A novel tracking analysis of the normalized least mean fourth algorithm

In this work, the tracking analysis of the Normalized Least Mean Fourth (NLMF) algorithm is investigated for a random walk channel under very weak assumptions. The novelty of this work resides in the fact that no restrictions are made on the dependence between the input successive regressors, the dependence among input regressor elements, the length of the adaptive filter, the distribution of noise and filter's input. Moreover, in our approach, there is no restriction made on the step size value and therefore the analysis holds for all the values of the step size in the range of stable NLMF algorithm. The analysis is based on a recently proposed performance measure called effective weight deviation vector which is the component of weight deviation vector in the direction of input regressor. In this paper, asymptotic time-averaged convergence for the mean square effective weight deviation, mean absolute excess estimation error, and the mean square excess estimation error for the NLMF algorithm are established. Finally, a number of simulation results are carried out to corroborate the theoretical findings.