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Tracking Signal Subspace Invariance for Blind Separation and Classification of Nonorthogonal Sources in Correlated Noise

Abstract

We investigate a new approach for the problem of source separation in correlated multichannel signal and noise environments. The framework targets the specific case when nonstationary correlated signal sources contaminated by additive correlated noise impinge on an array of sensors. Existing techniques targeting this problem usually assume signal sources to be independent, and the contaminating noise to be spatially and temporally white, thus enabling orthogonal signal and noise subspaces to be separated using conventional eigendecomposition. In our context, we propose a solution to the problem when the sources are nonorthogonal, and the noise is correlated with an unknown temporal and spatial covariance. The approach is based on projecting the observations onto a nested set of multiresolution spaces prior to eigendecomposition. An inherent invariance property of the signal subspace is observed in a subset of the multiresolution spaces that depends on the degree of approximation expressed by the orthogonal basis. This feature, among others revealed by the algorithm, is eventually used to separate the signal sources in the context of "best basis" selection. The technique shows robustness to source nonstationarities as well as anisotropic properties of the unknown signal propagation medium under no constraints on the array design, and with minimal assumptions about the underlying signal and noise processes. We illustrate the high performance of the technique on simulated and experimental multichannel neurophysiological data measurements.

References

  1. 1.

    Hero A, Messer H, Goldberg J, et al.: Highlights of statistical signal and array processing. IEEE Signal Processing Magazine 1998,15(5):21–64. 10.1109/79.708539

    Article  Google Scholar 

  2. 2.

    Bienvenu G, Kopp L: Optimality of high resolution array processing using the eigensystem approach. IEEE Transactions on Acoustics, Speech, and Signal Processing 1983,31(5):1235–1248. 10.1109/TASSP.1983.1164185

    Article  Google Scholar 

  3. 3.

    Zhang Y, Mu W, Amin MG: Subspace analysis of spatial time-frequency distribution matrices. IEEE Transactions on Signal Processing 2001,49(4):747–759. 10.1109/78.912919

    Article  Google Scholar 

  4. 4.

    Van Der Veen A-J, Deprettere EF, Swindlehurst AL: Subspace-based signal analysis using singular value decomposition. Proceedings of the IEEE 1993,81(9):1277–1308. 10.1109/5.237536

    Article  Google Scholar 

  5. 5.

    Stoica P, Besson O, Gershman AB: Direction-of-arrival estimation of an amplitude-distorted wavefront. IEEE Transactions on Signal Processing 2001,49(2):269–276. 10.1109/78.902109

    Article  Google Scholar 

  6. 6.

    Le Cadre JP: Parametric methods for spatial signal processing in the presence of unknown colored noise fields. IEEE Transactions on Acoustics, Speech, and Signal Processing 1989,37(7):965–983. 10.1109/29.32275

    Article  Google Scholar 

  7. 7.

    Stoica P, Viberg M, Ottersten B: Instrumental variable approach to array processing in spatially correlated noise fields. IEEE Transactions on Signal Processing 1994,42(1):121–133. 10.1109/78.258127

    Article  Google Scholar 

  8. 8.

    Ye H, DeGroat RD: Maximum likelihood DOA estimation and asymptotic Cramer-Rao bounds for additive unknown colored noise. IEEE Transactions on Signal Processing 1995,43(4):938–949. 10.1109/78.376846

    Article  Google Scholar 

  9. 9.

    Rieke F, Warland D, de Ruyter van Steveninck R, Bialek W: Spikes: Exploring the Neural Code. 3rd edition. MIT Press, Cambridge, Mass, USA; 1997.

    Google Scholar 

  10. 10.

    Oweiss KG: A systems approach for data compression and latency reduction in cortically controlled brain machine interfaces. IEEE Transactions on Biomedical Engineering 2006,53(7):1364–1377. 10.1109/TBME.2006.873749

    Article  Google Scholar 

  11. 11.

    Oweiss KG: Integration of the temporal, spectral and spatial information for classifying multi-unit extracellular neural recordings. IEEE Transactions on Biomedical Engineering in review

  12. 12.

