In this work, the noise performances of the fractal-fractional electrical circuits have been addressed. The nonlocal fractal calculus has been adopted as our mathematical basis. The fractal time component has also been included for the physical measurability of electrical quantities. The derivations of crucial stochastic parameters of circuit responses, which determine their noise performances, have been performed. Numerical simulations have also been conducted where the influences of Hausdorff dimension of the fractal set, orders of fractal-fractional reactive components, and other parameters on the noise performances have been studied. Regardless to any specific circuit, we have found that the noise performances can be improved by increasing the orders of fractal-fractional reactive components. The optimum Hausdorff dimensions, which the best noise performances can be achieved given the orders of fractal-fractional reactive components, have also been calculated. The results proposed in this work serve as the foundation for understanding noise in fractal-fractional electrical circuits and can be extensively applied to large-scaled circuits, for example, the infinite circuit networks and so forth.

中文翻译：

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分形分数电路的噪声性能

在这项工作中，分形分数电路的噪声性能已经得到解决。非局部分形微积分已被用作我们的数学基础。分形时间分量也包含在电量的物理可测量性中。已经执行了决定其噪声性能的电路响应的关键随机参数的推导。还进行了数值模拟，研究了分形集的 Hausdorff 维数、分形分数阶电抗分量的阶数以及其他参数对噪声性能的影响。无论对于任何特定的电路，我们都发现可以通过增加分形-分数型电抗元件的阶数来改善噪声性能。最佳豪斯多夫维度，还计算了在给定分形分数电抗分量的阶数的情况下可以实现的最佳噪声性能。这项工作中提出的结果是理解分形分数电路中噪声的基础，可以广泛应用于大规模电路，例如无限电路网络等。