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Fixed-Point Algorithms for the Blind Separation of Arbitrary Complex-Valued Non-Gaussian Signal Mixtures


We derive new fixed-point algorithms for the blind separation of complex-valued mixtures of independent, noncircularly symmetric, and non-Gaussian source signals. Leveraging recently developed results on the separability of complex-valued signal mixtures, we systematically construct iterative procedures on a kurtosis-based contrast whose evolutionary characteristics are identical to those of the FastICA algorithm of Hyvarinen and Oja in the real-valued mixture case. Thus, our methods inherit the fast convergence properties, computational simplicity, and ease of use of the FastICA algorithm while at the same time extending this class of techniques to complex signal mixtures. For extracting multiple sources, symmetric and asymmetric signal deflation procedures can be employed. Simulations for both noiseless and noisy mixtures indicate that the proposed algorithms have superior finite-sample performance in data-starved scenarios as compared to existing complex ICA methods while performing about as well as the best of these techniques for larger data-record lengths.


  1. Cichocki A, Unbehauen R, Rummert E: Robust learning algorithm for blind separation of signals. Electronics Letters 1994,30(17):1386-1387. 10.1049/el:19940956

    Article  Google Scholar 

  2. Bell AJ, Sejnowski TJ: An information-maximization approach to blind separation and blind deconvolution. Neural Computation 1995,7(6):1129-1159. 10.1162/neco.1995.7.6.1129

    Article  Google Scholar 

  3. Amari S, Cichocki A, Yang HH: A new learning algorithm for blind signal separation. In Advances in Neural Information Processing Systems. Volume 8. MIT Press, Cambridge, Mass, USA; 1996:757-763.

    Google Scholar 

  4. Pham DT: Blind separation of instantaneous mixture of sources via an independent component analysis. IEEE Transactions on Signal Processing 1996,44(11):2768-2779. 10.1109/78.542183

    Article  Google Scholar 

  5. Cardoso JF, Souloumiac A: Blind beamforming for non-Gaussian signals. IEE Proceedings, Part F: Radar and Signal Processing 1993,140(6):362-370. 10.1049/ip-f-2.1993.0054

    Google Scholar 

  6. Comon P: Independent component analysis. A new concept? Signal Processing 1994,36(3):287-314. 10.1016/0165-1684(94)90029-9

    Article  Google Scholar 

  7. Hyvärinen A, Oja E: A fast fixed-point algorithm for independent component analysis. Neural Computation 1997,9(7):1483-1492. 10.1162/neco.1997.9.7.1483

    Article  Google Scholar 

  8. Molgedey L, Schuster HG: Separation of a mixture of independent signals using time delayed correlations. Physical Review Letters 1994,72(23):3634-3637. 10.1103/PhysRevLett.72.3634

    Article  Google Scholar 

  9. Tong L, Liu R-W, Soon VC, Huang Y-F: Indeterminacy and identifiability of blind identification. IEEE Transactions on Circuits and Systems 1991,38(5):499-509. 10.1109/31.76486

    Article  Google Scholar 

  10. Belouchrani A, Abed-Meraim K, Cardoso J-F, Moulines E: A blind source separation technique using second-order statistics. IEEE Transactions on Signal Processing 1997,45(2):434-444. 10.1109/78.554307

    Article  Google Scholar 

  11. Amari S, Douglas SC, Cichocki A, Yang HH: Multichannel blind deconvolution and equalization using the natural gradient. Proceedings of the 1st IEEE Signal Processing Workshop on Signal Processing Advances in Wireless Communications (SPAWC '97), April 1997, Paris, France 101–104.

    Google Scholar 

  12. Ristaniemi T, Joutsensalo J: Advanced ICA-based receivers for block fading DS-CDMA channels. Signal Processing 2002,82(3):417-431. 10.1016/S0165-1684(01)00194-3

    Article  Google Scholar 

  13. Calhoun V, Adali T: Complex infomax: convergence and approximation of infomax with complex nonlinearities. Proceedings of the 12th IEEE Workshop on Neural Networks for Signal Processing, September 2002, Martigny, Switzerland 307–316.

