Complex backpropagation neural network using elementary transcendental activation functions

Designing a neural network (NN) for processing complex signals is a challenging task due to the lack of bounded and differentiable nonlinear activation functions in the entire complex domain C. To avoid this difficulty, 'splitting', i.e., using uncoupled real sigmoidal functions for the real and imaginary components has been the traditional approach, and a number of fully complex activation functions introduced can only correct for magnitude distortion but can not handle phase distortion. We have recently introduced a fully complex NN that uses a hyperbolic tangent function defined in the entire complex domain and showed that for most practical signal processing problems, it is sufficient to have an activation function that is bounded and differentiable almost everywhere in the complex domain. In this paper, the fully complex NN design is extended to employ other complex activation functions of the hyperbolic, circular, and their inverse function family. They are shown to successfully restore nonlinear amplitude and phase distortions of non-constant modulus modulated signals.