Digital Differentiators Research Articles

A shear wave elastography method based on 2D Savitzky-Golay digital differentiator

A shear wave elastography method based on 2D Savitzky-Golay digital differentiator

Direct Realization of Digital Differentiators in Discrete Domain for Active Damping of LCL -Type Grid-Connected Inverter

To damp the LCL-filter resonance in a grid-connected inverter, the feedback of capacitor current is usually adopted, and it can be replaced by the feedback of capacitor voltage as a low-cost solution, if an accurate digital differentiator can be made. The best way for realizing such a differentiator has so far proved to be an indirect nonideal generalized integrator (GI). As a simple alternative, this paper proposes two digital differentiators, which are directly developed in the discrete domain. They are a first-order differentiator based on backward Euler plus digital lead compensator and a second-order differentiator based on Tustin plus digital notch filter. The basic idea of the proposed methods is to correct their frequency responses to match the ideal differentiator with embedded digital filters. It is shown that the proposed differentiators exhibit the same derivative performance as the nonideal-GI differentiator, and they are more attractive for digital implementations due to their direct discrete natures, compact expressions, and easy algebraic manipulations. In particular, the proposed first-order differentiator is most competitive for its general representation and simplest implementation. Finally, a 12-kW prototype is built, and experiments are performed to verify the theoretical analysis.

Low-Delay Band-Pass Maximally Flat FIR Digital Differentiators

The present paper describes a closed-form transfer function of low-delay band-pass maximally flat FIR digital differentiators. Band-pass maximally flat FIR digital differentiators provide extremely high-accuracy differentiation around the center frequency which is adjusted arbitrarily. At the same time, they can reduce noise in the frequency except around the center frequency. However, the conventional method for designing band-pass maximally flat FIR digital differentiators requires linear phase characteristics. In contrast, the proposed method can realize low-delay characteristics as well as linear phase characteristics and, therefore, is a general expression of band-pass maximally flat FIR digital differentiators. The proposed transfer function is achieved as the sum of two maximally flat complex FIR digital differentiators, the coefficients of which are complex conjugates of each other. The transfer functions of these complex differentiators are derived as closed-form solutions, so that the proposed transfer function is also described as a closed-form solution. Through design examples, the effectiveness of the proposed method is confirmed.

Model-free fractional order differentiator based on fractional order Jacobi orthonormal functions

The aim of this paper is to design an algebraic and robust fractional order differentiator to estimate both the Riemann–Liouville and the Caputo fractional derivatives with an arbitrary order of an unknown signal in noisy environment, without knowing the model defining the signal. For this purpose, a new class of fractional order Jacobi orthonormal functions is firstly introduced. Secondly, the truncated fractional order Jacobi orthonormal series expansion is applied to filter the noisy signal, whose fractional derivative is used to estimate the desired one. Thus, the obtained differentiator is exactly given by an integral formula which depends on a set of design parameters. Thirdly, by applying the generalized Taylor's formula, some error analysis is provided. In particular, error bounds are given, which permit to study the design parameters' influence. Fourthly, a digital fractional order differentiator is deduced in discrete noisy case. Finally, by comparing with two existing fractional order differentiators, numerical results are given to illustrate the accuracy and the robustness of the proposed fractional order differentiator.

Modulating functions-based fractional order differentiator for fractional order linear systems with a biased output

This paper aims at designing a non-asymptotic fractional order differentiator for a class of fractional order linear systems with zero initial conditions, where the fractional orders can be commensurate or non-commensurate, and the output is corrupted by a non zero-mean noise. Firstly, a set of fractional differential equations are constructed, based on which the modulating functions method is applied, such that the fractional derivative with an arbitrary order of the output can be exactly given by an algebraic formula using a recursive way. In particular, this arbitrary order can be different to the fractional orders defining the considered system. Unlike the improper integral in the definition of the fractional derivatives, the obtained formula can be given by proper integrals by choosing appropriate modulating functions. Moreover, by taking an additional condition on the modulating functions, the term biasing the output can be eliminated in the obtained formula. After constructing the needed modulating functions, a digital fractional order differentiator is proposed in discrete noisy case with some error analysis. Finally, the efficiency and the robustness of the proposed fractional order differentiator is shown in numerical results.

