Order Digital Differentiators Research Articles

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

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.

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.

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 optimally designed digital differentiator based preprocessor for R-peak detection in electrocardiogram signal

Abstract Globally the human death rate is accelerating day by day due to the cardiovascular diseases (CVDs), and it will be elevated shortly. In this scenario, the QRS complex detection of electrocardiogram (ECG) signal is considered as a simple, non-invasive, inexpensive, and preliminary diagnosis method used to assess the cardiac health of a patient. In this paper, an optimally designed Integer Order Digital Differentiator (IODD) based preprocessor is proposed for the accurate estimation of R-peak locations in the ECG signal. IODD, one of the major constituents of the preprocessor, is designed most proficiently by using a metaheuristic evolutionary optimization method called Gravitational Search Algorithm (GSA). In GSA, as the number of iteration increases the exploration capability fades out, and the exploitation capability fades in, which help it to avoid the local optima stagnation problem and results in faster convergence. The IODD based preprocessor accentuates the QRS complexes of the ECG signal irrespective of its abnormal morphology. The employed detector is a simple threshold independent R-peak decision logic designed by utilizing the properties of the Hilbert transform. In order to emphasize the superiority of the proposed research work the proposed IODD based QRS detection approach is validated on the first channel records of MIT/BIH Arrhythmia database (MBAD), QT database (QTDB), MIT/BIH noise stress test database (NSTDB), atrial fibrillation termination challenge database (AFTDB), and MIT/BIH ST change database (STDB). The sensitivity (Se) and positive Predictivity (+P) values for MBAD, QTDB, NSTDB, AFTDB, and STDB are Se=99.92% and +P=99.92%, Se=99.98% and +P=99.96%, Se=95.23% and +P=94.41%, Se=99.03% and +P=99.76%, and Se=99.93% and +P=99.90%, respectively. These performance metrics ensure the accuracy of the proposed R-peak detection technique for a wide variety of QRS morphologies and thereby affirm the applicability of the proposed IODD for the efficient detection of R-peak locations. The performance of the proposed R-peak detector significantly outperforms the reported methods in terms of all the performance metrics. The enhanced QRS detection accuracy of the proposed approach is due to the better feature signal generating capability of the proposed IODD.

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.

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.

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.

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.

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.

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.

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 digital FIR variable fractional order integrator and differentiator

This paper presents an easy and simple method to design variable fractional order digital FIR integrators and differentiators based on fractional order systems. First, closed-form digital IIR fractional order integrators and differentiators have been obtained from the analog rational functions approximations, in a given frequency band, of the fractional order integrator s −m and differentiator s m (0 < m < 1) through the Tustin generating function. Then, closed-form digital FIR fractional order integrators and differentiators by truncation of the digital IIR ones have been derived. Next, polynomial interpolation has been used to design digital FIR variable fractional order integrators and differentiators that can be implemented by the Farrow structure. The main feature of variable fractional order operator is that its order can be changed without re-designing a new fractional order operator. Some examples have been presented through the paper to illustrate the performance and the effectiveness of the proposed design method. The results obtained have been discussed and have been compared to one of the most recent works in the literature using the same design parameters.

Design of Fractional Order Digital Differentiator Using Radial Basis Function

In this paper, the design of fractional order digital differentiator is investigated. First, the radial basis function interpolation method is described. Then, the non-integer delay sample estimation of discrete-time sequence is derived by using the radial basis function interpolation approach. Next, the Grünwald-Letnikov derivative and non-integer delay sample estimation are applied to obtain the transfer function of fractional order digital differentiator. Finally, the applications in digital image sharpening and parameter estimation of fractional noise process are studied to demonstrate the usefulness of this new design approach.

