Digital Differentiators Research Articles

Optimal design of 2-D FIR digital differentiator using $$L_1$$ L 1 -norm based cuckoo-search algorithm

In this article, an optimal design of two-dimensional finite impulse response digital differentiators (2-D FIR-DD) with quadrantally odd symmetric impulse response is presented. The design problem of 2-D FIR-DD is formulated as an optimization problem based on the $$L_1$$ -error fitness function. The novel error fitness function is based on the $$L_1$$ norm which is unique and is liable to produce a flat response. This design methodology incorporates advantages of $$L_1$$ -error approximating function and cuckoo-search algorithm (CSA) which is capable of attaining a global optimal solution. The optimized system coefficients are computed using $$L_1$$ -CSA and performance is measured in terms of magnitude response, phase response, absolute magnitude error and elapsed time. Simulation results have been compared with other optimization algorithms such as real-coded genetic algorithm and particle swarm optimization and it is observed that $$L_1$$ -CSA delivers optimal results for 2-D FIR-DD design problem. Further, performance of the $$L_1$$ -CSA based 2-D FIR-DD design is evaluated in terms of absolute magnitude error and algorithm execution time to demonstrate their effect with variation in the control parameters of CSA.

A background split algorithm for high speed pipelined ADCs

The non-ideal problem, include transfer function gain errors and timing offset between channels, seriously restrict pipelined analog-to-digital converters (ADCs) sampling rate upgrade. This letter analyzes these sources of errors and the effect they have on split architecture. Using Taylor approximation and the digital differentiator filter in a hybrid LMS algorithm, the new method does not need any additional analog circuitry, does not perturb the output samples, and does not require strictly well matched layout of signal path between two channels. Behavioral simulation results demonstrate that the proposed calibration technique can significantly improve signal-to-noise and distortion ratio and spurious free dynamic range performance from 63.8 to 79.8 dB and 65.7 to more than 90.0 dB, respectively, for a 13-bit 500 MS/s pipelined ADC.

Efficient Design of Digital FIR Differentiator using $L_1$-Method

In this paper, an efficient design of FIR digital differentiator using the L1-optimality criterion is proposed. We present a technique based on the modified Newton method to solve the design problem so that the optimal differentiator coefficients are obtained by minimizing the absolute error. The novel L1-error function leads to a flat response at low-frequencies. Extensive simulations are carried out to validate the proposed design. The superiority of the proposed design is evident by comparing it with other conventional design techniques such as, windowing, minimax and the least-squares approach.

Fractional Order Digital Differentiator Design Based on Power Function and Least squares

ABSTRACTIn 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.

Design of Wideband Fractional-Order Differentiator Using Interlaced Sampling Method

In this paper, the design of a wideband digital fractional-order differentiator (FOD) is investigated. First, conventional FOD designs are reviewed, and the reconstruction formula of the interlaced sampling method is used to design the proposed wideband FOD by index substitution and the Grunwald---Letnikov fractional derivative. Because a closed-form window design is obtained, the filter coefficients are easily computed. Then, the weighted least squares and convex optimization methods are applied to design non-sparse digital FODs that are optimal in the least squares or min---max sense. Next, the iterative hard thresholding and orthogonal matching pursuit methods are used to design sparse digital FODs to reduce the implementation complexity. Finally, several numerical examples are presented to show that the proposed FODs have smaller design errors in the high-frequency band than conventional digital FODs that do not use the auxiliary interlaced sampling signal.

低遅延特性を有する低域通過最大平たんFIRディジタル微分器の伝達関数

低遅延特性を有する低域通過最大平たんFIRディジタル微分器の伝達関数

Realization of Digital Differentiator Using Generalized Integrator For Power Converters

In power converters, the digital implementation of a differentiator is challenged by noise amplification and phase error. For example, the backward Euler differentiator loses its effectiveness at high frequency due to an introduced large-phase error. The Tustin differentiator, on the other hand, yields unacceptable noise amplification even though it produces the phase of an ideal differentiator perfectly. For an even more accurate performance, this letter proposes a new digital differentiator based on generalized integrator (GI). It will specifically be shown that the differentiation characteristic of a GI is an optimized compromise between those of backward Euler and Tustin differentiators. Several discretization techniques are then investigated for discretizing the GI-based differentiator. The effectiveness of the proposed differentiator has been verified by experimental results obtained with an LCL -filtered grid converter damped by feeding back the derivative of its capacitor voltage.

