Digital Fractional Order Differentiator Research Articles

Optimal wideband digital fractional-order differentiators using gradient based optimizer

In this paper, we propose a novel optimization approach to designing wideband infinite impulse response (IIR) digital fractional order differentiators (DFODs) with improved accuracy at low frequency bands. In the new method, the objective function is formulated as an optimization problem with two tuning parameters to control the error distribution over frequencies. The gradient based optimizer (GBO) is effectively employed on the proposed objective function. A wide range of design examples are presented to illustrate the effectiveness of the proposed approach. The proposed approximations are compared to those of recent literature in terms magnitude, phase, and group delay errors. The result reveal that our method can attain approximations have a favorable low frequency performance (with about 60% of relative magnitude error reduction) and maintain a comparable accuracy at most of the Nyquist band to those of the existing ones. Thus, our approximations can be attractive for low frequency applications.

Discrete cosine transform interpolation based design of two-dimensional FIR fractional order digital differentiator

Discrete cosine transform interpolation based design of two-dimensional FIR fractional order digital differentiator

Intra and inter-patient arrhythmia classification using feature fusion with novel feature set based on fractional-order and fibonacci series

Cardiac ailments are increasing at an alarming rate globally due to the sedentary lifestyle and increased desk-bound activities. For decades, the ECG signal is used in aiding the analysis of human heart. Arrhythmia is a disorder that alters the normal cardiac cycle of ECG signal. The automatic classification of arrhythmia is a highly desirable and tough task. Mainly, the traditional methods of arrhythmia classification, are evaluated on the intra-patient criterion that may not befit the inter-patient criterion. A complete classification method has been proposed in the paper which performs well in intra-patient, inter-patient criterion and it is effective for minority class of MIT-BIH arrhythmia database (MIT-BIH-AD). In the proposed method, after preliminary processing based on Riesz fractional-order digital differentiator and R-peak detection, various features are extracted from ECG beat. The feature set proposed is a fusion of time-domain, time–frequency domain features and the new and novel features based on the Fibonacci series and coefficients of fractional-order Riesz based derivative signals which are proposed in this work. The results are evaluated on the MIT-BIH-AD and the results for intra-patients’ criterion have achieved an overall accuracy, sensitivity and positive predictivity values of 99.85%, 99.16%, 99.93% respectively, and for the inter-patient scheme 92.5%, 89.89% and 95.54% an average value of accuracy, sensitivity and positive predictivity of six ECG classes. The obtained results in both criterions have outperformed other methods in the literature and the proposed work has also attained better results for minority class of MIT-BIH-AD in both the scheme.

Comparative study of DST interpolation approach of fractional orders

In this paper, comparative study of DST interpolation approach of various order by using different fractional derivatives are presented. First the definition of different fractional order derivatives like Grunwald-Letnikov, Weyl’s and Conformable are reviewed. Next, Fractional derivative of a discrete signal is determined after applying the DST interpolation approach. Next, the DST-IV method approach transfer function are obtained with the help of index-mapping technique. Lastly, some computative problems are discussed for checking the effectiveness of digital fractional order differentiators for design of proposed method using the integral square error formula. Error values of various fractional order derivatives have been presented in the form of table.

Closed-Form Analytical Formulation for Riemann–Liouville-Based Fractional-Order Digital Differentiator Using Fractional Sample Delay Interpolation

This paper intends to apply a new mathematical approach based on Riemann–Liouville fractional–differential operator to the design of fractional-order digital differentiators (FODDs). Under the research area of fractional-order calculus, Grunwald–Letnikov (GL) and Riemann–Liouville (RL) are most widely used fractional–differential operators. The GL-based methods have been extensively investigated by research community to design FODD, but there seems to be an improvement window for designing FIR filters based upon RL fractional–differential operator. Therefore, this paper establishes a generalized framework for the design of RL-based FODD (RL-FODD) and compares its performance with well-established GL-FODD designs, based upon their ability to yield ideal frequency response of FODD. Initially, closed-form analytical expression is formulated for computing FIR filter coefficients of RL-FODD. Then, the design accuracy of proposed filter in the high-frequency region is improved by incorporating non-integer sample delay into the design process. Several design examples are presented to illustrate the comparative analysis between conventional GL-FODD and proposed RL-FODD for varying fractional orders. The proposed method is evaluated, taking into account several amplitude-modulated and harmonic signals corrupted by AWGN and high-frequency chirp noise. Furthermore, the application in parameter estimation of fractional noise process is investigated. Compared with conventional GL-FODD, results of the proposed study validate its superiority and robustness based upon various experimental simulations.

