Design of Digital Differentiator Using the \(L_1\)-Method and Swarm Intelligence-Based Optimization Algorithms
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In this paper, a novel approach to design the digital linear-phase finite impulse response (FIR) differentiator is introduced. First, the differentiator design problem is formulated using the \(L_1\)-method. Then, the \(L_1\) optimality criterion is applied using the Bat algorithm (BA) and Particle swarm optimization (PSO) to further optimize the differentiator design. A novel fitness function is developed based on the \(L_1\)-error norm which is unique and is liable to produce a flat response. These techniques are developed in order to minimize the non-differentiable fitness function. Finally, the simulation results have been presented for 5th-, 7th- and 11th-order FIR differentiator using the \(L_1\)-method, PSO-\(L_1\) and BA-\(L_1\). The magnitude response of the designed differentiators is analyzed for different frequency bands on the basis of relative magnitude error computed with respect to the ideal response. All the reported techniques contribute toward superior results, when compared with the traditional gradient-based optimizations, such as the window method, minimax and least-squares approach. In addition, the \(L_1\)-method yields better results for higher-order designs. Furthermore, the proposed designs are tested on two input signals for their efficient response.
KeywordsFinite impulse response \(L_1\)-error criterion Digital differentiator Bat algorithm Particle swarm optimization
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