    Oweiss KG, Anderson DJ: A new technique for blind source separation using subband subspace analysis in correlated multichannel signal environments. Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP '01), May 2001, Salt Lake City, Utah, USA 5: 2813–2816.

    Google Scholar 

  13. 13.

    Hachem W, Desbouvries F, Loubaton P: MIMO channel blind identification in the presence of spatially correlated noise. IEEE Transactions on Signal Processing 2002,50(3):651–661. 10.1109/78.984756

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Kotoulas D, Koukoulas P, Kalouptsidis N: Subspace projection based blind channel order estimation of MIMO systems. IEEE Transactions on Signal Processing 2006,54(4):1351–1363.

    MATH  Article  Google Scholar 

  15. 15.

    Gold C, Henze DA, Koch C, Buzsáki G: On the origin of the extracellular action potential waveform: a modeling study. Journal of Neurophysiology 2006,95(5):3113–3128. 10.1152/jn.00979.2005

    Article  Google Scholar 

  16. 16.

    Johnson D, Dugeon D: Array Signal Processing: Concepts and Techniques. 1st edition. Prentice Hall, Englewood Cliffs, NJ, USA; 1993.

    Google Scholar 

  17. 17.

    Jagannatham AK, Rao BD: Whitening-rotation-based semi-blind MIMO channel estimation. IEEE Transactions on Signal Processing 2006,54(3):861–869.

    MATH  Article  Google Scholar 

  18. 18.

    Daubechies I: Ten Lectures on Wavelets. SIAM, Philadelphia, Pa, USA; 1992.

    Google Scholar 

  19. 19.

    Oweiss KG: Source detection in correlated multichannel signal and noise fields. Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP '03), April 2003, Hong Kong 5: 257–260.

    Google Scholar 

  20. 20.

    Coifman RR, Wickerhauser MV: Entropy-based algorithms for best basis selection. IEEE Transactions on Information Theory 1992,38(2, pt II):713–718. 10.1109/18.119732

    MATH  Article  Google Scholar 

  21. 21.

    Mallat HKS, Donoho D, Willsky AS: Best basis algorithm for signal enhancement. Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP '95), May 1995, Detroit, Mich, USA 3: 1561–1564.

    Google Scholar 

  22. 22.

    Pesquet J-C, Krim H, Carfantan H: Time-invariant orthonormal wavelet representations. IEEE Transactions on Signal Processing 1996,44(8):1964–1970. 10.1109/78.533717

    Article  Google Scholar 

  23. 23.

    Dragotti PL, Vetterli M: Wavelet footprints: theory, algorithms, and applications. IEEE Transactions on Signal Processing 2003,51(5):1306–1323. 10.1109/TSP.2003.810296

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    Crouse MS, Nowak RD, Baraniuk RG: Wavelet-based statistical signal processing using hidden Markov models. IEEE Transactions on Signal Processing 1998, 46: 886–902. 10.1109/78.668544

    MathSciNet  Article  Google Scholar 

  25. 25.

    Donoho DL: De-noising by soft-thresholding. IEEE Transactions on Information Theory 1995,41(3):613–627. 10.1109/18.382009

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Oweiss KG, Anderson DJ: A new approach to array denoising. Proceedings of the IEEE 34th Asilomar Conference on Signals, Systems and Computers (ASSC '00), October-November 2000, Pacific Grove, Calif, USA 2: 1403–1407.

    Google Scholar 

  27. 27.

    Williams DB, Johnson DH: Using the sphericity test for source detection with narrow-band passive arrays. IEEE Transactions on Acoustics, Speech, and Signal Processing 1990,38(11):2008–2014. 10.1109/29.103104

    Article  Google Scholar 

  28. 28.

    Sugiyama T: On the distribution of the largest latent root of the covariance matrix. The Annals of Mathematical Statistics 1967,38(4):1148–1151. 10.1214/aoms/1177698783

    MathSciNet  MATH  Article  Google Scholar 

  29. 29.

    Sugiyama T: On the distribution of the latent vectors for principal component analysis. The Annals of Mathematical Statistics 1965,36(6):1875–1876. 10.1214/aoms/1177699821

    MathSciNet  MATH  Article  Google Scholar 

  30. 30.

