The convergence of output error identification and adaptive IIR filtering algorithms in the presence of colored noise

The authors partially resolve the open problem of the global convergence and parameter consistency of the output error identification and adaptive IIR (infinite impulse response) filtering algorithms in the presence of independent additive colored noise. The algorithms considered include both the stochastic gradient and recursive least squares algorithms, employing a projection of the parameter estimates onto a compact convex set containing the true parameter. The colored noise is allowed to be a general nonstationary moving average noise of finite but unbounded order. The key idea in establishing self-optimality is the use of a backward recursion, combined with the use of the bounded growth rate of the regression vector. To establish the parameter consistency of the stochastic gradient-like algorithm, a simple general technique is developed