Unbiased adaptive system identification for correlated input and noise

We consider the problem of adaptive system identification when the additive noise is colored, following an ARMA model, and correlated with the input signal. By first assuming exact knowledge of the ARMA coefficients we use the Kalman filter theory to develop a prototype adaptive estimation algorithm which is optimum in the case of uncorrelated input and noise and outperforms, considerably, the classical RLS. We then apply the prototype algorithm in the case of correlated input and noise and show that it provides unbiased estimates as opposed to classical RLS which is highly biased. In the final part of our article, motivated by our prototype algorithm, we propose an RLS-type algorithmic variant which estimates the ARMA coefficients at the same time with the system identification part. Simulations show that this alternative version is only slightly inferior to the prototype algorithm, which requires exact knowledge of the ARMA model, inheriting all its notable advantages.