Diffusion LMS with Communication Delays: Stability and Performance Analysis

We study the problem of distributed estimation over adaptive networks where communication delays exist between nodes. In particular, we investigate the diffusion Least-Mean-Square (LMS) strategy where delayed intermediate estimates (due to the communication channels) are employed during the combination step. One important question is: Do the delays affect the stability condition and performance? To answer this question, we conduct a detailed performance analysis in the mean and in the mean-square-error sense of the diffusion LMS with delayed estimates. Stability conditions, transient and steady-state mean-square-deviation (MSD) expressions are provided. One of the main findings is that diffusion LMS with delays can still converge under the same step-sizes condition of the diffusion LMS without delays. Finally, simulation results illustrate the theoretical findings.