    Chapter  Google Scholar 

  14. Anemüller J, Sejnowski TJ, Makeig S: Complex independent component analysis of frequency-domain electroencephalographic data. Neural Networks 2003,16(9):1311-1323. 10.1016/j.neunet.2003.08.003

    Article  Google Scholar 

  15. Bingham E, Hyvärinen A: A fast fixed-point algorithm for independent component analysis of complex valued signals. International Journal of Neural Systems 2000,10(1):1-8.

    Article  Google Scholar 

  16. De Lathauwer L, De Moor B: On the blind separation of non-circular sources. Proceedings of the 11th European Signal Processing Conference (EUSIPCO '02), September 2002, Toulouse, France

    Google Scholar 

  17. Eriksson J, Koivunen V: Complex-valued ICA using second order statistics. Proceedings of the 14th IEEE Signal Processing Society Workshop on Machine Learning for Signal Processing, September-October 2004, Sao Luis, Brazil 183–191.

    Google Scholar 

  18. Eriksson J, Koivunen V: Complex random vectors and ICA models: identifiability, uniqueness, and separability. IEEE Transactions on Information Theory 2006,52(3):1017-1029.

    MathSciNet  Article  Google Scholar 

  19. Novey M, Adali T: ICA by maximization of nongaussianity using complex functions. Proceedings of IEEE Workshop on Machine Learning for Signal Processing, September 2005, Mystic, Conn, USA 21–26.

    Google Scholar 

  20. Eriksson J, Seppola A-M, Koivunen V: Complex ICA for circular and non-circular sources. Proceedings of the 13th European Signal Processing Conference (EUSIPCO '05), September 2005, Antalya, Turkey

    Google Scholar 

  21. Shalvi O, Weinstein E: Super-exponential methods for blind deconvolution. IEEE Transactions on Information Theory 1993,39(2):504-519. 10.1109/18.212280

    Article  Google Scholar 

  22. Kung S-Y: Independent component analysis in hybrid mixture: KuicNet learning algorithm and numerical analysis. Proceedings of International Symposium on Multimedia Information Processing, December 1997, Taipei, Taiwan 368–381.

    Google Scholar 

  23. Regalia PA, Mboup M: Undermodeled equalization: a characterization of stationary points for a family of blind criteria. IEEE Transactions on Signal Processing 1999,47(3):760-770. 10.1109/78.747781

    Article  Google Scholar 

  24. Douglas SC: On the convergence behavior of the FastICA algorithm. Proceedings of the 4th International Symposium on Independent Component Analysis and Blind Signal Separation, April 2003, Kyoto, Japan 409–414.

    Google Scholar 

  25. Douglas SC: A statistical convergence analysis of the FastICA algorithm for two-source mixtures. Proceedings of the 39th Asilomar Conference on Signals, Systems and Computers, October 2005, Pacific Grove, Calif, USA

    Google Scholar 

  26. Douglas SC, Yuan Z, Oja E: Average convergence behavior of the FastICA algorithm for blind source separation. Proceedings of the 6th International Conference on Independent Component Analysis and Blind Signal Separation (ICA '06), March 2006, Charleston, SC, USA 3889: 790–798.

    Article  Google Scholar 

  27. Cichocki A, Amari S: Adaptive Blind Signal and Image Processing: Learning Algorithms and Applications. John Wiley & Sons, New York, NY, USA; 2002.

    Book  Google Scholar 

  28. Bracewell R: The Fourier Transform and Its Applications. 3rd edition. McGraw-Hill, New York, NY, USA; 1999.

    MATH  Google Scholar 

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Correspondence to Scott C. Douglas.

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Douglas, S.C. Fixed-Point Algorithms for the Blind Separation of Arbitrary Complex-Valued Non-Gaussian Signal Mixtures. EURASIP J. Adv. Signal Process. 2007, 036525 (2007).

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  • Evolutionary Characteristic
  • Convergence Property
  • Fast Convergence
  • Computational Simplicity
  • Blind Separation