Design Of Digital Differentiators Research Articles

Design of IIR full-band differentiators with improved nearly linear phase

Linear, or almost linear phase in the design of digital differentiators depends both on the method of calculation of the filter coefficients and the adopted structural design. Acknowledging the importance of this design objective, we developed a method to derive infinite impulse response differentiators which tackles it from both perspectives and yields improved characteristics compared to our previous design. We have replaced the differentiator structure comprising an all-pass filter and a delay line with a configuration of two infinite impulse response all-pass filters. Furthermore, we provide a procedure for determining an improved initial solution to the optimization of filter coefficients, as well as a procedure for improving the differentiator's magnitude response. The final solution exhibits considerably lower error at low frequencies and is thus suitable for the common scenario where the relevant spectral content decreases with frequency. The proposed filters of third and fifth order outperform many existing solutions in literature, especially in terms of the phase characteristic and time-domain performance.

$$L_1$$-Norm-Based Optimal Design of Digital Differentiator Using Multiverse Optimization

$$L_1$$-Norm-Based Optimal Design of Digital Differentiator Using Multiverse Optimization

Guided Filtering Based Efficient Digital Differentiator Design for Electrocardiogram Signal Processing

Guided Filtering Based Efficient Digital Differentiator Design for Electrocardiogram Signal Processing

A novel approach to design optimal 2-D digital differentiator using vortex search optimization algorithm

The designing of 2-D digital differentiator is multimodal and high dimensional problem which requires large number of differentiator coefficients to be optimized, hence conventional design techniques will not lead to optimal 2-D differentiator design. Metaheuristic approaches is a good approach to handle multimodal and high dimensional problem under consideration. In this paper, a novel method for the design of FIR two-dimensional (2-D) digital differentiator with quadrantally odd symmetric properties is proposed. The coefficients of 2-D digital differentiator are computed and optimized using vortex search optimization (VSO) algorithm with the support of L1 error objective function. The unique combination of VSO and L1 error fitness function aids in achieving flatter magnitude response in designing of 2-D differentiator. Further, some comparative analysis with the well-known established techniques like cuckoo search algorithm (CSA), particle swarm optimization (PSO) and real coded genetic algorithm (RCGA) methodology is reported over full frequency bands, in order to demonstrate the advantage of the proposed novel 2-D digital differentiator design approach. To measure the performance of proposed system, some parameters are selected for illustration such as absolute magnitude error, mean absolute error, standard deviation and execution time. The excellence of the proposed approach is observed in approximating the ideal response of 2-D differentiator in contrast to other algorithms and to provide global optimal solution.

Design of Digital Differentiator Using a Combination of System Identification and Particle Swarm Optimization

Design of Digital Differentiator Using a Combination of System Identification and Particle Swarm Optimization

A Modified Annulus Sector Constraint for Constrained FIR Filter Designs

Optimal designs of finite impulse response (FIR) digital filters with prescribed magnitude and phase responses have found many applications in signal processing. To deal with the nonconvexity of the constraint region in the phase-error and magnitude-error constrained design, a new error constraint method based on a modified annulus sector (MAS) is proposed. With the new constraint method, the phase-error and magnitude-error constrained least squares, phase-error constrained minimax, and iterative reweighted phase-error constrained minimax designs of FIR filters are studied. Analyses and design examples demonstrate the higher accuracy of the MAS constraint and better filter performance by the constraint method than existing constraint methods. An application in the design of digital differentiators with flatness constraints is also provided.

Optimal SSA‐based wideband digital differentiator design for cardiac QRS complex detection application

AbstractIn this paper, a computationally efficient, highly accurate, wideband, stable, and minimum phase infinite impulse response type first‐order digital differentiator (DD) is designed by employing a swarm intelligence‐based search method called Salp Swarm Algorithm (SSA) for the QRS complex detection application. The optimal coefficients of the DD are computed by minimizing a suitable fitness function to meet the ideal differentiator magnitude response characteristics. The simulation results and the root mean square magnitude error metric justify the superiority of the proposed SSA‐based DD design as compared with all other differentiators employed in the QRS complex detection application, and the reported first‐order DDs based on the numerical methods and the other evolutionary algorithms. The electrocardiogram signal is preprocessed by the proposed DD to generate the feature signal corresponding to each QRS complex. The generated feature signal is used as a marker to identify the exact occurrence of the QRS complex by using an adaptive threshold‐based detection logic. The proposed DD‐based QRS detection approach achieves a sensitivity (Se), positive prediction (PP), detection error rate (DER), and accuracy (Acc) of 99.94%, 99.93%, 0.1279%, and 99.87%, respectively, when validated against MIT/BIH arrhythmia database. Also, against the QT database, the proposed QRS detector produces a Se of 99.93%, PP of 99.97%, DER of 0.09%, and Acc of 99.90%. The performance of the proposed QRS detection technique is compared with the methods already reported in the recent literature, and the superiority of the proposed approach is established with respect to different standard performance metrics. The noise tolerance capability of the proposed QRS detector is demonstrated against MIT/BIH noise stress test database.