Design of Flat Halfband Filters With Sharp Transition and Differentiators through Constrained Quadratic Optimization

An alternative method for the design of typeI Halfband FIR filters with flat magnitude and narrowtransition bands is presented. The methodology shownis based on the derivation of a quadratic programmingproblem with inequality constraints, which representsa set of linear equations obtained from flat and ripplerestrictions imposed over the frequency response of thefilter. The design is based on maximally flat constraints.The obtained filters have narrow transition bands ascompared to those presented in other maximally flatbased designs. The proposed method is not ripplefree as it does not take into account all the maximallyflat restrictions. Then, control of side lobes andtransition band is performed using a weighting matrixand inequality constraints as side lobes bounds. Thedesign of type IV FIR digital differentiators throughthe proposed method is also shown. Examples ofdesign, which compare the proposed method with otherspresented in the literature, are provided to verify theeffectiveness of the proposed method.

All-digital gated ring oscillator $$\Delta \Sigma$$ Δ Σ modulators

This paper presents all-digital time-mode $$\Delta \Sigma$$ modulators. The proposed modulators consist of a voltage-to-time integrator, a seven-stage gated ring oscillator functioning as a 3-bit quantizer, and seven digital differentiators. A detailed analysis of the nonlinear characteristics of the modulators is provided. Designed in IBM 130 nm 1.2 V CMOS technology with a 100 mV 100 kHz sinusoidal input and a 4.4 MHz frequency clock, the first-order modulator provides 47 dB SNR over 0–150 KHz bandwidth while consuming 1.1 mW while the second-order modulator provides 55 dB SNR over the same bandwidth while consuming consumes 1.45 mW.

A 6-b, 800-MS/s, 3.62-mW Nyquist Rate AC-Coupled VCO-Based ADC in 65-nm CMOS

A Nyquist voltage-controlled oscillator (VCO)-based analog-to-digital converter (ADC) architecture is proposed for ac-coupled systems that are commonly used in high-speed wireline and wireless communications. The proposed ADC utilizes a built-in high-pass filter as an analog differentiator, replacing the digital differentiator in conventional oversampling VCO-based ADCs. As a result, it avoids quantization noise shaping and achieves wideband Nyquist operation, inherited first-order antialiasing filtering, and improved VCO linearity without calibration. Analyses and simulations of various circuit nonidealities’ effects on the proposed ADC architecture are also provided. The ADC prototype achieves a peak SNDR of 34 dB and an SFDR of 50 dB with over 400-MHz input bandwidth and a sampling rate of 800 MS/s. It occupies an active area of 0.01 mm2 and consumes 3.62 mW in 65-nm CMOS.

Optimal design of wideband digital integrators and differentiators using hybrid flower pollination algorithm

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.

Optimal and accurate design of fractional‐order digital differentiator – an evolutionary approach

This study deals with the implementation of highly accurate, stable, minimum phase, and wideband fractional-order digital differentiators (FODDs) in terms of infinite impulse response filters using an efficient evolutionary optimisation algorithm called adaptive Gbest-guided gravitational search algorithm (GGSA). Performance evaluation of GGSA as compared with real coded genetic algorithm (RGA), particle swarm optimisation (PSO), and differential evolution (DE) based designs are carried out in terms of different magnitude and phase response error metrics, solution quality reliability, and convergence speed. Simulation results clearly demonstrate that GGSA significantly outperforms RGA, PSO, and DE in consistently achieving the most accurate FODDs in a computationally efficient manner. The proposed FODDs also significantly outperform all state-of-the-art designs in terms of magnitude responses.

