A robust and stable variable step-size design for the least-mean fourth algorithm using quotient form

Abstract

The least-mean fourth (LMF) algorithm is well-known to provide fast convergence and lower steady-state error, especially in non-Gaussian noise environments in contrast to the well-known least-mean square (LMS) algorithm. However, the standard LMF algorithm suffers from two main problems: initial instability due to a larger error term (as it involves a cube of the error term) and steady-state performance degradation due to input and noise statistics. This has motivated researchers to search for different variants of the LMF algorithm. But all the existing solutions are found to be either very sensitive to input and/or noise statistics or suffer from instability. Therefore, in this work, we propose a new variable step-size least-mean fourth (VSSLMF) algorithm with the aim to achieve both robust and stable design. The key idea of the design is based on employing a quotient form of the weighted error energies that achieves significant improvement in terms of the mean square error (MSE) while keeping its inherent superiority over the LMS algorithm in non-Gaussian environments. We also thoroughly investigate the performance of the proposed algorithm analytically in both stationary and non-stationary environments. Consequently, we derive the expressions for excess mean square error (EMSE) and mean square deviation (MSD) in both the transient and the steady-state scenarios. Extensive simulations are carried out to substantiate the theoretical findings.