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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.

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.

In this paper the design of 2-dimensional finite impulse response (2-D FIR) digital differentiator (DD) with quadrantally odd symmetric impulse response is presented. The L 1 -method is developed to design the 2-D system and minimize the L 1 -error. The design problem of 2-D FIR-DD is formulated as an optimization problem in order to compute the system coefficients with quadrantally odd symmetric properties. Finally, the design examples of 2-D differentiator of different order is demonstrated and analyzed in terms of the L 1 -error and elapsed time. The proposed 2-D FIR L 1 -DD is also compared with the existing 2-D differentiators designed using metaheuristic techniques and is observed to yield least L 1 -error.

In this paper, a framework for obtaining the higher order derivative of digital signals with an improved signal-to-quantization-noise ratio (SQNR) is developed. The framework consists of an oversampling analog-to-digital conversion unit with a dithering process and a higher order finite-impulse response (FIR) digital differentiator followed by a decimator. A simple and effective recursion is derived for designing the higher order FIR digital differentiator. In addition, the SQNR is analyzed. Computer simulations validate the proposed framework, the differentiator design method, and the results of the improved SQNR.

A new efficient algorithm for calculating the weighting coefficients of maximally linear, FIR digital differentiators is presented. Simple closed-form explicit and recursive formulas are derived in a very straightforward manner. Moreover, a simple recursive equation is established, relating coefficients of two digital differentiators of adjacent ranks.

In this paper, a new approximation is proposed to design fractional order digital differentiator using fractional delay filters. New approximation improves the design accuracy of conventional fractional differencing for wideband compared to existing approximations. Proposed new approximation is applied to see relatively small error for wideband.

An efficient algorithm for calculating the weighting coefficients of FIR digital differentiators and Hilbert transformers for midband frequency ranges is presented. New simple closed-form explicit and recursive formulas are derived in a straightforward manner. Moreover, a new simple recursive relation is established, relating the coefficients of two digital filters of adjacent ranks.

An efficient algorithm for calculating the coefficients of FIR digital differentiators for mid-band frequencies is presented. Simple closed-form explicit and recursive formulas are derived in a straightforward manner. Moreover, a simple recursive relation is obtained, relating the coefficients of two digital filters of adjacent ranks.

A Hopfield-type neural network is proposed for the design of nonrecursive digital differentiators and Hilbert transformers. Given the amplitude response, the all-analog network computes the filter coefficients in real time. The network is simulated and a few examples are included to show that this is an efficient way of solving the approximation problem and has a high potential for implementation in analog VLSI.

Explicit formulas for the tap-coefficients of Taylor-series-based full-band FIR digital differentiators have already been presented. However, those formulas were not derived mathematically from the Taylor series and were based on observations of different sets of results. The authors provide a mathematical proof of the formulas by deriving them mathematically from the Taylor series.

A simple way of calculating the weighting coefficients for the maximum flat digital FIR differentiator is presented. A relation with interpolating polynomials is also established.

Recursive as well as explicit mathematical formulas have been derived for the weighting coefficients of maximally linear, FIR digital differentiators for low frequencies.