Accurate Differentiation for Improving the Flicker Measurement in Wind Turbines

This paper analyzes the effect of digital differentiation on the measurement of flicker emission from grid-connected wind turbines (WTs). The flicker measurement procedure is defined in the IEC 61400-21 standard, which includes the estimation of a fictitious grid for the measurement of voltage fluctuations exclusively generated by a WT. The study focuses on two aspects of the implementation of the derivative of the line current that affect the estimation of the fictitious grid, namely the bandwidth and the phase response of the digital differentiator. The experiments were performed under two different frameworks, by using simulated simple waveforms and voltage and current recordings from an actual WT. The work cautions about the adverse effect of the phase response of the even-length finite impulse-response differentiators. It also recommends implementing digital differentiators with an accurate bandwidth to derive potential spectral components in the current in the whole bandwidth required by the procedure. This aim can be achieved by increasing either the length of the filter or the sampling rate of the processing. The conclusions should help in accurately implementing the procedure and reducing the deviations found in the flicker emission measurements for grid-connected WTs.

Optimal design of L 1 ‐norm based IIR digital differentiators and integrators using the bat algorithm

In this study, new infinite impulse response (IIR) digital differentiators of second, third and fourth orders based on optimising the L 1-error fitness function using the bat algorithm (BA) are proposed. The coefficients of numerator and denominator of the differentiators are computed by minimising the L 1-norm of the error fitness function along with imposing the constraint for the location of poles and zeros within the unit circle to ensure minimum phase. The transfer function of the differentiators are inverted and transformed into the digital integrators of the same orders. The results obtained for the solutions by the proposed L 1-based BA (L 1-BA) are superior to the designs using other techniques such as particle swarm optimisation and real-coded genetic algorithm. The designed optimal differentiator and integrator are compared with the existing models and are found to be of high accuracy and flatness in a wide frequency range along with minimum absolute magnitude error. The mean relative error (dB) is obtained as low as −67 dB and −73 dB for the proposed differentiators and integrators, respectively.

Non-asymptotic fractional order differentiator for a class of fractional order linear systems

This paper aims at designing a non-asymptotic fractional order differentiator for a class of fractional order linear systems to estimate the Riemann–Liouville fractional derivatives of the output in discrete noisy environment. The adopted method is a recent algebraic method originally introduced by Fliess and Sira-Ramirez. Firstly, the fractional derivative of the output of an arbitrary order is exactly given by a new algebraic formula in continuous noise free case without knowing the initial conditions of the considered system. Secondly, a digital fractional order differentiator is introduced in discrete noisy cases, which can provide robust estimations in finite-time. Then, some error analysis is given, where an error bound useful for the selection of the design parameter is provided. Finally, numerical examples illustrate the accuracy and the robustness of the proposed fractional order differentiator.

Enhanced colliding bodies optimisation-based optimal design of wideband digital integrators and differentiators

This paper presents an efficient approach to determine the optimal set of coefficients for the design of wideband infinite impulse response (IIR) digital integrators (DIs) and digital differentiators (DDs) of first, second, third, and fourth order, meeting the accurate magnitude response specification, using a recently proposed evolutionary optimisation algorithm called enhanced colliding bodies optimisation (ECBO). To demonstrate the effectiveness of the proposed approach, the results of the ECBO-based designs have been compared with those of eight other nature-inspired metaheuristic optimisation algorithms. Parametric and non-parametric statistical hypothesis tests are conducted to validate the consistency of the performance of the ECBO-based DIs and DDs. Simulation results demonstrate that ECBO-based designs achieve the least absolute magnitude error and demonstrate a competitive group delay response of the designed DIs and DDs of different orders as compared with the designs based on the competing algorithms. The proposed DIs and DDs also outperform those of the design approaches published in literature and achieve the best responses in terms of the maximum absolute magnitude error over a wide frequency range.

