. IEEE 63(12), 1692–1716 (1975). doi:10.1109/PROC.1975.10036

    Article  Google Scholar 

  49. B. Widrow, S.D. Stearns, Adaptive Signal Processing (Prentice Hall, New Jersey, 1985)

    MATH  Google Scholar 

  50. E. Walach, B. Widrow, The least mean fourth (LMF) adaptive algorithm and its family. IEEE Trans. Inf. Theory 30(2), 275–283 (1984). doi:10.1109/TIT.1984.1056886

    Article  Google Scholar 

  51. J. Weston, A. Elisseeff, B. Scholkopf, M. Tipping, Use of the zero-norm with linear models and kernel methods. JMLR Spec. Issue Var. Featur. Sel. 3, 1439–1461 (2003)

    MathSciNet  MATH  Google Scholar 

  52. X. Wang, X. Gui, D. Zhang, An effective fractal image compression algorithm based on plane fitting. Chin. Phys. B 21(9), 090507 (2012)

    Article  Google Scholar 

  53. X. Wang, F. Li, S. Wang, Fractal image compression based on spatial correlation and hybrid genetic algorithm. J. Vis. Commun. Image Represent. 20(8), 505–510 (2009). doi:10.1016/j.jvcir.2009.07.002

    Article  Google Scholar 

  54. X. Wang, S. Wang, An improved no-search fractal image coding method based on a modified gray-level transform. Comput. Gr. 32(4), 445–450 (2008). doi:10.1016/j.cag.2008.02.004

    Article  Google Scholar 

  55. X. Wang, J. Yun, Y. Zhang, An efficient adaptive arithmetic coding image compression technology. Chin. Phys. B 20(10), 104203 (2011). doi:10.1088/1674-1056/20/10/104203

    Article  Google Scholar 

  56. C. Ye, G. Gui, S. Matsushita, L. Xu, Robust stochastic gradient-based adaptive filtering algorithms to realize compressive sensing against impulsive interferences, in Chinese Control and Decision Conference (2016). doi:10.1109/CCDC.2016.7531301. ISSN: 1948-9447

  57. C. Ye, G. Gui, L. Xu, N. Shimoi, Improved adaptive sparse channel estimation using re-weighted L1-norm normalized least mean fourth algorithm, in Society of Instrument and Control Engineers of Japan (SICE) (2015). doi:10.1109/SICE.2015.7285363

  58. D. Yin, H. C. So, Y. Gu, Sparse constraint affine projection algorithm with parallel implementation and application in compressive sensing, in Proceedings of the IEEE ICASSP (2014), pp. 7238–7242. doi:10.1109/ICASSP.2014.6855005

  59. S. Zhang, J. Zhang, Modified variable step-size affine projection sign algorithm. Electron. Lett. 49(20), 1264–1265 (2013). doi:10.1049/el.2013.2337

    Article  Google Scholar 

  60. Y. Zhang, X. Wang, Fractal compression coding based on wavelet transform with diamond search. Nonlinear Anal. Real World Appl. 13(1), 106–112 (2012). doi:10.1016/j.nonrwa.2011.07.017

    Article  MathSciNet  MATH  Google Scholar 

  61. Z. Zhang, Z. Pi, B. Liu, TROIKA: a general framework for heart rate monitoring using wrist-type photoplethysmographic signals during intensive physical exercise. IEEE Trans. Biomed. Eng. 62(2), 522–531 (2015). doi:10.1109/TBME.2014.2359372

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported in part by the Japan Society for the Promotion of Science KAKENHI (No. 15K06072), National Natural Science Foundation of China Grants (Nos. 61401069, 61271240, 61501254), Jiangsu Special Appoint Professor Grant (RK002STP16001), High-level talent startup grant of Nanjing University of Posts and Telecommunications (XK0010915026), and “1311 Talent Plan” of Nanjing University of Posts and Telecommunications.

Author information

Authors and Affiliations

  1. Department of Electronics and Information Systems, Akita Prefectural University, Yurihonjo, Japan

    Chen Ye & Li Xu

  2. College of Telecommunication and Information Engineering, Nanjing University of Posts and Telecommunications, Nanjing, China

    Guan Gui

Authors
  1. Chen Ye
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guan Gui
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Li Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guan Gui.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ye, C., Gui, G. & Xu, L. Compressive Sensing Signal Reconstruction Using L0-Norm Normalized Least Mean Fourth Algorithms. Circuits Syst Signal Process 37, 1724–1752 (2018). https://doi.org/10.1007/s00034-017-0626-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-017-0626-2

Keywords

Search

Navigation