Design of Digital Differentiator Using the $$L_1$$ L 1 -Method and Swarm Intelligence-Based Optimization Algorithms

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 $$L_1$$ -method. Then, the $$L_1$$ 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 $$L_1$$ -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 $$L_1$$ -method, PSO- $$L_1$$ and BA- $$L_1$$ . 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 $$L_1$$ -method yields better results for higher-order designs. Furthermore, the proposed designs are tested on two input signals for their efficient response.

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 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.

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.

Efficient Design and Implementation of Variable Fractional Delay Filters Using Differentiators

In this paper, the power series expansion of exponential function is used to transform the design problem of variable fractional delay (VFD) filter into the designs of digital differentiators with various orders such that conventional digital differentiators can be applied to implement VFD filter efficiently. The proposed method is flexible because the VFD filter can be designed by making the trade-off among the storage requirement of filter coefficients, computational complexity and delay of filter. Finally, some numerical examples are demonstrated to show the effectiveness and flexibility of the proposed design methods.

Recursive wideband digital differentiator and integrator

A new design of a recursive wideband digital differentiator is obtained by optimising the pole-zero locations of existing recursive wideband digital differentiators. Further, a new integrator design is also obtained by inverting the transfer function of the proposed digital differentiator design with suitable modifications. The beauty of these designs is that they are only of second-order recursive systems and have not more than 0.48% relative error in magnitude responses almost over the full Nyquist band, thus being suitable for real-time circuit applications.

Two-dimensional fractional-order digital differentiator design by using differential evolution algorithm

Designing a fractional-order digital differentiator often requires considerably complex mathematical operations and numerical approximations. Thus this paper will propose a simple method to achieve the fractional-order digital differentiator design, particularly for two-dimensional fractional-order differentiators. A two-dimensional finite impulse response (FIR) digital filter structure is utilized and designed so that its corresponding magnitude response can satisfy that of a desired fractional-order differentiator of two variables. The algorithm used to design such two-dimensional digital differentiator is the differential evolution (DE), which is one of evolutionary computations and has excellent searching capacity. The efficiency of the proposed scheme can be confirmed by some illustrative examples.

Design of digital differentiator using difference formula and Richardson extrapolation

The design of digital differentiators is investigated. First, the backward difference formula in numerical differentiation is applied to derive the transfer function of a digital differentiator. In this design, the Richardson extrapolation is used to generate high-accuracy results while using low-order formulas. Then, conventional Lagrange finite impulse response (FIR) and Thiran infinite impulse response (IIR) allpass fractional delay filters are directly applied to implement the designed differentiator. Next, the proposed method is extended to design digital differentiators using central difference formula. Finally, several numerical examples are illustrated to demonstrate the effectiveness of this new design approach.

Narrow Transition Band FIR Hilbert Transformers With Flat Magnitude Response

Maximally flat (MAXFLAT) finite-impulse response digital filters, including the Hilbert transformers (HTs), have the smoothest magnitude responses among the available types of FIR digital filters. However, the transition bands of MAXFLAT filters are relatively wider, and this makes them unattractive in certain applications. We present new designs of even and odd length HTs by transforming a design of digital differentiators satisfying maximal linearity constraints at omega = pi/2. The resultant even and odd length HTs have highly smooth magnitude responses around omega = pi/2 and omega = (pi/4,3pi/4) respectively, and have relatively narrow transition bands compared to the existing MAXFLAT designs.

Design of a higher-order digital differentiator using a particle swarm optimization approach

This paper applies a novel optimization algorithm, particle swarm optimization (PSO), to design a higher-order differentiator that contains two different structures of even and odd orders. Four cases of linear phase finite impulse response (FIR) filters are designed to match the prescribed differentiation frequency response by using PSO algorithm. The algorithm with real-valued manipulations uses the velocity updating and position updating formulas to optimally solve the impulse response of the filter to the digital differentiator design problem. Simulation results reveals that the proposed method provides much better design performance than the well-known McClellan–Parks method.

Least-squares design of digital differentiators using neural networks with closed-form derivations

In this letter, a neural network implementation for the least-squares design of digital differentiators with closed-form derivations for Hopfield-related parameters are proposed. Using this technique, the optimal filter coefficients are obtained by iteratively updating the dynamic nonlinear equations of the network. Simulation results indicate that the proposed technique has the advantage of effectiveness and is suitable for hardware implementation in real time.

Digital differentiator design using fractional delay filter and limit computation

In this paper, the design of digital differentiator is investigated. First, the relation between fractional delay filter and first-order differentiator is established such that the differentiator can be obtained from fractional delay filter by using the computation of limit. Then, conventional finite-impulse response (FIR), allpass and Farrow-based fractional delay filters are directly applied to design first-order differentiator. Next, the proposed method is extended to design high-order differentiators. Finally, several design examples are illustrated to demonstrate the effectiveness of this new design approach.

Stable IIR digital differentiator design using iterative quadratic programming approach

In this paper, we present an iterative quadratic programming approach to design stable IIR digital differentiator. At each iteration, the cost function is transformed into a quadratic form by treating the denominator polynomial obtained from the preceding iteration as a part of the weighting function, and the pole radii are constrained to lie in the unit circle by using the implications of Rouche's theorem. After solving the standard quadratic programming problem at each iteration, the design algorithm converges to a stable and truly weighted least-squares solution. Design examples demonstrate that our method provides a better design results than the conventional quadratic programming method.