Performance Study on Complex-valued Function Approximation Problems

  • Sundaram Suresh
  • Narasimhan Sundararajan
  • Ramasamy Savitha
Part of the Studies in Computational Intelligence book series (SCI, volume 421)


In this chapter, we evaluate the approximation performances of the fully complex valued multi-layer perceptron network and the improved fully complex-valued multi-layer perceptron network described in Chapter 2, the fully complex-valued radial basis function network and the meta-cognitive fully complex-valued radial basis function network described in Chapter 3, and the fast learning fully complex valued relaxation network described in Chapter 4. The performances of these networks are studied in comparison with existing complex-valued learning algorithms like the complex-valued extreme learning machine and the complex-valued minimal resource allocation network using two synthetic, complex-valued function approximation problems and two real-world problems. The real world problems consist of a Quadrature Amplitude Modulation (QAM) channel equalization problem with circular signals and an adaptive beam-forming problem with non-circular signals.


Antenna Array Hide Neuron Radial Basis Function Network Quadrature Amplitude Modulation Transmitted Symbol 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Proakis, J.G., Salehi, M.: Digital Communication. McGraw-Hill Higher Education, New York (2008)Google Scholar
  2. 2.
    Patra, J.C., Pal, R.N., Baliarsingh, R., Panda, G.: Nonlinear channel equalization for QAM constellation using artificial neural networks. IEEE Transactions on System, Man and Cybernetics, Part B: Cybernetics 29(2), 262–271 (1999)CrossRefGoogle Scholar
  3. 3.
    Chen, S., McLaughlin, S., Mulgrew, B.: Complex valued radial basis function network, part II: Application to digital communications channel equalization. Signal Processing 36(2), 175–188 (1994)CrossRefzbMATHGoogle Scholar
  4. 4.
    Cha, I., Kassam, S.A.: Channel equalization using adaptive complex radial basis function networks. IEEE Journal on Selected Areas in Communications 13(1), 122–131 (1995)CrossRefGoogle Scholar
  5. 5.
    Li, M.B., Huang, G.-B., Saratchandran, P., Sundararajan, N.: Fully complex extreme learning machine. Neurocomputing 68(1-4), 306–314 (2005)CrossRefGoogle Scholar
  6. 6.
    Suresh, S., Omkar, S.N., Mani, V., Prakash, T.N.G.: Lift coefficient prediction at high angle of attack using recurrent neural network. Aerospace Science and Technology 7(8), 595–602 (2003)CrossRefzbMATHGoogle Scholar
  7. 7.
    Leung, H., Haykin, S.: The complex backpropagation algorithm. IEEE Transactions on Signal Processing 39(9), 2101–2104 (1991)CrossRefGoogle Scholar
  8. 8.
    Kim, T., Adali, T.: Fully complex multi-layer perceptron network for nonlinear signal processing. Journal of VLSI Signal Processing 32(1/2), 29–43 (2002)zbMATHGoogle Scholar
  9. 9.
    Savitha, R., Suresh, S., Sundararajan, N., Saratchandran, P.: A new learning algorithm with logarithmic performance index for complex-valued neural networks. Neurocomputing 72(16-18), 3771–3781 (2009)CrossRefGoogle Scholar
  10. 10.
    Chen, S., McLaughlin, S., Mulgrew, B.: Complex valued radial basis function network, part I: Network architecture and learning algorithms. EURASIP Signal Processing Journal 35(1), 19–31 (1994)CrossRefzbMATHGoogle Scholar
  11. 11.
    Savitha, R., Suresh, S., Sundararajan, N.: A fully complex-valued radial basis function network and its learning algorithm. International Journal of Neural Systems 19(4), 253–267 (2009)CrossRefGoogle Scholar
  12. 12.
