Dependence of the Stability of the Least Mean Fourth Algorithm on Target Weights Non-Stationarity

The paper investigates a new stability problem of the least mean fourth (LMF) algorithm, which is the dependence of the algorithm stability on the time-variation of the target weights of the adaptive filter. The analysis is done in the context of tracking a Markov plant with a stationary white Gaussian input. It is found that the algorithm diverges if the mean square increment of the plant parameter vector exceeds a threshold value that depends on the step-size, input variance, and noise moments. The paper also derives a closed form of the steady-state mean square deviation without the usual assumption of a strong noise. Comparison of the tracking capabilities of the LMF and LMS algorithms is provided. The comparison is done in terms of the minimum mean square deviation attained by each algorithm over the stability range of its step-size. Gaussian, uniform, and binary distributions of the noise are considered. Conditions that make one algorithm outperform the other are determined. Analytical results are supported by simulations.