New Insights into Convergence Theory of Constrained Frequency-Domain Adaptive Filters

Two kinds of update equations are commonly used for the constrained frequency-domain adaptive filter (FDAF), namely the gradient-constrained version and the weight-constrained version. The constraint is imposed only on the stochastic gradient vector in the first version, while it is imposed on the whole weight vector in the second version. It was already found that the two versions have different convergence behaviors, but a rigors analysis of the convergence behavior of the gradient-constrained FDAF is still lacking so far. This paper presents a comprehensive statistical analysis of the gradient-constrained FDAF. We set up an equivalent update equation of the gradient-constrained FDAF, which provides a close link with that of the weight-constrained version. Then, the mean and mean-square convergence behaviors of the gradient-constrained FDAF are analyzed using the new update equation, and the corresponding steady-state solutions are provided. Theoretical results confirm that the gradient-constrained FDAF will converge to a biased solution and exhibits a larger mean-square error than the weight-constrained version when, for instance, the weight vector is not initialized properly. Simulation results agree with our theoretical predictions very well.