Hilbert Transform Design Based on Fractional Derivatives and Swarm Optimization

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 (<inline-formula> <tex-math notation="LaTeX">$\boldsymbol {\omega }_{\mathbf {0}}$ </tex-math></inline-formula>), 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 <inline-formula> <tex-math notation="LaTeX">$\boldsymbol {\omega }_{\mathbf {0}}$ </tex-math></inline-formula>, 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 <inline-formula> <tex-math notation="LaTeX">$\boldsymbol {\omega }_{\mathbf {0}}$ </tex-math></inline-formula>. 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.