A complex Echo State Network for nonlinear adaptive filtering

The operation of echo state networks (ESNs) is extended to the complex domain, in order to perform nonlinear complex valued adaptive filtering of nonlinear and nonstationary signals. This is achieved by introducing a nonlinear output layer into an ESN, whereby full adaptivity is provided by introducing an adaptive amplitude into the nonlinear activation function within the output layer of ESN. This allows us to control and track the degree of nonlinearity, which facilitates real-world adaptive filtering applications. Learning algorithms for such ESN are derived, and the benefits of the combination of sparse connections and nonlinear adaptive output layer are illustrated by simulations of both benchmark and real world signals.