Fractional-Order Digital Filters of Complex Orders

This work introduces a new class of fractional-order digital filters of complex orders. The complex-order dynamics are first approximated by frequency-dependent minimum-phase continuous transfer functions within a frequency band. The indirect discretization method is then used to discretize the approximated form of the fractional-order Laplacian integro-differential variable of complex orders, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$s^{\pm\alpha\pm j\beta}$</tex>, to obtain a rational z-transfer function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$H(\mathrm{z},\ \pm\alpha,\ \pm\beta)$</tex>. The positive (negative) complex orders of the fractional-order differentiators (integrators) provide a leading phase at high frequency, respectively, which add new features for a new class of filters and controllers of complex orders. The main points of this work are verified via numerical simulations.