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Digital differentiators enable the computation of the derivative of a continuous-time signal at discrete time instances, and they are used in many signal processing applications. This paper derives a unified filter order estimate for digital differentiators that are realized with linear-phase finite-length impulse response filters and designed in the minimax sense. The estimate is useful at the high-level system design when assessing the implementation complexity and it enables fewer designs when finding the minimal filter order required to satisfy a prescribed tolerable approximation error. The proposed unified estimate covers both wideband and lowpass differentiators of integer degrees up to ten. Furthermore, degree-individual filter order estimates are derived which improve and extend previous results. The performance of both the unified and degree-individual order estimates is evaluated through simulation examples and compared with previous estimates. [ABSTRACT FROM AUTHOR]

Inherent control delays and changeable grid operational conditions pose significant challenges to the stability of the Modular multilevel converter-based high voltage direct current (MMC-HVDC) system, inducing the risk of high-frequency resonances (HFR). This paper proposes a passivity enhancement based on the second-order digital differentiator (DD) and lead-lag compensator, ensuring reliable and robust HFR suppression performance even under the resonance frequency drift and fault earthing scenes. First, the passivity-based stability analysis is provided based on the system output impedance model, elucidating the HFR mechanisms and frequency drift problems. Second, the proposed passivity enhancement scheme's control structure and impedance reshaping principle are analyzed to verify resonance suppression ability. Third, the control parameters tuning process and direct realization of differentiator are detailed, reinforcing the stable margin and harmonic rejection ability. Finally, several case studies and simulation results are provided to validate the effectiveness of the proposed passivity-based HFR suppression scheme. • The proposed scheme integrates a second-order digital differentiator plus a lead-lag compensator. • Superior passivity and harmonic rejection ability are ensured, with frequency drift risk-solving ability. • Parameters tuning and differentiator discretization realization are elaborated. • Large phase lag and noise amplification problems are solved by impedance reshaping. [ABSTRACT FROM AUTHOR]

Design of full-band infinite impulse response (IIR) digital differentiator (DD) and digital integrator (DI) based on the Lk-norm and the min-max optimality criteria have been demonstrated in literature. However, the designed IIR DDs and DIs using these criteria do not have linear phase or have unsatisfactory magnitude response. In this paper, a novel approach to design, optimize and improve the magnitude response and the linearity of the phase response of IIR DD and DI using the cuckoo search (CS) optimization algorithm is proposed. The proposed approach is based on a varied Lk-norm where $k$ linearly increases with the number of the frequency samples between the values 1 and 2. The design and optimization of DDs and DIs are carried out using the CS optimization method due to its simplicity, efficiency, and robustness in solving general multidimensional optimization problems. The performance of the designed DDs and DIs using the proposed approach is compared with designs based on different criteria and algorithms available in literature. The comparison shows that the designed DDs and DIs based on the proposed approach have better phase responses and better or at least comparable magnitude responses compared to those of DDs and DIs designed using other methods in the literature.

In this paper, a modified Finite Impulse Response based linear Pass integrator for centered frequency, ranging between 0.1 π to 0.9 π has been realized. Both the cases have been considered i.e. for what values the phase response is of use and where the phase response has zero value. An iterative formula has been used to calculate the weights depending upon the Transfer Functions, and applying differentiation method. A flat output approximation for the desired frequency ω has been applied for which the results overlap with the ideal integrator. Performance comparison of the proposed integrator has been done with the previous one and relative percentage errors have been observed for both cases implemented. Graphical analysis has also been carried out for frequency responses having degree greater than one (i.e. k = 2, 3, 4) for both cases of proposed integrator and compared with the ideal integrator's response. [ABSTRACT FROM AUTHOR]

In this paper, a novel design of third and fifth order differentiator based on lattice wave digital filter (LWDF), established on optimizing L_1-error approximation function using cuckoo search algorithm (CSA) is proposed. We present a novel realization of minimum multiplier differentiator using LWD structure leading to requirement of optimizing only N coefficients for Nth order differentiator. The gamma coefficients of lattice wave digital differentiator (LWDD) are computed by minimizing the L_1-norm fitness function leading to a flat response. The superiority of the proposed LWDD is evident by comparing it with other differentiators mentioned in the literature. The magnitude response of the designed LWDD is found to be of high accuracy with flat response in a wide frequency range. The simulation and statistical results validates that the designed minimum multiplier LWDD circumvents the existing one in terms of minimum absolute magnitude error, mean relative error (dB) and efficient structural realization, thereby making the proposed LWDD a promising approach to digital differentiator design.

