Proposed work deals with the design of a family of stable IIR digital integrators via use of minimax and pole, zero, and constant optimization methods. First the minimax optimization method is used to design a family of second-, third-, and fourth-order digital integrators by optimizing the magnitude response in a min-max sense under the satisfactory condition of constant group delay. Then the magnitude and group delay response is further improved using pole, zero, and constant optimization method. Subsequently, by modifying the transfer function of all of the designed integrators appropriately, new differentiators are obtained. Simulation results show that proposed approach outperforms existing design methods in terms of both magnitude and phase response. 1. Introduction Digital integrators and differentiators are integral parts of many systems like digital signal processing, control, audio, and video processing, communication, and medical applications. Frequency response of an ideal digital integrator is and of ideal digital differentiator is , where and is the angular frequency in radians/second. Due to increasing number of applications involving digital signal processing, variety of requirements that have to be met by digital integrators and differentiators have increased as well. Digital integrators and differentiators can be classified as finite impulse response (FIR) and infinite impulse response (IIR), respectively. For a large number of applications, where high selectivity and efficient processing of digital signal are required, IIR digital integrators and differentiators are preferred. Generally IIR digital integrators and differentiators designs have been proposed by using the Newton-Cotes integration rule [1–13]. These digital operators can be designed directly or by transformation of analog integrators and differentiators like impulse invariance, bilinear transformation, forward difference equation, and backward difference equation [1–3]. Rectangular, Trapezoidal, Simpson 1/3, Simpson 3/8, and Boole are the basic integrators proposed [1–3]. Schneider et al. [4] have used parabolic and cubic numerical integration rules in design of digital integrator. Al-Alaoui [5] has designed digital integrator by using linear interpolation method (mixing of trapezoidal and rectangular integrators); since then, this method has gained immense popularity. Ngo [6] has designed an integrator by applying the -transform on one of the closed-form Newton-Cotes integration rule. Tseng and Lee [7] have used fractional delay in design of digital integrator. Gupta et
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