Skip to content


  • Research Article
  • Open Access

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

EURASIP Journal on Advances in Signal Processing20062007:037485

  • Received: 1 October 2005
  • Accepted: 27 May 2006
  • Published:


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.


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

Authors’ Affiliations

Electrical & Computer Engineering Department, Michigan State University, East Lansing, MI 48824-1226, USA
Electrical Engineering & Computer Science Department, University of Michigan, Ann Arbor, MI 48109-2122, USA


  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. 10.1109/79.708539View ArticleGoogle Scholar
  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.1983.1164185View ArticleGoogle Scholar
  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. 10.1109/78.912919View ArticleGoogle Scholar
  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. 10.1109/5.237536View ArticleGoogle Scholar
  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. 10.1109/78.902109View ArticleGoogle Scholar
  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. 10.1109/29.32275View ArticleGoogle Scholar
  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. 10.1109/78.258127View ArticleGoogle Scholar
  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. 10.1109/78.376846View ArticleGoogle Scholar
  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.MATHGoogle Scholar
  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.2006.873749View ArticleGoogle Scholar
  11. Oweiss KG: Integration of the temporal, spectral and spatial information for classifying multi-unit extracellular neural recordings. IEEE Transactions on Biomedical Engineering in reviewGoogle Scholar
  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. 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. 10.1109/78.984756MathSciNetMATHView ArticleGoogle Scholar
  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.MATHView ArticleGoogle Scholar
  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.00979.2005View ArticleGoogle Scholar
  16. Johnson D, Dugeon D: Array Signal Processing: Concepts and Techniques. 1st edition. Prentice Hall, Englewood Cliffs, NJ, USA; 1993.Google Scholar
  17. Jagannatham AK, Rao BD: Whitening-rotation-based semi-blind MIMO channel estimation. IEEE Transactions on Signal Processing 2006,54(3):861–869.MATHView ArticleGoogle Scholar
  18. Daubechies I: Ten Lectures on Wavelets. SIAM, Philadelphia, Pa, USA; 1992.MATHView ArticleGoogle Scholar
  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. Coifman RR, Wickerhauser MV: Entropy-based algorithms for best basis selection. IEEE Transactions on Information Theory 1992,38(2, pt II):713–718. 10.1109/18.119732MATHView ArticleGoogle Scholar
  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. Pesquet J-C, Krim H, Carfantan H: Time-invariant orthonormal wavelet representations. IEEE Transactions on Signal Processing 1996,44(8):1964–1970. 10.1109/78.533717View ArticleGoogle Scholar
  23. Dragotti PL, Vetterli M: Wavelet footprints: theory, algorithms, and applications. IEEE Transactions on Signal Processing 2003,51(5):1306–1323. 10.1109/TSP.2003.810296MathSciNetMATHView ArticleGoogle Scholar
  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. 10.1109/78.668544MathSciNetView ArticleGoogle Scholar
  25. Donoho DL: De-noising by soft-thresholding. IEEE Transactions on Information Theory 1995,41(3):613–627. 10.1109/18.382009MathSciNetMATHView ArticleGoogle Scholar
  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. 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. 10.1109/29.103104View ArticleGoogle Scholar
  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/1177698783MathSciNetMATHView ArticleGoogle Scholar
  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/1177699821MathSciNetMATHView ArticleGoogle Scholar
  30. Vetterli M, Herley C: Wavelets and filter banks: theory and design. IEEE Transactions on Signal Processing 1992,40(9):2207–2232. 10.1109/78.157221MATHView ArticleGoogle Scholar
  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.View ArticleGoogle Scholar
  32. Daubechies I, Sweldens W: Factoring wavelet transforms into lifting steps. Journal of Fourier Analysis and Applications 1998,4(3):247–269. 10.1007/BF02476026MathSciNetMATHView ArticleGoogle Scholar
  33. Berry M-W: Large-scale sparse singular value computations. International Journal of Super-Computer Applications 1992,6(1):13–49.Google Scholar
  34. Nicolelis M (Ed): Methods for Neural Ensemble Recordings. CRC Press, Boca Raton, Fla, USA; 1998.Google Scholar
  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.2003.820544View ArticleGoogle Scholar
  36. Kandel ER, Schwartz JH, Jessell TM (Eds): Principles of Neural Science. 3rd edition. Appleton & Lange, Amsterdam, The Netherlands; 1991. chapter 2Google Scholar
  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/9/4/001MathSciNetMATHView ArticleGoogle Scholar
  38. Fee MS, Mitra PP, Kleinfeld D: Variability of extracellular spike waveforms of cortical neurons. Journal of Neurophysiology 1996,76(6):3823–3833.View ArticleGoogle Scholar
  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)00050-7View ArticleGoogle Scholar
  40. Rissanen J: Modeling by shortest data description. Automatica 1978,14(5):465–471. 10.1016/0005-1098(78)90005-5MATHView ArticleGoogle Scholar
  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. 10.1109/78.80959View ArticleGoogle Scholar
  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. 10.1109/78.726815View ArticleGoogle Scholar
  43. Buzsáki G: Large-scale recording of neuronal ensembles. Nature Neuroscience 2004,7(5):446–451. 10.1038/nn1233View ArticleGoogle Scholar
  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.View ArticleGoogle Scholar


© K. G. Oweiss and D. J. Anderson. 2007

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.