Comparative study of DST interpolation approach of fractional orders

In this paper, comparative study of DST interpolation approach of various order by using different fractional derivatives are presented. First the definition of different fractional order derivatives like Grunwald-Letnikov, Weyl’s and Conformable are reviewed. Next, Fractional derivative of a discrete signal is determined after applying the DST interpolation approach. Next, the DST-IV method approach transfer function are obtained with the help of index-mapping technique. Lastly, some computative problems are discussed for checking the effectiveness of digital fractional order differentiators for design of proposed method using the integral square error formula. Error values of various fractional order derivatives have been presented in the form of table.

Optimal design of zero-phase digital Riesz FIR fractional-order differentiator

This article introduces an optimal design of finite impulse response-type Riesz fractional-order digital differentiator (FODD) of orders p = 0.3, 0.5, and 0.75. A very little study on Riesz-based differentiator has been reported so far. Moreover, none of the published Riesz FODD-based literature has studied the optimal design of Riesz FODD, which is explored in this article. In the reported literature, the Riesz FODD is realized by adopting conventional mathematical approaches, which are not efficient for the Riesz FODD-type design problem where the nature of the search landscape is multidimensional, multimodal, non-uniform, and nonlinear. To solve this problem, the design parameters of the proposed Riesz FODDs are estimated by using three independent metaheuristic optimization algorithms called salp swarm algorithm (SSA), gases Brownian motion optimization (GBMO), and gravitational search algorithm (GSA). In SSA, the adaptive variation of the iteration-dependent parameters helps the search agents not only to avoid the suboptimal solutions, but also ensures the faster convergence to reach the near-global optimal solution compared to GSA- and GBMO-based design approaches. The maximum phase error (MPE) and maximum absolute magnitude error (MAME) of the proposed SSA-based Riesz FODD are as low as 3.7119E−014 degree and − 9.6824 dB, respectively, for p = 0.3; 3.1523E−014 degree and − 15.6443 dB, respectively, for p = 0.5; and 3.5780E−014 degree and − 18.7525 dB, respectively, for p = 0.75. In terms of MPE and MAME, the proposed SSA-based Riesz FODD significantly outperforms the GSA- and GBMO-based Riesz FODD designs. The proposed SSA-based Riesz FODD of order p = 0.5 is also implemented on the TMS320C6713 digital signal processor, and its precise zero-phase performance is tested on real electrocardiogram (ECG) signal. Moreover, the proposed Riesz FODD is utilized in the ECG QRS complex identification system to demonstrate its real-time application.

Efficient design of wideband digital fractional order differentiators and integrators using multi-verse optimizer

In this paper, a novel method is proposed based on combining L1-norm optimally criterion with a recently-proposed metaheuristic called multi-verse optimizer (MVO) to design 2nd–4th order stable, minimum phase and wideband infinite impulse response (IIR) digital fractional order differentiators (DFODs) for the fractional order differentiators (FODs) of one-half, one-third and one-fourth order. To confirm the superiority of the proposed approach, we conduct comparisons of the MVO-based designs with the real-coded genetic algorithm (RCGA) and particle swarm optimization (PSO)-based designs in terms of accuracy, robustness, consistency, and efficiency. The transfer functions of the proposed designs are inverted to obtain new models of digital fractional order integrators (DFOIs) of the same order. A comparative study of the frequency responses of the proposed digital fractional order differentiators and integrators with the ones of the existing models is then conducted. The results demonstrate that the proposed designs yield the optimal magnitude responses in terms of absolute magnitude error (AME) with flat response profiles.

