This paper presents a new efficient method for implementing the Hilbert transform using an all-pass filter, based on fractional derivatives (FDs) and swarm optimization. In the proposed method, the squared error difference between the desired and designed responses of a filter is minimized. FDs are introduced to achieve higher accuracy at the reference frequency ( ω0 ), which helps to reduce the overall phase error. In this paper, two approaches are used for finding the appropriate values of the FDs and reference frequencies. In the first approach, these values are estimated from a series of experiments, which require more computation time but produce less accurate results. These experiments, however, justify the behavior of the error function, with respect to the FD and ω0 , as a multimodal and nonconvex problem. In the second approach, a variant of the swarm-intelligence-based multimodal search space technique, known as the constraint-factor particle swarm optimization, is exploited for finding the suitable values for the FD and ω0 . The performance of the proposed FD-based method is measured in terms of fidelity aspects, such as the maximum phase error, total squared phase error, maximum group delay error, and total squared group delay error. The FD-based approach is found to reduce the total phase error by 57% by exploiting only two FDs.

中文翻译：

---

基于分数导数和群优化的希尔伯特变换设计。

本文提出了一种基于分数导数（FDs）和群体优化的全通滤波器实现希尔伯特变换的有效方法。在所提出的方法中，滤波器的期望响应与设计响应之间的平方误差差被最小化。引入FD可以在参考频率（ω0）上实现更高的精度，这有助于降低整体相位误差。在本文中，使用两种方法来找到适当的FD值和参考频率。在第一种方法中，这些值是从一系列实验中估算出来的，这些实验需要更多的计算时间，但产生的结果却不那么准确。然而，这些实验证明了误差函数相对于FD和ω0的行为是多峰和非凸问题。在第二种方法中 利用基于群智能的多峰搜索空间技术的一种变体，即约束因子粒子群优化，来找到FD和ω0的合适值。所提出的基于FD的方法的性能是从保真度方面进行衡量的，例如最大相位误差，总平方相位误差，最大群时延误差和总平方群时延误差。发现仅使用两个FD，基于FD的方法可将总相位误差降低57％。最大群延迟误差和总平方群延迟误差。发现仅使用两个FD，基于FD的方法可将总相位误差降低57％。最大群延迟误差和总平方群延迟误差。发现仅使用两个FD，基于FD的方法可将总相位误差降低57％。