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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]

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]

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]

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 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]

Recently, the Taylor–Fourier transform (TFT) was proposed to analyze the spectrum of signals with oscillating harmonics. The coefficients of this linear transformation were obtained through the calculation of the pseudoinverse matrix, which provides the classical solution to the normal equations of the least-squares (LS) approximation. This paper presents a filtering design technique that obtains the coefficients of the filters at each harmonic by imposing the maximally flat conditions to the polynomials defining their frequency responses. This condition can be used to solve the LS problem at each particular harmonic frequency, without the need of obtaining the whole set, as in the classical pseudoinverse solution. In addition, the filter passband central frequency can follow the fluctuations of the fundamental frequency. Besides, the method offers a reduction of the computational burden of the pseudoinverse solution. An implementation of the proposed estimator as an adaptive algorithm using its own instantaneous frequency estimate to relocate its bands is shown, and several tests are used to compare its performance with that of the ordinary TFT. [ABSTRACT FROM AUTHOR]

New IIR digital differintegrators (differentiator and integrator) with very minor absolute relative errors are presented in this paper. The digital integrator is designed by interpolating some of the existing integrators. The optimum value of the interpolation ratio is obtained through linear programming optimization. Subsequently, by modifying the transfer function of the proposed integrator appropriately, new digital differentiator is obtained. Simulation results demonstrate that the proposed differintegrator are a more accurate approximation of ideal ones, than the existing differintegrators. Furthermore, the proposed differentiator has been tested in an image processing application. Edges characterize boundaries and are, therefore, a problem of fundamental importance in image processing. For comparison purpose Prewitt, Sobel, Roberts, Canny, Laplacian of Gaussian (LOG), Zerocross operators were used and their results are displayed. The results of edge detection by some of the existing differentiators are also provided. The simulation results have shown the superiority of the proposed approach over existing ones. [ABSTRACT FROM AUTHOR]

In this paper, the one-dimension digital fractional-order Savitzky-Golay differentiator (DFOSGD), which generalizes the Savitzky-Golay filter from the integer order to the fractional order, is introduced and extended to 2-D by a group of direction operators. Then, a new image-enhancing algorithm is proposed based on the 2-D DFOSGD, and an unsupervised optimization algorithm is proposed for choosing the fractional-order parameter. Four numerical experiments are used to assess the performance of 2-D DFOSGD-based image-enhancing algorithm, and the results demonstrate its validity. [ABSTRACT FROM AUTHOR]

Novel designs of third-order recursive wideband digital integrator and differentiator are presented. The integrator is obtained by interpolating two digital integration techniques, the Schneider-Kaneshige-Groutage (SKG) rule and the Trapezoidal rule. The differentiator is obtained by inverting the transfer function of the designed integrator after necessary modifications. The proposed integrator and differentiator approximate their ideal counterparts with absolute magnitude errors less than 0.02 and 0.14, respectively, over the entire frequency spectrum. Their performance compares favorably with existing integrators and differentiators. [ABSTRACT FROM AUTHOR]

This brief proposes a design method for a digital fractional order Savitzky-Golay differentiator (DFOSGD), which generalizes the Savitzky-Golay filter from the integer order to the fractional order for estimating the fractional order derivative of the contaminated signal. The proposed method calculates the moving window's weights using the polynomial least-squares method and the Riemann-Liouville fractional order derivative definition, and then computes the fractional order derivative of the given signal using the convolution between the weights and the signal, instead of the complex mathematical deduction. Hence, the computation time is greatly improved. Frequency-domain analysis reveals that the proposed differentiator is essentially a fractional order low-pass filter. Experiments demonstrate that the proposed DFOSGD can accurately estimate the fractional order derivatives of both noise-free signal and contaminated signal. [ABSTRACT FROM PUBLISHER]

In this correspondence, we propose a fractional bilinear transform used in analog-to-digital (A/D) conversion. This transfer function is derived by means of fractional delay filters. According to this transformation, first-order differentiator can be easily designed. Next, some design examples are illustrated to show the linear frequency mapping property when doing A/D conversion. Finally, the designed tow-order digitat differentiator is compared with the existing differentiators. [ABSTRACT FROM AUTHOR]

This brief presents a general theory of the Newton-Cotes digital integrators which is derived by applying the z-transform technique to the closed-form Newton-Cotes integration formula. Based on this developed theory, a new wideband third-order trapezoidal digital integrator is found to be a class of trapezoidal digital integrators. The novel wideband third-order trapezoidal integrator accurately approximates the ideal integrator over the whole Nyquist frequency range and compares favourably with existing integrators. Based on the designed wideband third-order trapezoidal integrator, a new wideband digital differentiator is designed, which approximates the ideal differentiator reasonably well over the whole Nyquist frequency range and compares favourably with existing differentiators. The low orders and high accuracies of the novel wideband trapezoidal integrator and the new wideband differentiator make them attractive for real-time applications. [ABSTRACT FROM AUTHOR]