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Fully Complex Multi-Layer Perceptron Network for Nonlinear Signal Processing

  • Taehwan Kim
  • Tülay Adali
Article

Abstract

Designing a neural network (NN) to process complex-valued signals is a challenging task since a complex nonlinear activation function (AF) cannot be both analytic and bounded everywhere in the complex plane ℂ. To avoid this difficulty, ‘splitting’, i.e., using a pair of real sigmoidal functions for the real and imaginary components has been the traditional approach. However, this ‘ad hoc’ compromise to avoid the unbounded nature of nonlinear complex functions results in a nowhere analytic AF that performs the error back-propagation (BP) using the split derivatives of the real and imaginary components instead of relying on well-defined fully complex derivatives. In this paper, a fully complex multi-layer perceptron (MLP) structure that yields a simplified complex-valued back-propagation (BP) algorithm is presented. The simplified BP verifies that the fully complex BP weight update formula is the complex conjugate form of real BP formula and the split complex BP is a special case of the fully complex BP. This generalization is possible by employing elementary transcendental functions (ETFs) that are almost everywhere (a.e.) bounded and analytic in ℂ. The properties of fully complex MLP are investigated and the advantage of ETFs over split complex AF is shown in numerical examples where nonlinear magnitude and phase distortions of non-constant modulus modulated signals are successfully restored.

nonlinear adaptive signal processing fully complex neural network split complex neural network elementary transcendental functions bounded almost everywhere analytic almost everywhere 

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References

  1. 1.
    H. Silverman, Complex Variables, Houghton, Newark, USA, 1975.zbMATHGoogle Scholar
  2. 2.
    T. Clarke, “Generalization of Neural Network to the Complex Plane,” in Proc. of IJCNN, vol. 2, 1990, pp. 435-440.Google Scholar
  3. 3.
    G. Georgiou and C. Koutsougeras, “Complex Backpropagation,” IEEE Trans. on Circuits and Systems II, vol. 39,no. 5, 1992, pp. 330-334.CrossRefzbMATHGoogle Scholar
  4. 4.
    C. You and D. Hong, “Nonlinear Blind Equalization Schemes Using Complex-Valued Multilayer Feedforward Neural Networks,” IEEE Trans. on Neural Networks, vol. 9,no. 6, 1998, pp. 1442-1455.CrossRefGoogle Scholar
  5. 5.
    D. Mandic and J. Chambers, Recurrent Neural Networks for Prediction, John Wiley and Sons, 2001.Google Scholar
  6. 6.
    A. Hirose, “Continuous Complex-Valued Back-Propagation Learning,” Electronics Letters, vol. 28,no. 20, 1992, pp. 1854-1855.CrossRefGoogle Scholar
  7. 7.
    H. Leung and S. Haykin, “The Complex Backpropagation Algorithm,” IEEE Trans. on Signal Proc., vol. 3,no. 9, 1991, pp. 2101-2104.CrossRefGoogle Scholar
  8. 8.
    N. Benvenuto, M. Marchesi, F. Piazza, and A. Uncini, “Non Linear Satellite Radio Links Equalized Using Blind Neural Networks,” in Proc. of ICASSP, vol. 3, 1991, pp. 1521-1524.Google Scholar
  9. 9.
    N. Benvenuto and F. Piazza, “On the Complex Backpropagation Algorithm,” IEEE Trans. on Signal Processing, vol. 40,no. 4, 1992, pp. 967-969.CrossRefGoogle Scholar
  10. 10.
    M. Ibnkahla and F. Castanie, “Vector Neural Networks for Digital Satellite Communications,” in Proc. of ICC, vol. 3, 1995, pp. 1865-1869.Google Scholar
  11. 11.
    A. Uncini, L. Vecci, P. Campolucci, and F. Piazza, “Complex-Valued Neural Networks with Adaptive Spline Activation Functions,” IEEE Trans. on Signal Processing, vol. 47,no. 2, 1999.Google Scholar
  12. 12.
    S. Bandito and E. Biglieri, “Nonlinear Equalization of Digital Satellite Channels,” IEEE Jour. on SAC., vol. SAC-1., 1983, pp. 57-62.Google Scholar
  13. 13.
    G. Kechriotis and E. Manolakos, “Training Fully Recurrent Neural Networks with Complex Weights,” IEEE Trans. on Circuits and Systems—II: Analog and Digital Signal Processing, vol. 41,no. 3, 1994, pp. 235-238.CrossRefGoogle Scholar
  14. 14.
    J. Deng, N. Sundararajan, and P. Saratchandran, “Communication Channel Equalization Using Complex-Valued Minimal Radial Basis Functions Neural Network,” in Proc. of IEEE IJCNN 2000, vol. 5, 2000, pp. 372-377.Google Scholar
  15. 15.
    K.Y. Lee and S. Jung, “Extended Complex RBF and its Application to M-QAM in Presence of Co-Channel Interference,” Electronics Letters, vol. 35,no. 1, 1999, pp. 17-19.CrossRefGoogle Scholar
  16. 16.
    S. Chen, P.M. Grant, S. McLaughlin, and B. Mulgrew, “Complex-Valued Radial Basis Function Networks,” in Proc. of third IEEE International Conference on Artificial Neural Networks,” 1993, pp. 148-152.Google Scholar
  17. 17.
    T. Kim and T. Adah, “Fully Complex Backpropagation for Constant Envelop Signal Processing,” in Proc. of IEEE Workshop on Neural Networks for Sig. Proc., Sydney, Dec. 2000, pp. 231-240.Google Scholar
  18. 18.
    T. Kim and T. Adah, “Complex Backpropagation Neural Network Using Elementary Transcendental Activation Functions,” in Proc. of IEEE ICASSP, Proc. vol. II, Salt Lake City, May 2001.Google Scholar
  19. 19.
    T. Kim and T. Adah, “Nonlinear Satellite Channel Equalization Using Fully Complex Feed-Forward Neural Networks,” in Proc. of IEEE Workshop on Nonlinear Signal and Image Processing, Baltimore, June, 2001, pp. 141-150.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Taehwan Kim
    • 1
    • 2
  • Tülay Adali
    • 2
  1. 1.Center for Advanced Aviation System DevelopmentThe MITRE Corporation, M/S N670McLeanUSA
  2. 2.Information Technology Laboratory, Department of Computer Science and Electrical EngineeringUniversity of Maryland Baltimore CountyBaltimoreUSA

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