Optimal design of zero-phase digital Riesz FIR fractional-order differentiator

Abstract

This article introduces an optimal design of finite impulse response-type Riesz fractional-order digital differentiator (FODD) of orders p = 0.3, 0.5, and 0.75. A very little study on Riesz-based differentiator has been reported so far. Moreover, none of the published Riesz FODD-based literature has studied the optimal design of Riesz FODD, which is explored in this article. In the reported literature, the Riesz FODD is realized by adopting conventional mathematical approaches, which are not efficient for the Riesz FODD-type design problem where the nature of the search landscape is multidimensional, multimodal, non-uniform, and nonlinear. To solve this problem, the design parameters of the proposed Riesz FODDs are estimated by using three independent metaheuristic optimization algorithms called salp swarm algorithm (SSA), gases Brownian motion optimization (GBMO), and gravitational search algorithm (GSA). In SSA, the adaptive variation of the iteration-dependent parameters helps the search agents not only to avoid the suboptimal solutions, but also ensures the faster convergence to reach the near-global optimal solution compared to GSA- and GBMO-based design approaches. The maximum phase error (MPE) and maximum absolute magnitude error (MAME) of the proposed SSA-based Riesz FODD are as low as 3.7119E−014 degree and − 9.6824 dB, respectively, for p = 0.3; 3.1523E−014 degree and − 15.6443 dB, respectively, for p = 0.5; and 3.5780E−014 degree and − 18.7525 dB, respectively, for p = 0.75. In terms of MPE and MAME, the proposed SSA-based Riesz FODD significantly outperforms the GSA- and GBMO-based Riesz FODD designs. The proposed SSA-based Riesz FODD of order p = 0.5 is also implemented on the TMS320C6713 digital signal processor, and its precise zero-phase performance is tested on real electrocardiogram (ECG) signal. Moreover, the proposed Riesz FODD is utilized in the ECG QRS complex identification system to demonstrate its real-time application.