Optimal variable step-size diffusion LMS algorithms

We derive theoretical expressions of the optimum step-size for diffusion least-mean squares (LMS) algorithms. The resulting optimal step-size leads to the largest correction for the distributed LMS adaptive filter from iteration i to iteration i + 1. For practical computation, we use time-averaging filters and establish the mean-square stability for adapt-then-combine (ATC) and combine-then-adapt (CTA) strategies. We introduce optimal variable step-size diffusion LMS algorithms with detailed and practical guidelines for their implementation. Simulation results support the analysis and prove that the proposed algorithms significantly improve the performance in both transient phase and steady state. The numerical experiments reveal that, compared with the existing approaches, the proposed adaptive algorithms are less sensitive to control parameters and more robust with respect to statistical variations of the environment.