Abstract—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. I. INTRODUCTION ost of the signal processing tools have been developed to operate on signals with real values. However, there exist areas where signal processing in the complex domain is required. For example, in digital communication, the symbols sent through a communication channel are usually implemented as points in complex domain. In phase shift keying (PSK) the binary information is conveyed by changes in the phase of the reference signal [1]. In quadrature PSK, four different phase values are used to encode two bits of data. In quadrature amplitude modulation (QAM), both the amplitude and the phase of the reference signal are utilized to encode information more efficiently with increased data transfer rate. The most commonly used QAM are 32QAM, 64QAM, 128QAM and 256QAM where 5 to 8 bits of data are transferred per symbol. Phase plays an important role in these systems and the symbols are represented by complex numbers. The straightforward method to deal with complex numbers is to process the real and complex parts of the signal separately. However, this is suboptimal because it neglects the interdependency between the real and complex part of the signal [2]. Alternatively, the existing tools of signal processing, which are originally designed to deal with real signals, have been effectively modified to operate in Manuscript received April 13, 2007. This work was supported in part by the National Science Foundation under Grants CISE-0541241, ECS- 0422718, CNS-0540304 and by Office of Naval Research N00014-1-1- 0405. S. Seth, M. C. Ozturk and J. C. Principe are with the Computational NeuroEngineering Laboratory of the University of Florida, Gainesville, FL, 32611 (phone: 352-392-2682; fax: 352-392-0044; e-mail: sohan@ ufl.edu). complex domain. In the neural networks literature, the popular back propagation algorithm (BP) for multilayer perceptron (MLP) was originally developed in real domain and in the past two decades it has been effectively used in many diverse areas [2, 3, 4]. Kim and Adali [2] have rigorously developed the complex back propagation (CBP) algorithm for complex MLP (CMLP). CMLP uses fully complex activation functions instead of split complex functions that treat real and imaginary parts of the signal separately, and achieve consistently better performance. However, the difficulty with the MLP training in complex domain arises mostly from the selection of activation functions that can operate in complex domain. The nonlinear functions have to be both analytic and bounded since the backpropagation algorithm requires derivatives of the nonlinearities to calculate the weight updates [2, 10]. Moreover MLPs are trained in an iterative approach which is usually slow and computationally intensive. The model parameter such as the input dimension and number of hidden units must be selected through exhaustive testing as they play a crucial role in faster convergence and minimal training error. In this paper, we propose a new utilization of recurrent neural network topologies, called echo state networks (ESN), for signal processing in the complex domain. ESNs have a recurrent topology of nonlinear processing elements (PEs); the state of which is called echo states [5, 6, 7]. The parameters of the recurrent topology is never trained but fixed a priori. The input to the system is fed to the recurrent network of fixed weights and echo states are computed. Then, a memoryless readout network is used to generate the system output from echo states. The readout network is usually a simple linear combiner, which allows the use of simple linear regression to train the readout weights. The use of ESNs for complex domain is very convenient since system training is equivalent to simple linear regression, which is trivial in the complex domain. The derivatives of the nonlinear activation functions are never necessary since the recurrent part is fixed apriori. The nonlinear activation functions have to be modified similar to [2] to ensure bounded output. However, being analytic is not anymore required since ESN does not require the computation of derivatives for training. We compare CESNs with CMLP and a linear network trained with complex least mean squares (CLMS) algorithm on a complex channel equalization experiment. We demonstrate that CESN can achieve lower symbol error rates with a simple and fast learning. Signal Processing with Echo State Networks in the Complex Domain Sohan Seth, Mustafa C. Ozturk, Josè C. Principe, Fellow, IEEE M