Influence of Methods Approximating Fractional-Order Differentiation on the Output Signal Illustrated by Three Variants of Oustaloup Filter

Fractional-order (FO) differential equations are more and more frequently applied to describe real-world applications or models of phenomena. Despite such models exhibiting high flexibility and good fits to experimental data, they introduce their inherent inaccuracy related to the order of approximation. This article shows that the chosen model influences the dynamic properties of signals. First, we calculated symbolically the steady-state values of an FO inertia using three variants of the Oustaloup filter approximation. Then, we showed how the models influence the Nyquist plots in the frequency domain. The unit step responses calculated using different models also have different plots. An example of FO control system evidenced different trajectories dependent on applied models. We concluded that publicized parameters of FO models should also consist of the name of the model used in calculations in order to correctly reproduce described phenomena. For this reason, the inappropriate use of FO models may lead to drawing incorrect conclusions about the described system.

Design of Variable Fractional Delay Filter using FIR Filter Approximation

Beamforming plays an important role in the field of wireless communication. Beamforming means combination of a radio frequency (RF) signals from multiple antennas to form a single direction beam. This technique improves the quality of communication and reduces the interference of signal. In beam forming technique, different phases signals can be achieved with different signals and the received phase delay signals are converted into same phase, multiply with weight factor and combined this signals to form a beam in desired direction. The required phase delays are generated by using a Variable fractional delay filter. Variable fractional delay filter is design by using a direct form of a FIR filter structure. Variable fractional delay filter is calculated by two different phase signals from digital antennas and those two different phase signals are converted to in- phase and added together to form a beam forming. As the order of the filter increases, the delay also increases. The filter coefficients of the variable fractional delay filter are calculated my using a Lagrange interpolation method. The variable fractional delay filter is designed by using software Xilinx version 14.3