New insights into the normalization of the least mean fourth algorithm

The available normalization of the least mean fourth algorithm is investigated. It is shown that that normalization does not protect the algorithm from divergence when the input power of the adaptive filter increases. The reason of this drawback is that the normalization is done by dividing the weight vector update term by the squared norm of the regressor, while the update term is a fourth order polynomial in the regressor. The paper presents a normalized LMF algorithm that is based on dividing the weight vector update term by the fourth power of the norm of the regressor. This normalization protects the algorithm from divergence when the input power increases. An approximate stability step-size bound of the proposed algorithm is derived. The step-size bound depends on the weight initialization, while it does not depend on the input power of the adaptive filter for non-small signal-to-noise ratio. Simulation results support the analytical results of the paper.