Design of recursive digital integrators and differentiators using particle swarm optimization

SummaryThis paper presents a new method for designing digital recursive integrators and differentiators by using nonlinear stochastic global optimization based on particle swarm optimization (PSO). A modified PSO is used to optimize the unknown coefficients of second‐order and third‐order recursive integrators in order to obtain a better magnitude response closer to the ideal integrator. The coefficients of the state‐of‐the‐art available filters in the literature are chosen as initial points in the PSO process. By choosing a good starting point, convergence to an optimal solution is greatly facilitated. A dynamic modification of the fitness function used in the PSO process leads to the design of a set of digital integrators each optimized for a specific frequency range. Then, second‐order and third‐order recursive digital differentiators are designed by inverting and stabilizing the transfer functions of the designed recursive integrators. The obtained stabilized differentiators are further optimized using PSO to further improve their performance. The magnitude responses of the designed filters outperform the existing integrators and differentiators. Copyright © 2015 John Wiley & Sons, Ltd.

An algebraic fractional order differentiator for a class of signals satisfying a linear differential equation

This paper aims at designing a digital fractional order differentiator for a class of signals satisfying a linear differential equation to estimate fractional derivatives with an arbitrary order in noisy case, where the input can be unknown or known with noises. Firstly, an integer order differentiator for the input is constructed using a truncated Jacobi orthogonal series expansion. Then, a new algebraic formula for the Riemann–Liouville derivative is derived, which is enlightened by the algebraic parametric method. Secondly, a digital fractional order differentiator is proposed using a numerical integration method in discrete noisy case. Then, the noise error contribution is analyzed, 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.

Fractional Euler analog-to-digital transform

In this paper, the design of fractional Euler transform which is the regular Euler transform where the digital delay operator z−1 is replaced by the ideal digital fractional delay operator z−α is presented. This introduced transform will be used in analog-to-digital transform to improve the accuracy of a digital filter equivalent to a given analog filter. However, because of the fractional delay operator z−α the proposed fractional Euler transform cannot be exactly implemented; only a limited implementation by means of infinite impulse response (IIR) digital filter can be achieved using approximation techniques. First, the fractional order α of the digital fractional delay operator z−α, for 0<α<0.5, is selected such that the relative error between the analog differentiator s and its equivalent digital one obtained by the fractional Euler transform is minimum. Then, the ideal digital fractional delay operator z−α is approximated by a digital IIR filter based on approximation of analog fractional order system leading to an IIR digital filter implementation of the fractional Euler transform. Illustrative examples are given to show the effectiveness of the fractional Euler analog-to-digital transform design. Finally, the design of low order digital differentiator using the proposed fractional Euler transform has been compared to the most recent designed analog-to-digital transforms.

Design of Digital Differentiators Using Interior Search Algorithm

Abstract 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.

Wideband Differentiator and Integrator With Ideal Phase Response

In this paper, we consider the design of a digital differentiator and integrator using the natural logarithm series. The proposed class of differentiators and integrators matches the ideal phase response and the magnitude response as closely as possible at low frequencies for orders lower than those reported in the literature. Furthermore, the bandwidth of the digital differentiator and integrator is extended to cover the entire Nyquist range using a simple multirate technique. In principle, the approach yields the results that are similar to those obtained by the design of fractional order differentiators and integrators.

FIR Linear Phase Fractional Order Digital Differentiator Design using Convex Optimization

In this paper, design of linear phase FIR digital differentiators is investigated using convex optimization. The problem of differentiator design is first described in terms of convex optimization with different optimization variables’ options, taken one at a time. The method is then used to design first order low pass differentiators and results are compared with Salesnick’s technique and Parks McClellan algorithm. The designed FIR low pass differentiator has improvement in transition width and flexibility to optimize different parameters. The concept of low pass differentiation is further generalized to fractional order differentiators. Fractional order differentiators are designed by using minmax technique on mean square error. Design examples demonstrate easy design procedure and flexibility in the process as well as improvement over existing fractional order differentiators in terms of mean square error in passband. Finally, fractional order differentiators are designed and used for texture enhancement of color images. Better texture enhancement than existing filtering approaches is established based on average gradient and entropy values. General Terms FIR Filter, Fractional order differentiator, Low pass differentiator, Full band differentiator, Image Enhancement.

Improved Digital Rational Approximation of the Operator $$S^{\alpha }$$ S α Using Second-Order s-to-z Transform and Signal Modeling

In this paper, an improved stable digital rational approximation of the fractional-order operator $$s^\alpha ,\alpha \in \,R$$s?,??R is developed. First, a novel efficient second-order digital differentiator is derived from the transfer function of the digital integrator proposed by Tseng. Then, the fractional power of the new s-to-z transform is expanded using power series expansion (PSE)-signal-modeling technique to obtain stable rational approximation of $$s^\alpha $$s?. Simulation results show that the proposed rational approximation has better frequency characteristics in almost the whole frequency range than that of existing first-order s-to-z transforms based approximations for different values of the fractional-order $$\alpha $$?. This paper also shows the benefit of using PSE-signal-modeling approach with first- or second-order mapping functions over PSE-truncation approach that is used in recent works for rational approximation of the operator $$s^\alpha $$s?, and highlights the major disadvantage of the latter approach that leads to undesirable rational models with complex conjugate poles and zeros.