    Trees, H.L.V.: Optimum Array Processing, Detection, Estimation and Modulation Theory: Part IV. John Wiley and Sons, New York (2001)CrossRefGoogle Scholar
  13. 13.
    Widrow, B., Mantey, P.E., Griffiths, L.J., Goode, B.B.: Adaptive antenna systems. Proceedings of the IEEE 55(12), 2143–2159 (1967)CrossRefGoogle Scholar
  14. 14.
    Godara, L.C., Gray, D.A.: A structured gradient algorithm for adaptive beamforming. Journal of the Acoustic Society of America 86(3), 1040–1046 (1989)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Papadias, C.B., Paulraj, A.: A constant modulus algorithm for multiuser signal separation in presence of delay spread using antenna arrays. IEEE Signal Processing Letters 4(6), 178–181 (1997)CrossRefGoogle Scholar
  16. 16.
    Mandyam, G.D., Ahmed, N., Srinath, M.D.: Adaptive beamforming based on the conjugate gradient algorithm. IEEE Transactions on Aerospace and Electronic Systems 33(1), 343–347 (1997)CrossRefGoogle Scholar
  17. 17.
    El Zooghby, A.H., Christodoulou, C.G., Georgiopoulos, M.: Neural network based adaptive beamforming for one- and two-dimensional antenna arrays. IEEE Transactions on Antennas Propagation 46(12), 1891–1893 (1998)CrossRefGoogle Scholar
  18. 18.
    Suksmono, A.B., Hirose, A.: Intelligent beamforming by using a complex-valued neural network. Journal of Intelligent and Fuzzy Systems 15(3-4), 139–147 (2004)Google Scholar
  19. 19.
    Godara, L.C.: Performance analysis of structured gradient algorithm antenna array processing. IEEE Transactions on Antennas Propagation 38(7), 1078–1083 (1990)CrossRefGoogle Scholar
  20. 20.
    Godara, L.C.: Improved LMS algorithm for adaptive beamforming. IEEE Transactions on Antennas Propagation 38(10), 1631–1635 (1990)CrossRefGoogle Scholar
  21. 21.
    Song, X., Wang, J., Niu, X.: Robust adaptive beamforming algorithm based on neural network. In: IEEE International Conference on Automation and Logistics (ICAL 2008), pp. 1844–1849 (2008)Google Scholar
  22. 22.
    Godara, L.C.: Application of antenna arrays to mobile communications, part II: Beam-forming and direction-of-arrival considerations. Proceedings of the IEEE 85(8), 1195–1245 (1997)CrossRefGoogle Scholar
  23. 23.
    Du, K.L., Lai, A.K.Y., Cheng, K.K.M., Swamy, M.N.S.: Neural methods for antenna array signal processing: A review. Signal Processing 82(4), 547–561 (2002)CrossRefzbMATHGoogle Scholar
  24. 24.
    Savitha, R., Suresh, S., Sundararajan, N., Saratchandran, P.: Complex-valued function approximation using an improved BP learning algorithm for feed-forward networks. In: IEEE International Joint Conference on Neural Networks (IJCNN 2008), June 1-8, pp. 2251–2258 (2008)Google Scholar
  25. 25.
    Savitha, R., Suresh, S., Sundararajan, N.: Complex-valued function approximation using a fully complex-valued RBF (FC-RBF) learning algorithm. In: International Joint Conference on Neural Networks (IJCNN 2009), pp. 2819–2825 (2009)Google Scholar
  26. 26.
    Chen, S., Hong, X., Harris, C.J., Hanzo, L.: Fully complex-valued radial basis function networks: Orthogonal least squares regression and classification. Neurocomputing 71(16-18), 3421–3433 (2008)CrossRefGoogle Scholar
  27. 27.
    Monzingo, R.A., Miller, T.W.: Introduction to Adaptive Arrays. SciTech. Publishing, Raleigh (2004)Google Scholar

Copyright information

© Springer-Verlag GmbH Berlin Heidelberg 2013

Authors and Affiliations

  • Sundaram Suresh
    • 1
  • Narasimhan Sundararajan
    • 2
  • Ramasamy Savitha
    • 1
  1. 1.School of Computer EngineeringNanyang Technological UniversitySingaporeSingapore
  2. 2.School of Electrical and Electronics EngineeringNanyang Technological UniversitySingaporeSingapore

Personalised recommendations