Fully Complex Multi-Layer Perceptron Network for Nonlinear Signal Processing
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
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.
- Silverman, H. (1975) Complex Variables. Houghton, Newark, USA
- Clarke, T. (1990) Generalization of Neural Network to the Complex Plane. Proc. of IJCNN 2: pp. 435-440
- Georgiou, G., Koutsougeras, C. (1992) Complex Backpropagation. IEEE Trans. on Circuits and Systems II 39: pp. 330-334
- You, C., Hong, D. (1998) Nonlinear Blind Equalization Schemes Using Complex-Valued Multilayer Feedforward Neural Networks. IEEE Trans. on Neural Networks 9: pp. 1442-1455
- D. Mandic and J. Chambers, Recurrent Neural Networks for Prediction, John Wiley and Sons, 2001.
- Hirose, A. (1992) Continuous Complex-Valued Back-Propagation Learning. Electronics Letters 28: pp. 1854-1855
- Leung, H., Haykin, S. (1991) The Complex Backpropagation Algorithm. IEEE Trans. on Signal Proc. 3: pp. 2101-2104
- Benvenuto, N., Marchesi, M., Piazza, F., Uncini, A. (1991) Non Linear Satellite Radio Links Equalized Using Blind Neural Networks. Proc. of ICASSP 3: pp. 1521-1524
- Benvenuto, N., Piazza, F. (1992) On the Complex Backpropagation Algorithm. IEEE Trans. on Signal Processing 40: pp. 967-969
- Ibnkahla, M., Castanie, F. (1995) Vector Neural Networks for Digital Satellite Communications. Proc. of ICC 3: pp. 1865-1869
- 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.
- Bandito, S., Biglieri, E. (1983) Nonlinear Equalization of Digital Satellite Channels. IEEE Jour. on SAC. SAC-1: pp. 57-62
- Kechriotis, G., Manolakos, E. (1994) Training Fully Recurrent Neural Networks with Complex Weights. IEEE Trans. on Circuits and Systems—II: Analog and Digital Signal Processing 41: pp. 235-238
- Deng, J., Sundararajan, N., Saratchandran, P. (2000) Communication Channel Equalization Using Complex-Valued Minimal Radial Basis Functions Neural Network. Proc. of IEEE IJCNN 2000 5: pp. 372-377
- Lee, K.Y., Jung, S. (1999) Extended Complex RBF and its Application to M-QAM in Presence of Co-Channel Interference. Electronics Letters 35: pp. 17-19
- 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.
- 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.
- 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.
- 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.
- Fully Complex Multi-Layer Perceptron Network for Nonlinear Signal Processing
Journal of VLSI signal processing systems for signal, image and video technology
Volume 32, Issue 1-2 , pp 29-43
- Cover Date
- Print ISSN
- Online ISSN
- Kluwer Academic Publishers
- Additional Links
- nonlinear adaptive signal processing
- fully complex neural network
- split complex neural network
- elementary transcendental functions
- bounded almost everywhere
- analytic almost everywhere
- Industry Sectors
- Author Affiliations
- 1. Center for Advanced Aviation System Development, The MITRE Corporation, M/S N670, 7515 Colshire Drive, McLean, Virginia, 22102, USA
- 2. Information Technology Laboratory, Department of Computer Science and Electrical Engineering, University of Maryland Baltimore County, Baltimore, Maryland, 21250, USA