Linear phase maximally flat digital differentiators (DDs) with stopbands obtained by minimizing the Lp norm are filters with important practical applications, as they can differentiate input signals without distortion. Stopbands designed by minimizing the Lp norm can be used to control the relationship between the steepness in the transition band and the ripple scale. However, linear phase DDs are unsuitable for real-time processing because each group delay is half of the filter order. In this paper, we proposed a design method for a low-delay maximally flat low-pass/band-pass FIR DDs with stopbands obtained by minimizing the Lp norm. The proposed DDs have low-delay characteristics that approximate the linear phase characteristics only in the passband. The proposed transfer function is composed of two functions, one with flat characteristics in the passband and one that ensures the transfer function has Lp approximated characteristics in the stopband. In the optimization of the latter function, Newton's method is employed.

Maximally flat digital differentiators (MFDDs) are widely used in many applications. By using MFDDs, we obtain the derivative of an input signal with high accuracy around their center frequency of flat property. Moreover, to avoid the influence of noise, it is desirable to attenuate the magnitude property of MFDDs expect for the vicinity of the center frequency. In this paper, we introduce a design method of linear phase FIR band-pass MFDDs with an arbitrary center frequency. The proposed transfer function for both of TYPE III and TYPE IV can be achieved as a closed form function using Jacobi polynomial. Furthermore, we can easily derive the weighting coefficients of the proposed MFDDs using recursive formula. Through some design examples, we confirm that the proposed method can adjust the center frequency arbitrarily and the band width having flat property.

An explicit expression for the impulse response coefficients of the predictive FIR digital filters is derived. The formula specifies a four-parameter family of smoothing FIR digital filters containing the Savitsky-Goaly filters, the Heinonen-Neuvo polynomial predictors, and the smoothing differentiators of arbitrary integer orders. The Hahn polynomials, which are orthogonal with respect to a discrete variable, are the main tool employed in the derivation of the formula. A recursive formula for the computation of the transfer function of the filters, which is the z-transform of a terminated sequence of polynomial ordinates, is also introduced. The formula can be used to design structures with low computational complexity for filters of any order.

Classical designs of maximally flat finite impulse response digital filters need to perform inverse discrete Fourier transformation of the frequency responses, in order to calculate the impulse response coefficients. Several attempts have been made to simplify the designs by obtaining explicit formulas for the impulse response coefficients. Such formulas have been derived for digital differentiators having maximal linearity at zero frequency, using different techniques including interpolating polynomials and the Taylor series etc. We show that these formulas can be obtained directly by application of maximal linearity constraints on the frequency response. The design problem is formulated as a system of linear equations, which can be solved to achieve maximal linearity at an arbitrary frequency. Certain special characteristics of the determinant of the coefficients matrix of these equations are explored for designs centered at zero frequency, and are used in derivation of explicit formulas for the impulse response coefficients of digital differentiators of both odd and even lengths.

Explicit formulas for the tap-coefficients of Taylor series based type III FIR digital differentiators have already been presented. However, those formulas were not derived mathematically from the Taylor series and were based on observation of different sets of the results. In this paper, we provide a mathematical proof of the formulas by deriving them mathematically from the Taylor series.

The explicit formula for the coefficients of maximally linear digital differentiators is derived by the use of Taylor series. A modification in the formula is proposed to extend the effective passband of the differentiator with the same number of coefficients.