This paper aims to develop an accurate stochastic model for the diffusion normalized least-mean-square (dNLMS) algorithm operating with both combine-then-adapt (CTA) and adapt-then-combine (ATC) strategies, aiming to provide a theoretical basis for supporting the study of this algorithm. In particular, considering uncorrelated and correlated Gaussian input data, model expressions are derived for predicting the mean and mean-square behavior of either an individual node or the whole adaptive network for both transient and steady-state phases. Based on these expressions, the impact of the diffusion strategy along with a combination rule on the algorithm performance is assessed and discussed. In addition, examples are presented to demonstrate how model expressions can help the designer in the adjustment of the algorithm parameters without the need of extensive trial-and-error procedures, making performance comparisons less laborious. The effectiveness of the proposed model is assessed through simulation results covering different operating conditions, network topologies, combination rules, step-size values, as well as for a wide range of eigenvalue spreads of the input data correlation matrix and signal-to-noise ratio (SNR) values.

本文旨在为同时采用组合先适应（CTA）和先适应再组合（ATC）策略的扩散归一化最小均方（dNLMS）算法开发精确的随机模型，以提供理论基础支持该算法的研究。特别是，考虑到不相关和相关的高斯输入数据，可导出模型表达式，以预测单个节点或整个自适应网络在瞬态和稳态阶段的均值和均方行为。基于这些表达式，评估并讨论了扩散策略以及组合规则对算法性能的影响。此外，给出了一些示例，以说明模型表达式如何可以帮助设计人员调整算法参数，而无需大量的反复试验过程，从而使性能比较省力。通过仿真结果评估了所提出模型的有效性，仿真结果涵盖了不同的工作条件，网络拓扑，组合规则，步长值，以及输入数据相关矩阵和信噪比的特征值分布范围广（SNR）值。