Weighted Least Square Design Technique for Hilbert Transformer using Fractional Derivative

In this paper, a new design method for realization of Hilbert transformer (HT) by an all-pass infinite impulse response (IIR) filter is proposed. The design problem is devised as minimization of the weighted least squared errors of the phase response in the frequency domain, where the phase response of denominator polynomial of HT acts as a weighting function. The least squared design techniques usually suffer from large errors at the band edge frequencies. To resolve this issue, our aim is to have the phase response of the all-pass IIR filter closely matched with the desired phase response of HT. The approximation with desired phase response of HT has been achieved by using fractional derivatives constraints (FDCs). In our experiments, it is observed that the design problem using single fractional derivative (FD) has a multimodal behavior in nature. Moreover, highly precise value of order of FD is required, whose exploration task is computationally expensive. Therefore, recently developed heuristic search techniques, also known as swarm-based optimization techniques (SOTs), such as particle swarm optimization (PSO), and its variants, cuckoo search (CS) algorithm, and artificial bee colony (ABC) algorithm, are used for finding the required values. However, these methods are capable of solving non-differentiable and multimodal problems due to their multi-dimensional randomly guided search mechanism. The proposed methodology has gained 59% improvement in phase approximation error as compared to conventional reported techniques.