A Fractional-Order Adaptive Filtering Algorithm in Impulsive Noise Environments
Yongjiang Luo, Jiali Yang, Qiang Zhang, Changlong Wang
Abstract
As classical adaptive filtering algorithms, least mean <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> -norm (LMP) and its variants have good convergence performance in impulsive noise obeying <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> -stable distribution with the characteristic exponent <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha \in (1,2]$ </tex-math></inline-formula> . However, when dealing with the noise of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha \in (0,1]$ </tex-math></inline-formula> , such as Cauchy noise, the performance of these algorithms is greatly degraded or even fail, because the cost functions of which are not first-order differentiable everywhere. To solve such problem, we present a fractional-order LMP (FOLMP) and its normalized version in this brief. By optimizing LMP with the fractional-order gradient, the cost function in FOLMP will be fractional-order differentiable everywhere. Moreover, the mean square stability is analyzed to get the ranges of fractional order and step size for ensuring the stability of FOLMP. Experimental results show that the proposed algorithms have faster convergence speed and better tracking performance than previous algorithms in impulsive noise environments regardless of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> values.