Design of Bandpass and Bandstop Infinite Impulse Response Filters Using Fractional Derivative

Abstract

In this paper, a new design method for digital bandpass and bandstop infinite impulse response filters with nearly linear-phase response is proposed. In this method, the phase response of an all-pass filter is optimized in the frequency domain to yield less passband error Err and stopband error Er <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">9</sub> with optimal stopband attenuation A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">9</sub> . To achieve high accuracy in passband and stopband regions, fractional derivative (FD) constraints are evaluated in the respective regions, and the filter coefficients are computed using the Lagrange multiplier method. The behavior of fidelity parameters measured in terms of E <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">rr</sub> , Er <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">9</sub> , and phase error Er <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ph</sub> is multimodel w.r.t. FD values. Therefore, modern heuristic technique, known as cuckoo search optimization is used for determining the optimal value of FDs and reference frequency simultaneously to minimize the fitness function, which is constructed as a sum of the squared error in passband and stopband. The designed filter yields up to 60% reduction in Err and Erph in the case of bandpass filter. Meanwhile, the response of filter is not degraded due to the finite word length effect.