A Fractional-Order Adaptive Filtering Algorithm in Impulsive Noise Environments

As classical adaptive filtering algorithms, least mean p-norm (LMP) and its variants have good convergence performance in impulsive noise obeying α-stable distribution with the characteristic exponent α(1,2. However, when dealing with the noise of α(0,1, 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 α values.