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In this paper a simple and an efficient approach for approximating the fractional delay operator z-a (0 <; a <; 0.5) using digital infinite impulse response (IIR) filters is proposed. In this technique, the coefficients of the closed form digital IIR filter derived for the approximation of the fractional delay operator, in a given frequency band, are based on the approximation of fractional order systems. First, analog rational function approximation, for a given frequency band, of the fractional power pole (FPP) is given. Then the Tustin (bilinear) generating function is used to digitize the FPP to obtain a closed form IIR digital filter which approximates the digital fractional delay operator z-a for 0 <; a <; 0.5. Finally, an example has been presented to illustrate the effectiveness of the proposed design technique.