Journal of Communications and Networks

The Korean Institute of Commucations and Information Sciences (한국통신학회)
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The Expectation and Sparse Maximization Algorithm

  • Received : 2010.02.26
  • Accepted : 2010.06.29
  • Published : 2010.08.31
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Abstract

In recent years, many sparse estimation methods, also known as compressed sensing, have been developed. However, most of these methods presume that the measurement matrix is completely known. We develop a new blind maximum likelihood method-the expectation-sparse-maximization (ESpaM) algorithm-for models where the measurement matrix is the product of one unknown and one known matrix. This method is a variant of the expectation-maximization algorithm to deal with the resulting problem that the maximization step is no longer unique. The ESpaM algorithm is justified theoretically. We present as well numerical results for two concrete examples of blind channel identification in digital communications, a doubly-selective channel model and linear time invariant sparse channel model.

Keywords

References

  1. E. J. Candes and T. Tao, "Decoding by linear programming," IEEE Trans. Inf. Theory, vol. 51, no. 12, pp. 4203–4215, Dec. 2005. https://doi.org/10.1109/TIT.2005.858979
  2. E. J. Candes and T. Tao, "Near-optimal signal recovery from random projections: Universal encoding strategies?," IEEE Trans. Inf. Theory, vol. 52, no. 12, pp. 5406–5425, Dec. 2006
  3. A. P. Dempster, N. M. Laird, and D. B. Rubin, "Maximum-likelihood from incomplete data via the EM algorithm,"J. Roy. Stat. Soc., vol. B39, pp. 1– 38, 1977.
  4. O. Cappe, E.Moulines, and T. Ryde, Inference in Hidden Markov Models, Springer, 2nd ed., 2007.
  5. Springer, 2nd ed., 2007. [5] L. E. Baum, T. Petrie, G. Soules, and N.Weiss, "A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains," Ann. Math. Stat., vol. 41, pp. 164–171, 1970. https://doi.org/10.1214/aoms/1177697196
  6. A. Doucet, S. Godsill, and C. Andrieu, "On sequential Monte Carlo sampling methods for Bayesian filtering," Statistics and Computing, vol. 10, no. 3, pp. 197–208, 2000. https://doi.org/10.1023/A:1008935410038
  7. E. Punskaya, Sequential Monte Carlo Methods for Digital Communications, Ph.D. thesis, Cambridge Univ., Cambridge, U.K., 2003.
  8. T. Ghirmai, M. F. Bugallo, J. Miguez, and P. M. Djuric, "A sequential Monte Carlo method for adaptive blind timing estimation and data detection," IEEE Trans. Signal Process., vol. 53, no. 8, pp. 2855–2865, 2005.
  9. M. Briers, A. Doucet, and S. R. Maskell, "Smoothing algorithms for statespace models," Tech. Rep., Cambridge University Engineering Department Technical Report, CUED/F-INFENG/TR.498, 2004.
  10. S. Barembruch, A. Garivier, and E.Moulines, "On approximate maximum likelihood methods for blind identification: How to cope with the curse of dimensionality," IEEE Trans. Signal Process., vol. 57, no. 11, pp. 4247 – 4259, 2009.
  11. P. Fearnhead, D.Wyncoll, and J. Tawn, "A sequential smoothing algorithm with linear computational cost," submitted, 2008.
  12. S. G. Mallat and Z. Zhang, "Matching pursuits with time-frequency dictionaries," IEEE Trans. Signal Process., vol. 41, no. 12, pp. 3397–3415, 1993. https://doi.org/10.1109/78.258082
  13. Y. C. Pati, R. Rezaiifar, and P. S. Krishnaprasad, "Orthogonal matching pursuit: Recursive function approximation with applications to wavelet decomposition," in Proc. ACSSC, Nov. 1993, vol. 1, pp. 40–44.
  14. J. A. Tropp, "Greed is good: Algorithmic results for sparse approximation," IEEE Trans. Inf. Theory, vol. 50, no. 10, pp. 2231–2242, Oct. 2004. https://doi.org/10.1109/TIT.2004.834793
  15. S. Gleichman and Y. C. Eldar, "Blind compressed sensing," submitted to IEEE Trans. Inf. Theory, CCIT Report; 759 Feb. 2010, EE Pub No. 1716, EE Dept., Technion–Israel Institute of Technology, [Online] arXiv 1002.2586.
