Fully Complex MultiLayer Perceptron Network for Nonlinear Signal Processing
 Taehwan Kim,
 Tülay Adali
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Designing a neural network (NN) to process complexvalued 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 backpropagation (BP) using the split derivatives of the real and imaginary components instead of relying on welldefined fully complex derivatives. In this paper, a fully complex multilayer perceptron (MLP) structure that yields a simplified complexvalued backpropagation (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 nonconstant modulus modulated signals are successfully restored.
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 Title
 Fully Complex MultiLayer Perceptron Network for Nonlinear Signal Processing
 Journal

Journal of VLSI signal processing systems for signal, image and video technology
Volume 32, Issue 12 , pp 2943
 Cover Date
 20020801
 DOI
 10.1023/A:1016359216961
 Print ISSN
 09225773
 Online ISSN
 1573109X
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

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

 Taehwan Kim ^{(1)} ^{(2)}
 Tülay Adali ^{(2)}
 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