A fractional-order digital differentiator is employed for the calculation of a time-derivative of the applied signal. In the recent few decades, this particular concept of a fractional derivative has been gaining a lot of attention in various applications concerning engineering, technology and science that includes image processing along with automatic control. Once there has been an effective use for this continuous-time Fractional-Order Differentiator (FOD), the trend in its research is primarily toward using a discrete-time fractional differentiator. All these conventional techniques tend to make use of a unimodal function for approximating an ideal FOD. For these techniques, there is a minimization of the fitness function that is accomplished by the algorithms which are based on the gradient. The fractional-order circuits along with their systems include an emerging area that has a high level of potential in aspects such as the biomedical instrumentation, control or signal processing. A digital differentiator is a tool that is extremely helpful in the determination and estimation of time derivatives of any given signal. Irrespective of the actual type of filter chosen (the Finite Impulse Response (FIR) or the Infinite-Length Impulse Response (IIR)), it is critical to bring down the complexity of computation needed for the implementation of the filter for a certain bandwidth and error of approximation. A metaheuristic algorithm normally has some advantages in the solving of problems which are Non-Deterministic Polynomial (NP)-hard. The Shuffled Frog Leaping Algorithm (SFLA) has been a new heuristic algorithm proposed in this work for the determination of optimal coefficients of the problem of FIR-FOD. A design for fractional-order-based digital differentiator is not a very important topic in research and signal processing. [ABSTRACT FROM AUTHOR]

• If the frequency is contemplated in models, the estimates in signals are better. • The importance to increase and decrease the sampling rate in dynamic signals. • Update parameters in models achieve better performance in dynamic signals. This paper presents a method to obtain good synchronous and instantaneous estimates in position, velocity and acceleration; using a position measurement only. The proposed method shows better estimates in position, velocity and acceleration than other methods such as nonlinear tracking differentiator (TD), extended state estimation (ESO) and digital differentiator based on Taylor series (DDBTS). The proposed method allows for an increased accuracy of estimates by generating a feedback frequency obtained to the position measurement signals. The model and method proposed in this paper reduce the error when the measurement signal presents a change in frequency. The model update and adjusts simultaneously to changes in sample rates using a feedback frequency estimate. The proposed method was validated with the QNET DC Motor Control Trained (DCMCT). Since the method requires the measurement position only from an encoder, it eliminates the need for more sensors for velocity and acceleration, thus begin less costly. [ABSTRACT FROM AUTHOR]

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 L1-method. Then, the L1 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 L1-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 L1-method, PSO-L1 and BA-L1. 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 L1-method yields better results for higher-order designs. Furthermore, the proposed designs are tested on two input signals for their efficient response. [ABSTRACT FROM AUTHOR]

Heart rate variability (HRV) analysis is considered as a preliminary diagnosis method to check the cardiac health of the human heart. The reliability of the HRV analysis system solely depends on the accuracy of the QRS complex detector. Hence, in this paper, an optimally designed digital differentiator (DD) for precise detection of QRS complex is proposed. The proposed DD is designed by using an efficient evolutionary optimization technique called gases Brownian motion optimization (GBMO) algorithm and is used in the preprocessing stage of the QRS detector. In GBMO algorithm, a balanced trade-off is maintained between both the exploration and the exploitation phases to find the global optimum solution. The electrocardiogram signal is preprocessed by using the proposed DD to generate the feature signals corresponding to the R-peaks only. The detection technique utilizes the principle of Hilbert transform and zeroes crossing detection. The proposed approach is verified against all the first channel records of MIT/BIH arrhythmia database by considering the standard QRS detection performance metrics and produces a sensitivity (Se) of 99.92%, positive predictivity (+P) of 99.92%, detection error rate (DER) of 0.1562%, QRS detection rate of 99.92%, accuracy (Acc) of 99.84%, and F score of 0.9992%. With respect to the standard performance metrics, the proposed QRS detector outperforms all the recently reported QRS detection techniques. [ABSTRACT FROM AUTHOR]

