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.

中文翻译：

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具有通信延迟的扩散 LMS：稳定性和性能分析

我们研究了在节点之间存在通信延迟的自适应网络上的分布式估计问题。特别是，我们研究了扩散最小均方 (LMS) 策略，其中在组合步骤中采用了延迟中间估计（由于通信渠道）。一个重要的问题是：延迟是否会影响稳定性条件和性能？为了回答这个问题，我们在具有延迟估计的扩散 LMS 的均值和均方误差意义上进行了详细的性能分析。提供了稳定性条件、瞬态和稳态均方偏差 (MSD) 表达式。主要发现之一是具有延迟的扩散 LMS 仍然可以在没有延迟的扩散 LMS 的相同步长条件下收敛。最后，