Simulation of Fractional-Order Low-Pass Filters

The attenuation of standard analog low-pass filters corresponds to a multiple value of -6 decibels per octave. This quantified value is related to the order of the filter. The issue addressed here is concerned with the extension of integer orders to non integer orders, such that the attenuation of a low-pass filter can be continuously adjusted. Fractional differential systems are known to provide such asymptotic behaviors and many results about their simulation are available. But even for a fixed cutoff frequency, their combination does not generate an additive group with respect to the order and they involve stability problems. In this paper, a class of low-pass filters with orders between 0 (the filter is a unit gain) and 1 (standard one-pole filter) is defined to restore these properties. These infinite dimensional filters are not fractional differential but admit some well-posed representations into weighted integrals of standard one-pole filters. Based on this, finite dimensional approximations are proposed and recast into the framework of statespace representations-space representations. A special care is given to reduce the computational complexity, through the dimension of the state. In practice, this objective is reached for the complete family, without damaging the perceptive quality, with dimension 13. Then, an accurate low-cost digital version of this family is built in the time-domain. The accuracy of the digital filters is verified on the complete range of parameters (cutoff frequencies and fractional orders). Moreover, the stability is guaranteed, even for time-varying parameters. As an application, a plugin has been implemented which provides a new audio tool for tuning the cutoff frequency and the asymptotic slope in a continuous way. As a very special application, choosing a one-half order combined with a low cutoff frequency (20 Hz or less), the filter fed with a white noise provides a pink noise generator.