Efficient Design of Zero-Phase Riesz Fractional Order Digital Differentiator Using Manta-Ray Foraging Optimisation for Precise Electrocardiogram QRS Detection

This article introduces a new design of an optimised zero-phase response fractional order digital differentiator (FODD) called Riesz FODD of half order by employing a recently developed Manta-Ray Foraging Optimisation (MRFO) for improved and precise electrocardiogram QRS detection. A new weighted cost function is developed that precisely considers the characteristics of an ideal Riesz fractional-order differentiator. Simulation tests of the rational approximations of the Riesz FODD based on the MRFO algorithm ensure close correspondence with the theoretical zero-phase and near-ideal magnitude responses and sensibly supersede the Least-squares method, Convex optimisation, Flower Pollination Algorithm, and Artificial Bee Colony based Riesz FODD designs. The Zero-phase response of the proposed Riesz FODD assists in preserving the features of the differentiated ECG precisely at the same time instant where they occur in the unfiltered signal. Additionally, the optimal magnitude characteristics of the proposed Riesz FODD help to obtain highly accurate differentiated ECG. These two indigenous properties of the proposed Riesz FODD accommodate the QRS detector to precisely identify the R-peak location and thus sensibly escalate the QRS detection accuracy when tested against several real electrocardiogram (ECG) signals of the PhysioNet ECG repository.

Highly-Accurate Fractional-Order Microwave Differentiators for Band-Specific UWB Applications

Highly accurate designs of band-specific fractional-order microwave differentiators are proposed to operate in lower and upper frequency bands of UWB range. Firstly, fractional-order digital differentiators are obtained by minimizing the magnitude error function of a single-zero multi-pole transfer function. Absolute relative error (ARE) analysis clearly demonstrates that the proposed lower and upper band fractional-order digital differentiator designs are having highly accurate magnitude response and outperform the existing fractional-order differentiators with maximum ARE values of − 44.14 dB and − 48.79 dB respectively. To operate in microwave range, these designs are realized using Z-domain chain scattering parameters of equal-electrical length line elements. Microstrip configurations obtained for lower and upper band fractional-order microwave differentiators consist of a short-circuited shunt stub and three serial transmission line sections. Implementation of designs is done on Rogers RO3003 substrate having thickness 0.75 mm. Measured results for transmission coefficient of lower and upper frequency band designs are found to be in good agreement with simulated ones over the frequency range 1.2 to 6.2 GHz and 6 to 10.9 GHz respectively.

Algebraic Fractional Order Differentiator Based on the Pseudo-State Space Representation

Abstract The aim of this paper is to design an algebraic fractional order differentiator for a class of commensurate fractional order linear systems modeled by the pseudo-state space representation. For this purpose, a new algebraic method is introduced by designing an operator which can transform the considered system into a fractional order integral equation by eliminating unknown initial conditions. Based on the obtained equation, the desired fractional derivative is exactly given by a new algebraic formula using a recursive way. Then, a digital fractional order differentiator is introduced in discrete noisy cases. Finally, numerical results are given to illustrate the accuracy and the robustness of the proposed method.

An Efficient R-Peak Detection Using Riesz Fractional-Order Digital Differentiator

Clinically, electrocardiogram (ECG) is a powerful tool for determining the health and functioning of the human heart. Faster detection and diagnosis of heart functioning would aid cardiologists to provide appropriate treatment to the patients (subjects). In this paper, the concept of fractional-order calculus is employed for noise cancellation and artifacts removal in ECG signal as fractional-order differentiator proved to provide more peculiar details about signals than an integer-order differentiator. In the proposed method, R-peaks are detected using Riesz fractional-order digital differentiator (RFODD) based on the differencing method. The differentiation operation enhances the high-frequency components of the signal. So, QRS complex which is a high-frequency component in ECG is accentuated and the R-peaks are detected using appropriate threshold technique. The proposed method is tested on Massachusetts Institute of Technology-Beth Israel Hospital (MIT-BIH) arrhythmia database, and the experimental results of the proposed method have achieved a sensitivity of 99.95%, positive predictivity of 99.949% and an error rate of 0.095%. ECG waveforms are analyzed on various fractional orders of RFODD, and their performance parameters, i.e., sensitivity, positive predictivity and error rate, are also calculated.

Design of Low Frequency Suitable Fractional Order Differintegrators

Design of low frequency applicable fractional order digital differentiators and integrators using direct discretization is the main objective of this paper. Model order reduction procedure is used to obtain the novel first order s-to-z transform. Continued fraction expansion (CFE) technique is used to discretize that transform. The design of one-half digital differintegrators based on proposed transforms will be compared with the well known existing first order transforms namely, Bi-linear and Al-Alaoui transforms. The efficacy of the designed methods presented interms of normalized magnitude error (NME). The designed fractional order digital filter coefficients are tabulated and all simulation results are carried out by using MATLAB software.

