A Fractional-Order Logarithmic Adaptive Filtering Algorithm Under $\alpha$-Stable Noise

In practical applications, noise is more appropriately modelled as an a-stable random process than as a Gaussian process. To cope with a-stable noise, a fractional-order stochastic gradient descent (SGD) method is invented to adaptive filtering (AF) algorithms. When both input signal and noise follow α-stable random processes, the AF algorithms based on mean square error (MSE) criteria can be affected. Nevertheless, the fractional-order AF (FoAF) algorithm can effectively address this issue. For the purpose of better stability performance of the FoAF algorithm, different cost functions (CFs) are introduced into fractional-order SGD method. Based on the fractional-order derivative of a logarithmic CF (LCF), a fractional-order logarithmic adaptive filtering (FoLAF) algorithm is formed in this condition. Trials show that the FoLAF algorithm achieves faster convergence, excellent steady-state performance and superior tracking performance.