Design of Second Order Digital FIR Full-Band Differentiators Using Weighting Coefficients

Most methods for the FIR differentiator design, published so far, have rather complex algorithms, requiring significant computing time. In addition, their design procedures and equations differ for the cases of even and odd differentiator length. FIR differentiators (filters) designed by these methods mostly have the passband group delay level corresponding to that of the linear phase FIR filters. In that sense, the design procedure of the first and higher order FIR full-band differentiators, proposed in this paper, is not only more efficient, but general too. Its efficiency consists in the direct (by this means - faster) determination of numerical values of their impulse response coefficients. Its generality consists in the fact that this method uses the same procedure and formulas for the design of differentiators with both even and odd length (without any mutual difference). (In order to avoid confusion, in the following text, the order of differentiation will be denoted by k, while the length of the FIR structure will be denoted by N). In addition, FIR differentiators designed by this method have approximately constant passband group delay level, which is lower than that of the corresponding linear phase FIR differentiators. These characteristics are enabled due to the fact that full-band differentiators, designed by the presented method, possess neither the (anti) symmetric feature of their impulse response coefficients, nor the strictly linear phase. The proposed method is based on the simultaneous approximation of the prescribed magnitude and group delay responses. The method presents an approach for the FIR differentiator frequency response approximation, directly in the complex, and not in the real domain. Two weighting coefficients, αR and αI, for the approximation of the real and imaginary part of the frequency response, respectively, are introduced. In the convenient manner, they are incorporated into properly and specifically defined quadratic measure error of the FIR differentiator’s frequency response approximation. This manner enables adjusting (decreasing) contributions of the frequency response real and imaginary part approximation errors to the total approximation error. Performing a proper analysis of the real and imaginary part approximation errors (of designed FIR differentiator frequency responses), values of these two parameters, giving a small variation of the group delay response, are obtained simultaneously with very small magnitude response error in the passband. In order to illustrate the proposed method effectiveness, numerical design examples of the second order FIR full-band differentiators are given.

Concerning continued fractions representation of noninteger order digital differentiators

The discretization of a fractional-order differentiator snu , where nu is a real number 0<nu<1, is an important issue in digital signal processing. This letter gives the analytical expression of the convergents of a continued fraction expansion of the binomial (1+s)nu to get a rapidly convergent, stable, minimum-phase approximation of digital differentiators

Neural least-squares design of higher order digital differentiators

The least-squares (LS) design of higher order digital differentiators can be formulated as to solve a system of linear equations. This is completed by solving a matrix inversion or by using some matrix properties to simplify the optimization. Here, a Hopfield neural network (NN) with closed-form evaluation for the related parameters is utilized to solve the mentioned set of linear equations in real-time. Simulation results are given to illustrate the excellent performance of the proposed neural-based method.

Design of fractional order digital FIR differentiators

In this paper, the design of a fractional order FIR differentiator is investigated. First, the fractional derivative of the power function is defined. Then, the impulse response of the fractional order differentiator is obtained by solving linear equations of the Vandermonde form. Finally, one example is used to demonstrate that the fractional derivatives of digital signals are easily computed by using the proposed filtering technique.

Computation of fractional derivatives using Fourier transform and digital FIR differentiator

In this paper, the computation of a fractional derivative using the Fourier transform and a digital FIR differentiator is investigated. First, the Cauchy integral formula is generalized to define the fractional derivative of functions. Then the fractional differentiation property of the Fourier transform of functions is presented. Using this property, the fractional derivative of a function can be computed in the frequency domain. Next, we develop a least-squares method to design the fractional order digital differentiator. When a signal passes through the designed differentiator, the output will be its fractional derivative. One design example is included to illustrate the effectiveness of this approach. Finally, the designed fractional order differentiator is used to generate a random fractal process which is better than the process obtained by the conventional method.

Analytic closed-form matrix for designing higher order digital differentiators using eigen-approach

The eigenfilter method has been proposed for designing higher order differentiators very effectively. The design method is based on the computation of an eigenvector of an appropriate real, symmetric, and positive-definite matrix. The elements of this matrix are usually evaluated by very time-consuming numerical integration. We present simple analytic closed-form formulas to compute these matrix-elements very efficiently. Hence, the eigenfilter approach for differentiators becomes much easier and more accurate than before and design time is reduced greatly for designing long filters.