4018 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 8, AUGUST 2007 A Mean-Square Stability Analysis of the Least Mean Fourth Adaptive Algorithm Pedro Inácio Hübscher, José Carlos M. Bermudez, Senior Member, IEEE, and Vítor H. Nascimento, Member, IEEE Abstract—This paper presents a new convergence analysis of the least mean fourth (LMF) adaptive algorithm, in the mean square sense. The analysis improves previous results, in that it is valid for non-Gaussian noise distributions and explicitly shows the depen- dence of algorithm stability on the initial conditions of the weights. Analytical expressions are derived presenting the relationship be- tween the step size, the initial weight error vector, and mean-square stability. The analysis assumes a white zero-mean Gaussian refer- ence signal and an independent, identically distributed (i.i.d.) mea- surement noise with any even probability density function (pdf). It has been shown by Nascimento and Bermudez [“Probability of Divergence for the Least-Mean Fourth (LMF) Algorithm,” IEEE Transactions on Signal Processing, vol 54, no. 4, pp. 1376–1385, Apr. 2006] that the LMF algorithm is not mean-square stable for reference signals whose pdfs have infinite support. However, the probability of divergence as a function of the step size value tends to rise abruptly only when it moves past a given threshold. Our analysis provides a simple (and yet precise) estimate of the region of quick rise in the probability of divergence. Hence, the present analysis is useful for predicting algorithm instability in most prac- tical applications. Index Terms—Adaptive estimation, adaptive filters, adaptive systems, convergence analysis, least mean fourth, stability. I. INTRODUCTION A DAPTIVE algorithms based on higher order moments of the error signal have been shown to perform better mean square estimation than the well-known least mean square (LMS) algorithm in some important applications. The least-mean fourth (LMF) is one of such algorithms [2]. It seeks to minimize the mean fourth error, which is a convex (and, thus, unimodal) function of the adaptive weight vector [2], [3]. Over the years, LMF has been shown to have desirable properties for different applications [2], [4]–[8]. It has been shown that the LMF algorithm can outperform LMS for Gaussian, uniform, and sinusoidal noise distributions [2], [9]. These results have increased the interest in a more detailed analysis of the LMF Manuscript received June 9, 2006; revised October 24, 2006. This work was supported in part by CNPq by Grants No. 308095/2003-0 and 303361/2004-2, and by the FAPESP, by Grant No. 2004/15114-2. The associate editor coordi- nating the review of this manuscript and approving it for publication was Prof. J. Chambers. P. I. Hübscher is with the INPE—National Institute for Space Research, 12227-010, São José dos Campos, SP, Brazil (e-mail: pedro.hubscher@sir. inpe.br). J. C. M. Bermudez is with the Department of Electrical Engineering, Fed- eral University of Santa Catarina, Florianópolis 88040-900, SC, Brazil (e-mail: j.bermudez@ieee.org). V. H. Nascimento is with the Department of Electronic Systems Engineering, University of São Paulo, São Paulo, SP, Brazil (e-mail: vitor@lps.usp.br). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSP.2007.894423 algorithm behavior, since its practical use has been limited in great part due to the lack of good analytical models to predict its performance. In [9], a statistical analysis has been presented, which led to accurate analytical models for the mean and mean-square behavior of the LMF algorithm for small step sizes. Another important aspect of the algorithm’s behavior, which was not addressed in [9], is its stability. There are several approaches to analyze the convergence of adaptive algorithms: deterministic (worst-case) [10], [11] and stochastic (in the mean, in the mean-square [11], and almost- sure [12], [13]). Deterministic approaches such as in [10] tend to be very conservative, requiring the step size to be quite small in order to guarantee stability, while almost-sure analysis may exaggerate, and conclude that an algorithm is stable when its performance is not good at all (an explanation for this can be found in [14]). Walach and Widrow [2] studied the convergence properties of the LMF algorithm in the mean-square sense. Their analysis was restricted to steady state, and the stability limit was not expressed as a function of the initial conditions, even though the reported simulation results indicated this dependence. In [12], the ODE method was used to analyze general fixed-step adaptive algorithms, including LMF. However, no analytical ex- pression is given for the LMF stability conditions. In [15], the authors comment on the dependence of LMF’s stability on its initial conditions. An expression is provided for the maximum adaptation constant for convergence in the mean. However, the analysis in [15] does not consider the mean-square case, and assumes that both the input signal and the measurement noise are Gaussian. Reference [16] has shown that the stability of the LMF algorithm depends on the initial conditions, but such de- pendence was not explicitly determined. In [17] it is shown that LMF stability depends on the initial conditions, and this de- pendence is described analytically, using an elegant argument. However, the analysis in [17] is restricted to Gaussian noise. More recently, a different approach to stability analysis of the LMF algorithm was proposed in [1], which shows that there is always a nonzero probability of divergence in any given real- ization when the input signal has a probability density function (pdf) with infinite support. The probability of divergence was approximated for white Gaussian inputs. This paper presents a new convergence analysis of the LMF algorithm in the mean-square sense. 1 The analysis considers a white zero-mean Gaussian reference signal and an independent, identically distributed (i.i.d.) zero-mean measurement noise with any even pdf. Thus, the derived results are also valid for the important applications in which the LMF algorithm is employed with non-Gaussian measurement noise [2]. Our 1 Initial results on this work have been presented in [18]. 1053-587X/$25.00 © 2007 IEEE