Incremental augmented complex adaptive IIR algorithm for training widely linear ARMA model

In this paper, we propose a distributed adaptive learning algorithm to train the coefficients of a widely linear autoregressive moving average model by measurements collected by the nodes of a network. We assume that each node uses the augmented complex adaptive infinite impulse response (ACA-IIR) filter as the learning rule, and nodes interact with each other under an incremental mode of cooperation. To derive the proposed algorithm, called the incremental ACAIIR (IACA-IIR), we firstly formulate the distributed adaptive learning problem as an unconstrained minimization problem. Then, we apply stochastic gradient optimization argument to solve it and derive the proposed algorithm. We further find the step size range where the stability of the proposed algorithm is guaranteed. We also introduce a reduced-complexity version of the IACA-IIR algorithm. Since the proposed algorithm relies on the augmented complex statistics, it can be used to model both types of complex-valued signals (proper and improper signals). To evaluate the performance of the proposed algorithm, we use both synthetic and real-world complex signals in our simulations. The results exhibit superior performance of the proposed algorithm over the non-cooperative ACA-IIR algorithm.