    Vetterli M, Herley C: Wavelets and filter banks: theory and design. IEEE Transactions on Signal Processing 1992,40(9):2207–2232. 10.1109/78.157221

    MATH  Article  Google Scholar 

  31. 31.

    Suhail Y, Oweiss KG: A reduced complexity integer lifting wavelet-based module for real-time processing in implantable neural interface devices. Proceedings of Annual International Conference of the IEEE Engineering in Medicine and Biology, September 2004, San Francisco, Calif, USA 2: 4552–4555.

    Article  Google Scholar 

  32. 32.

    Daubechies I, Sweldens W: Factoring wavelet transforms into lifting steps. Journal of Fourier Analysis and Applications 1998,4(3):247–269. 10.1007/BF02476026

    MathSciNet  MATH  Article  Google Scholar 

  33. 33.

    Berry M-W: Large-scale sparse singular value computations. International Journal of Super-Computer Applications 1992,6(1):13–49.

    Google Scholar 

  34. 34.

    Nicolelis M (Ed): Methods for Neural Ensemble Recordings. CRC Press, Boca Raton, Fla, USA; 1998.

    Google Scholar 

  35. 35.

    Wise K, Anderson D, Hetke J, Kipke D, Najafi K: Wireless implantable microsystems: high-density electronic interfaces to the nervous system. Proceedings of the IEEE 2004,92(1):76–97. 10.1109/JPROC.2003.820544

    Article  Google Scholar 

  36. 36.

    Kandel ER, Schwartz JH, Jessell TM (Eds): Principles of Neural Science. 3rd edition. Appleton & Lange, Amsterdam, The Netherlands; 1991. chapter 2

    Google Scholar 

  37. 37.

    Lewicki MS: A review of methods for spike sorting: the detection and classification of neural action potentials. Network: Computation in Neural Systems 1998,9(4):53–78. 10.1088/0954-898X/9/4/001

    MathSciNet  MATH  Article  Google Scholar 

  38. 38.

    Fee MS, Mitra PP, Kleinfeld D: Variability of extracellular spike waveforms of cortical neurons. Journal of Neurophysiology 1996,76(6):3823–3833.

    Article  Google Scholar 

  39. 39.

    Fee MS, Mitra PP, Kleinfeld D: Automatic sorting of multiple unit neuronal signals in the presence of anisotropic and non-Gaussian variability. Journal of Neuroscience Methods 1996,69(2):175–188. 10.1016/S0165-0270(96)00050-7

    Article  Google Scholar 

  40. 40.

    Rissanen J: Modeling by shortest data description. Automatica 1978,14(5):465–471. 10.1016/0005-1098(78)90005-5

    MATH  Article  Google Scholar 

  41. 41.

    Chen W, Wong KM, Reilly JP: Detection of the number of signals: a predicted eigen-threshold approach. IEEE Transactions on Signal Processing 1991,39(5):1088–1098. 10.1109/78.80959

    Article  Google Scholar 

  42. 42.

    Wu Y, Tam K-W: On determination of the number of signals in spatially correlated noise. IEEE Transactions on Signal Processing 1998,46(11):3023–3029. 10.1109/78.726815

    Article  Google Scholar 

  43. 43.

    Buzsáki G: Large-scale recording of neuronal ensembles. Nature Neuroscience 2004,7(5):446–451. 10.1038/nn1233

    Article  Google Scholar 

  44. 44.

    Harris KD, Henze DA, Csicsvari J, Hirase H, Buzsáki G: Accuracy of tetrode spike separation as determined by simultaneous intracellular and extracellular measurements. Journal of Neurophysiology 2000,84(1):401–414.

    Article  Google Scholar 

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Correspondence to Karim G. Oweiss.

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Oweiss, K.G., Anderson, D.J. Tracking Signal Subspace Invariance for Blind Separation and Classification of Nonorthogonal Sources in Correlated Noise. EURASIP J. Adv. Signal Process. 2007, 037485 (2006). https://doi.org/10.1155/2007/37485

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Keywords

  • Signal Source
  • Subspace Invariance
  • Noise Environment
  • Source Separation
  • Correlate Noise