  16. W. U. Bajwa, J. Haupt, A. M. Sayeed, and R. Nowak, "Compressed channel sensing: A new approach to estimating sparse multipath channels," to appear in Proc. IEEE, 2010.
  17. J.-J. Fuchs, "Multipath time-delay estimation," in Proc. ICASSP, Apr. 1997, vol. 1, pp. 527–530.
  18. W. U. Bajwa, J. Haupt, G. Raz, and R. Nowak, "Compressed channel sensing," in Proc. CISS, Mar. 2008, pp. 5–10.
  19. S. F. Cotter and B. D. Rao, "Sparse channel estimation via matching pursuit with application to equalization," IEEE Trans. Commun., vol. 50, no. 3, pp. 374–377, Mar. 2002. https://doi.org/10.1109/26.990897
  20. W. U. Bajwa, A.M. Sayeed, and R. Nowak, "Learning sparse doublyselective channels," in Proc. ACCCC, Sept. 2008, pp. 575–582.
  21. G. Taubock, F. Hlawatsch, D. Eiwen, and H. Rauhut, "Compressive estimation of doubly selective channels in multicarrier systems: Leakage effects and sparsity-enhancing processing," IEEE J. Sel. Topics Signal Process., vol. 4, no. 2, pp. 255–271, Apr. 2010.
  22. W. U. Bajwa, A. Sayeed, and R. Nowak, "Compressed sensing of wireless channels in time, frequency, and space," in Proc. ACSSC, Oct. 2008, pp. 2048–2052.
  23. Y. Lui and D. K. Borah, "Estimation of time-varying frequency-selective channels using a matching pursuit technique," in Proc. IEEE WCNC, Mar. 2003, vol. 2, pp. 941–946.
  24. W. Li and J. C. Preisig, "Estimation of rapidly time-varying sparse channels," IEEE J. Ocean. Eng., vol. 32, no. 4, pp. 927–939, Oct. 2007.
  25. M. Sharp and A. Scaglione, "Estimation of sparse multipath channels," in Proc. MILCOM, Nov. 2008, pp. 1–7.
  26. G. B. Giannakis and C. Tepedelenlioglu, "Basis expansion models and diversity techniques for blind identification and equalization of time-varying channels," Proc. IEEE, vol. 86, no. 10, pp. 1969–1986, Oct. 1998. https://doi.org/10.1109/5.720248
  27. F. B. Salem and G. Salut, "Deterministic particle receiver for multipath fading channels in wireless communications. part I: FDMA," Traitement du Signal, vol. 21, no. 4, pp. 347–358, 2004.
  28. W. Turin, "MAP decoding in channels with memory," IEEE Trans. Commun., vol. 48, no. 5, pp. 757–763, May 2000. https://doi.org/10.1109/26.843188
  29. C. N. Georghiades and J. C. Han, "Sequence estimation in the presence of random parameters via the em algorithm," IEEE Trans. Commun., vol. 45, no. 3, pp. 300–308, Mar. 1997. https://doi.org/10.1109/26.558691
  30. M. S. Asif, W. Mantzel, and J. Romberg, "Random channel coding and blind deconvolution," in Proc. ACCCC, 2009.
  31. H. Nguyen and B. C. Levy, "The expectation-maximization Viterbi algorithm for blind adaptive channel equalization," IEEE Trans. Commun., vol. 53, no. 10, pp. 1671–1678, Oct. 2005. https://doi.org/10.1109/TCOMM.2005.857162
  32. R. Tibshirani, "Regression shrinkage and selection via the Lasso," J. Roy. Stat. Soc B., vol. 58, pp. 267–288, 1996.
  33. D. L. Donoho, "For most large underdetermined systems of equations, the minimal L1-norm near-solution approximates the sparsest near-solution," Comm. Pure Appl. Math, vol. 59, pp. 907–934, 2006. https://doi.org/10.1002/cpa.20131
  34. S. S. Chen, D. L. Donoho, and M. L. Saunders, "Atomic decomposition by basis pursuit," SIAM J. Sci. Comput., vol. 20, no. 1, pp. 33–61, 1998. https://doi.org/10.1137/S1064827596304010
  35. A. Viterbi, "Error bounds for convolutional codes and an asymptotically optimum decoding algorithm," IEEE Trans. Inf. Theory, vol. 13, no. 2, pp. 260–269, Apr. 1967. https://doi.org/10.1109/TIT.1967.1054010
  36. P. Fearnhead and P. Clifford, "On-line inference for hidden Markov models via particle filters," J. Roy. Stat. Soc. B, vol. 65, no. 4, pp. 887–899, 2003. https://doi.org/10.1111/1467-9868.00421
  37. J. K. Tugnait, "Detection and estimation for abruptly changing systems," in Proc. Decision and Control including the Symposium on Adaptive Processes, vol. 20, Dec., 1981, pp. 1357–1362.