Signal Processing with Echo State Networks in the Complex Domain

Signal processing in the complex domain is an essential part of signal processing particularly in digital communication systems. Considerable efforts have been made to convert the well established tools of real signal processing such as backpropagation training algorithm for multilayer perceptron to perform in complex domain. In this paper, we propose the echo state network (ESN) approach for complex domain signal processing. The complex ESN (CESN) replaces the real connection weights for the reservoir and readout with complex numbers and the real activation functions with fully complex nonlinearities. CESNs provide much faster and simpler learning compared to other techniques in the literature since ESN does not backpropagate the errors through complex nonlinearities for training but only adapts a feedforward linear readout. Experiment on nonlinear channel equalization show superiority of CESN in terms of lower symbol error rates in addition to the fast and simple training.