Digital Signal Processing 87 (2019) 132–144 Contents lists available at ScienceDirect Digital Signal Processing www.elsevier.com/locate/dsp Design of IIR fullband differentiators using parallel all-pass structure Goran Stanˇ ci´ c a , Ivan Krsti´ c b,∗ , Miloš Živkovi´ c a a University of Niš, Faculty of Electronic Engineering, 18000 Niš, Serbia b University of Priština, Faculty of Technical Sciences, 38220 Kosovska Mitrovica, Serbia a r t i c l e i n f o a b s t r a c t Article history: Available online 6 February 2019 Keywords: All-pass digital filter Parallel connection Weighted Chebyshev approximation Fullband differentiator Low-pass differentiator A new approach for the design of recursive fullband digital differentiators using parallel all-pass structure is discussed in this paper. While magnitude response of designed fullband differentiator approximates the ideal one in the weighted Chebyshev sense, its phase response is a nearly-linear function of frequency at low frequencies. The low-pass differentiators, presented in this paper, are obtained by cascading the proposed recursive fullband differentiators with the corresponding low-pass filters. The phase response linearity error of such low-pass differentiators is shown to be primarily affected by the phase response nonlinearity of the utilized low-pass filter. A comparison with some of the existing fullband and low-pass differentiators shows that proposed differentiators require less multiplications, while their phase and magnitude responses are either better or slightly worse than those of existing differentiators.  2019 Elsevier Inc. All rights reserved. 1. Introduction Digital fullband differentiators, needed in a wide range of dig- ital signal processing applications [1–5] where the time derivative of input signal needs to be computed, can be designed either as an infinite impulse response (IIR) [6–20] or finite impulse response filters [21–23]. While the order of the IIR fullband differentiator is significantly lower compared to its finite impulse response filter counterpart, the perfectly linear phase response of IIR fullband dif- ferentiator cannot be achieved. On the other hand, this is not an issue in most practical applications if obtained phase response is nearly-linear function of frequency. There are several approaches to the IIR fullband differentia- tor design. Conventional approach to the IIR differentiator design is based on inversion of the IIR integrator transfer function fol- lowed by reflection of the unstable poles inside the unit circle and compensation of the amplitude [6–9]. In other words, conven- tional approach reduces to IIR integrator design problem. Starting point of design methods of the second approach is the IIR differen- tiator transfer function, obtained either by conventional approach or by any other design method, which is than optimized utiliz- ing classical [10] or evolutionary [10,16] optimization techniques. Methods of the third approach determine the unknown coeffi- cients of the recursive transfer function such that some objective function is minimized. Design method presented in [12] formu- * Corresponding author. E-mail addresses: goran.stancic@elfak.ni.ac.rs (G. Stanˇ ci´ c), ivan.krstic@pr.ac.rs (I. Krsti´ c), miskoz@elfak.rs (M. Živkovi´ c). lates the IIR fullband differentiators’ design problem as convex constrained optimization problem in unknown zeros’ and poles’ radiuses and phase angles, such that its solution minimizes the group delay-deviation under the constraint that maximum magni- tude response error is below some prescribed value. Coefficients of the direct form differentiators’ coefficients are determined by min- imizing the L 2 norm of the magnitude response error by means of metaheuristic optimization techniques in [8,14,17], and by iter- ative quadratic programming approach in [18]. On the other hand, a noniterative method presented in [20] formulates the fullband differentiators’ design problem as quadratic programming problem such that the magnitude and phase response specifications are si- multaneously approximated. Another method of the third approach is given in [15] where coefficients of the lattice wave digital filter representation of the third and the fifth order fullband differen- tiators are determined by minimizing the L 1 norm error using the metaheuristic optimization technique. In this paper, a new approach for the design of IIR fullband digital differentiators using parallel all-pass structure is presented. Magnitude response of obtained fullband differentiators approxi- mate the ideal one in the weighted Chebyshev sense. On the other hand, although phase response linearity of proposed IIR fullband differentiators cannot be controlled, it is a nearly-linear function of frequency at low frequencies. Thus, phase response linearity of low-pass differentiator, obtained by cascading proposed IIR full- band differentiator with the corresponding low-pass filter, is pri- marily affected by the phase response linearity of the utilized low-pass filter. To the best of our knowledge, except differentia- tors presented in [15], design of IIR fullband differentiators using https://doi.org/10.1016/j.dsp.2019.01.026 1051-2004/ 2019 Elsevier Inc. All rights reserved.