In this paper, a recently proposed metaheuristic optimization technique called hybrid flower pollination algorithm (HFPA) is applied to design wideband infinite impulse response digital differentiators (DDs) and digital integrators (DIs). In recent years, benchmark nature-inspired optimization algorithms such as particle swarm optimization (PSO), simulated annealing, and genetic algorithm have been employed for the design of wideband DDs and DIs. However, individually, these algorithms show major drawbacks such as premature convergence, thus leading to a sub-optimal solution. HFPA, however, is a hybrid approach which combines the efficient exploitation and exploration capabilities of two different metaheuristics, namely PSO and flower pollination algorithm (FPA), respectively. The HFPA-based designs have been compared with real-coded genetic algorithm, PSO, differential evolution, success-history-based adaptive differential evolution with linear population size reduction (L-SHADE), self-adaptive differential evolution (jDE), and FPA-based designs with respect to the solution quality, robustness, convergence, and optimization time. Simulation results demonstrate that among all the algorithms, the HFPA-based designs consistently achieve superior performances in the least number of function evaluations. Exhaustive experimentations are conducted to determine the best values of the control parameters of HFPA for the optimal design of DDs and DIs. The proposed designs also outperform the recently reported designs based on non-optimal, classical, and nature-inspired optimization approaches in terms of magnitude response. The lower orders of the proposed designs render them suitable for real-time applications. [ABSTRACT FROM AUTHOR]

This paper presents an efficient approach to design stable, wideband, and infinite impulse response digital integrators (DIs) and digital differentiators (DDs) of first, second, third, and fourth order using an evolutionary optimization algorithm called harmony search (HS). In recent years, although wideband DIs and DDs have been designed using metaheuristic optimization techniques such as simulated annealing, genetic algorithm, and particle swarm optimization (PSO), these algorithms lead to sub-optimal solutions because of stagnation and premature convergence. HS algorithm, however, promises an enhanced frequency response for DIs and DDs because of the better exploration and exploitation of the search space. Simulation results demonstrate the superiority of HS-based designs as compared with three well-known benchmark evolutionary optimization algorithms, namely real coded genetic algorithm (RGA), PSO, and differential evolution (DE) based designs by yielding the least values of different magnitude response error metrics. Parametric and non-parametric statistical hypothesis tests are also conducted to compare the consistency in the performance of HS-based DIs and DDs with those of the designs based on RGA, PSO, and DE. The proposed HS-based designs also outperform those of the designs based on both classical and evolutionary optimization approaches reported in recent literature in terms of the maximum absolute magnitude error metric. [ABSTRACT FROM AUTHOR]

In this paper, a novel design of third and fifth order differentiator based on lattice wave digital filter (LWDF), established on optimizing L1-error approximation function using cuckoo search algorithm (CSA) is proposed. We present a novel realization of minimum multiplier differentiator using LWD structure leading to requirement of optimizing only N coefficients for N th order differentiator. The γ coefficients of lattice wave digital differentiator (LWDD) are computed by minimizing the L1 -norm fitness function leading to a flat response. The superiority of the proposed LWDD is evident by comparing it with other differentiators mentioned in the literature. The magnitude response of the designed LWDD is found to be of high accuracy with flat response in a wide frequency range. The simulation and statistical results validates that the designed minimum multiplier LWDD circumvents the existing one in terms of minimum absolute magnitude error, mean relative error (dB) and efficient structural realization, thereby making the proposed LWDD a promising approach to digital differentiator design. [ABSTRACT FROM AUTHOR]

In this article, we propose the use of power function and least squares method for designing of a fractional order digital differentiator. The input signal is transformed into a power function by using Taylor series expansion, and its fractional derivative is computed using the Grunwald–Letnikov (G–L) definition. Next, the fractional order digital differentiator is modelled as a finite impulse response (FIR) system that yields fractional order derivative of the G–L type for a power function. The FIR system coefficients are obtained by using the least squares method. Two examples are used to demonstrate that the fractional derivative of the digital signals is computed by using the proposed technique. The results of the third and fourth examples reveal that the proposed technique gives superior performance in comparison with the existing techniques. [ABSTRACT FROM PUBLISHER]

This paper presents a novel metaheuristic algorithm called Interior Search Algorithm (ISA), which is applied for digital differentiator design problem. ISA is based on the principles of aesthetic techniques commonly used in interior design and decoration. ISA has a very quick convergence rate and only one control parameter. The approach presented here has alleviated from the problems of premature convergence, stagnation and revisiting of the same solution over and over again, which is common in other optimization techniques. Statistical and simulation results have been compared with already existing differentiator design methods such as segment rule, genetic algorithm (GA), simulated annealing (SA), pole zero method (PZ) and particle swarm optimization (PSO). The results affirm that the proposed method outperforms its counterparts in terms of absolute magnitude error and phase error. [ABSTRACT FROM AUTHOR]