Enhanced colliding bodies optimisation-based optimal design of wideband digital integrators and differentiators

This paper presents an efficient approach to determine the optimal set of coefficients for the design of wideband infinite impulse response (IIR) digital integrators (DIs) and digital differentiators (DDs) of first, second, third, and fourth order, meeting the accurate magnitude response specification, using a recently proposed evolutionary optimisation algorithm called enhanced colliding bodies optimisation (ECBO). To demonstrate the effectiveness of the proposed approach, the results of the ECBO-based designs have been compared with those of eight other nature-inspired metaheuristic optimisation algorithms. Parametric and non-parametric statistical hypothesis tests are conducted to validate the consistency of the performance of the ECBO-based DIs and DDs. Simulation results demonstrate that ECBO-based designs achieve the least absolute magnitude error and demonstrate a competitive group delay response of the designed DIs and DDs of different orders as compared with the designs based on the competing algorithms. The proposed DIs and DDs also outperform those of the design approaches published in literature and achieve the best responses in terms of the maximum absolute magnitude error over a wide frequency range.

Design Method for Low-Delay Maximally Flat FIR Digital Differentiators with Variable Stopbands Obtained by Minimizing <i>L<sub>p</sub></i> Norm

Design Method for Low-Delay Maximally Flat FIR Digital Differentiators with Variable Stopbands Obtained by Minimizing <i>L<sub>p</sub></i> Norm

Design and Implementation of Second order Microwave Differentiator

Objective: To design and implementation of second order digital differentiator at microwave frequencies. Method: We proposed simple accurate formulations for DiScrete-time Processing (DSP) of second order digital differentiators in the Z-domain. The transmission lines are constructed by using chain scattering parameters to map with second order digital differentiator in order work at microwave frequencies. So the developed equations were instrumented by using micro strip lines to form a second order microwave differentiator and simulated by using Advanced Design System (ADS) software.Findings: A simplified mathematical expressions are derived to design the digital differentiator at microwave frequencies. The designed second order microwave differentiator magnitude response is similar to second order digital differentiator. Applications: It can be used in target detection of marine radar.

Optimal design of wideband digital integrators and differentiators using harmony search algorithm

AbstractThis 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.

Design of minimum multiplier fractional order differentiator based on lattice wave digital filter

In this paper, a novel design of fractional order differentiator (FOD) based on lattice wave digital filter (LWDF) is proposed which requires minimum number of multiplier for its structural realization. Firstly, the FOD design problem is formulated as an optimization problem using the transfer function of lattice wave digital filter. Then, three optimization algorithms, namely, genetic algorithm (GA), particle swarm optimization (PSO) and cuckoo search algorithm (CSA) are applied to determine the optimal LWDF coefficients. The realization of FOD using LWD structure increases the design accuracy, as only N number of coefficients are to be optimized for Nth order FOD. Finally, two design examples of 3rd and 5th order lattice wave digital fractional order differentiator (LWDFOD) are demonstrated to justify the design accuracy. The performance analysis of the proposed design is carried out based on magnitude response, absolute magnitude error (dB), root mean square (RMS) magnitude error, arithmetic complexity, convergence profile and computation time. Simulation results are attained to show the comparison of the proposed LWDFOD with the published works and it is observed that an improvement of 29% is obtained in the proposed design. The proposed LWDFOD approximates the ideal FOD and surpasses the existing ones reasonably well in mid and high frequency range, thereby making the proposed LWDFOD a promising technique for the design of digital FODs.

Design of fractional order differentiator using type-III and type-IV discrete cosine transform

In this paper, an interpolation method based on discrete cosine transform (DCT) is employed for digital finite impulse response-fractional order differentiator (FIR-FOD) design. Here, a fractional order digital differentiator is modeled as finite impulse response (FIR) system to get an optimized frequency response that approximates the ideal response of a fractional order differentiator. Next, DCT-III and DCT-IV are utilized to determine the filter coefficients of FIR filter that compute the Fractional derivative of a given signal. To improve the frequency response of the proposed FIR-FOD, the filter coefficients are further modified using windows. Several design examples are presented to demonstrate the superiority of the proposed method. The simulation results have also been compared with the existing FIR-FOD design methods such as DFT interpolation, radial basis function (RBF) interpolation, DCT-II interpolation and DST interpolation methods. The result reveals that the proposed FIR-FOD design technique using DCT-III and DCT-IV outperforms DFT interpolation, RBF interpolation, DCT-II interpolation and DST interpolation methods in terms of magnitude error.