In this chapter, the complex Backpropagation (BP) algorithm for the complex backpropagation neural networks (BPN) consisting of the suitable node activation functions having multi-saturated output regions is presented and analyzed by the benchmark testing. And then the complex BPN is utilized as nonlinear adaptive equalizers that can deal with both quadrature amplitude modulation (QAM) and phase shift key (PSK) signals of any constellation sizes. In addition, four nonlinear blind equalization schemes using complex BPN for M-ary QAM signals are described and their learning algorithms are presented. The presented complex BP equalizer (CBPE) gives, compared with conventional linear complex equalizers, an outstanding improvement with respect to bit error rate (BER) when channel distortions are nonlinear.
Finite-time projective synchronization of stochastic complex-valued neural networks with probabilistic time-varying delays