Design of Multi-Band Digital Filters and Full-Band Digital Differentiators Without Frequency Sampling and Iterative Optimization

Most design problems of digital filters (or differentiators) are formulated with a set of grid point in the frequency region (frequency sampling). These problems are usually difficult to solve, and often require iterative optimization. The objective of this paper is to provide an efficient and simplified design approach to multi-band filters (including low-pass filters or high-pass filters) as well as full-band differentiators. The proposed method does not require frequency sampling and iterative optimization to compute the coefficients of the filters or that of the differentiators. The magnitude and phase specifications are simultaneously approximated, and the errors in the specified frequency bands are controlled by using frequency-weighting factors. In addition, a maximum pole radius, which corresponds to a stability margin, can be specified to robustly ensure the stability of the filters or the differentiators. To evaluate the efficiency of proposed method, we compare the proposed method with several established methods. Simulation results show that, although the propose method does not utilize frequency sampling and iterative optimization, the designed filters and differentiators have sufficient performance.

Wideband digital integrators and differentiators designed using particle swarm optimisation

Digital integrators (DIs) and digital differentiators (DDs) of second, third and fourth-order based on particle swarm optimisation (PSO) algorithm are presented. A modified particle swarm optimisation (MPSO) algorithm with reducing maximum velocity has been used to optimise the mean square error of the digital operators. Statistical and simulation results have been presented for comparing quality of optimal operators obtained by MPSO, genetic algorithm (GA), two variants of PSO and PSO-GA hybrid techniques. The results obtained for best solutions by the proposed algorithm are either superior or at par with the basic PSO variants and hybrid techniques. The proposed digital operators have also been simulated using MATLAB, and the results have been compared with that of existing DIs and DDs derived by different optimisation algorithms, to demonstrate the effectiveness of the use of proposed MPSO. The relative magnitude errors (dB) obtained for digital integrators and differentiators are as low as -40 and -35 dB, respectively, which are valid for almost the full band of normalised frequency.

Conceptual design of a selectable fractional-order differentiator for industrial applications

Abstract In the past decade, researchers working on fractional-order systems modeling and control have been considering working on the design and development of analog and digital fractional-order differentiators, i.e. circuits that can perform non-integer-order differentiation. It has been one of the major research areas under such field due to proven advantages over its integer-order counterparts. In particular, traditional integer-order proportional-integral-derivative (PID) controllers seem to be outperformed by fractional-order PID (FOPID or PIλDμ) controllers. Many researches have emerged presenting the possibility of designing analog and digital fractional-order differentiators, but only restricted to a fixed order. In this paper, we present the conceptual design of a variable fractional-order differentiator in which the order can be selected from 0 to 1 with an increment of 0.05. The analog conceptual design utilizes operational amplifiers and resistor-capacitor ladders as main components, while a generic microcontroller is introduced for switching purposes. Simulation results through Matlab and LTSpiceIV show that the designed resistor-capacitor ladders can perform as analog fractional-order differentiation.

Robust fractional order differentiators using generalized modulating functions method

This paper aims at designing a fractional order differentiator for a class of signals satisfying a linear differential equation with unknown parameters. A generalized modulating functions method is proposed first to estimate the unknown parameters, then to derive accurate integral formulae for the left-sided Riemann–Liouville fractional derivatives of the studied signal. Unlike the improper integral in the definition of the left-sided Riemann–Liouville fractional derivative, the integrals in the proposed formulae can be proper and be considered as a low-pass filter by choosing appropriate modulating functions. Hence, digital fractional order differentiators applicable for on-line applications are deduced using a numerical integration method in discrete noisy case. Moreover, some error analysis are given for noise error contributions due to a class of stochastic processes. Finally, numerical examples are given to show the accuracy and robustness of the proposed fractional order differentiators.

Improving the time domain response of fractional order digital differentiators by windowing

In this paper, the limitation of using (jω)ν as the ideal frequency response of fractional order digital differentiators for non-bandlimited signals is discussed. High frequency error enhanced by (jω)ν, along with the cause of time domain response degradation, is presented. Windows are proposed, which can help to improve the time domain response of fractional order digital differentiator and greatly reduce the filter order if the differentiator is approximated by finite impulse response (FIR) filter. Simulation results show that windowing the output of fractional order digital differentiator in frequency domain is effective in improving the time domain response of signals.

Design of IIR Digital Differentiators Using Constrained Optimization

A new optimization method for the design of fullband and lowpass IIR digital differentiators is proposed. In the new method, the passband phase-response error is minimized under the constraint that the maximum passband amplitude-response relative error is below a prescribed level. For lowpass IIR differentiators, an additional constraint is introduced to limit the average squared amplitude response in the stopband so as to minimize any high-frequency noise that may be present. Extensive experimental results are included, which show that the differentiators designed using the proposed method have much smaller maximum phase-response error for the same passband amplitude-response error and stopband constraints when compared with several differentiators designed using state-of-the-art competing methods.