An Efficient and Robust Digital Fractional Order Differentiator Based ECG Pre-Processor Design for QRS Detection.

This paper presents an efficient infinite impulse response type digital fractional order differentiator (DFOD) based electrocardiogram (ECG) pre-processor to detect QRS complexes. First, an efficient optimizer namely, Antlion optimization algorithm is employed to solve the proposed DFOD design problem. Then, the designed DFOD is deployed in the pre-processing stage of a threshold independent R-peak detection technique. Finally, the proposed QRS complex detector is thoroughly assessed on the standard ECG datasets of MIT/BIH Arrhythmia, MIT/BIH ST Change, MIT/BIH Supraventricular Arrhythmia, European ST-T, QT, and T-Wave Alternans Challenge databases to show the wide sense practicability of the proposed DFOD-based QRS detector. The root-means-square magnitude error (RMSME) and the average group delay (τDD) metrics of the proposed DFOD are as low as -38.17 dB and 0.04 samples, respectively. The percentage of improvement in terms of RMSME metric compared to the best-reported approach is 15%. The overall sensitivity of 99.89% and positive predictivity of 99.88% are incurred by considering all the six databases. To the best of the authors' knowledge, it is the first time when the evolutionary algorithm based IIR-type DFOD is employed for the QRS complex detection and establishing its performance superiority. The results so obtained are compared with the results of all the recently reported QRS detectors. The proposed DFOD based ECG pre-processor has a great potential to robustly generate the feature signal related to the ECG QRS complex irrespective of the ECG morphology. Thus, the proposed DFOD based QRS detector can be employed in clinical ECG monitoring devices to augment the QRS detection performance.

Design and FPGA implementation of lattice wave fractional order digital differentiator

This paper deals with the design and FPGA implementation of a fractional order digital differentiator. It is realized using computationally efficient lattice wave digital filter (LWDF) which requires minimum multipliers in comparison to the direct form structure. A nature inspired ant lion optimization (ALO) algorithm is utilized to compute optimal coefficients of the LWDF based digital differentiator. The proposed LWDF based digital differentiator is compared with the existing literature in terms of magnitude response, percentage magnitude error and root mean square magnitude error. LWDF structure comprises of constant coefficient multipliers, adders and delay units which are implemented on FPGA with the help of Xilinx system generator for DSP design tool. For efficient implementation of multipliers, multiplier-less logic through digit recoding techniques such as canonic signed digit (CSD) representation and radix-2r encoding are described through Verilog HDL and incorporated in the implementation model as black box. Post-implementation results show that implementation based on the radix-2r encoding is found to be more efficient than that of the CSD encoding. The design and implementation results are also reported to highlight the improvements.

An efficient design of finite impulse response — Fractional-order differentiator using shuffled frog leaping algorithm heuristic

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.

Design and Implementation of Fractional Order Differintegrators Using Reduced s to z Transforms

The Design and implementation of fractional order digital differentiators and integrators is the main objective of this paper. Identify the novel reduced s to z transforms calculated using model order reduction techniques. Thus the transforms are discretized directly using Continued Fraction Expansion (CFE).The designed differentiators and integrators are implemented on Xilinx Spartan 3E field programmable gate arrays (FPGA) and are tested using the sinusoidal, square and triangular waveforms. The practical results agree with the theoretical ones.

Optimal Design of Fractional-Order Digital Differentiator Using Flower Pollination Algorithm

This paper presents an efficient approach to design wideband, accurate, stable, and minimum-phase fractional-order digital differentiators (FODDs) in terms of the infinite impulse response (IIR) filters using an evolutionary optimization technique called flower pollination algorithm (FPA). The efficiency comparisons of FPA with real-coded genetic algorithm (RGA), particle swarm optimization (PSO), and differential evolution (DE)-based designs are conducted with respect to different magnitude and phase response error metrics, parametric and nonparametric statistical hypotheses tests, computational time, and fitness convergence. Exhaustive simulation results clearly demonstrate that FPA significantly outperforms RGA, PSO, and DE in attaining the best solution quality consistently. Extensive analysis is also conducted in order to determine the role of control parameters of FPA on the performance of the designed FODDs. The proposed FPA-based FODDs outperform all the designs published in the recent literature with respect to the magnitude responses and also achieve a competitive performance in terms of the phase response.

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.