In this work, a new class of stochastic gradient algorithm is developed based on fractional calculus. Unlike the existing algorithms, the concept of complex fractional gradient is introduced by employing Caputo’s fractional derivative which results in a fractional steepest descent algorithm and a fractional-order complex LMS (FoCLMS) algorithm. We demonstrate that with the Caputo’s fractional gradient definition, the Weiner solution remains invariant. Convergence analysis of the proposed FoCLMS algorithm is presented for both transient and steady state scenarios. Consequently, expressions for the learning curves and steady state EMSE are derived. Our theoretical developments are validated by simulation experiments. Extensive simulations are presented to investigate all possible scenarios: channel with negative weights and real input data, channel with positive weights and complex input data, and channel with complex weights and complex input data.

在这项工作中，基于分数阶微积分开发了一类新的随机梯度算法。与现有算法不同的是，通过使用Caputo的分数阶导数引入了复分数梯度的概念，这导致分数最速下降算法和分数阶复数LMS（FoCLMS）算法。我们证明了使用 Caputo 的分数梯度定义，Weiner 解保持不变。所提出的 FoCLMS 算法的收敛分析针对瞬态和稳态情况进行了介绍。因此，导出了学习曲线和稳态 EMSE 的表达式。我们的理论发展得到了模拟实验的验证。提供了广泛的模拟来研究所有可能的场景：具有负权重和真实输入数据的通道，