A unifying view of error nonlinearities in LMS adaptation

This paper presents a unifying view of various error nonlinearities that are used in least mean square (LMS) adaptation such as the least mean fourth (LMF) algorithm and its family and the least-mean mixed-norm algorithm. Speci cally, it is shown that the LMS algorithm and its errormodi ed variants are approximations of two recently developed optimum nonlinearities which are expressed in terms of the additive noise probability density function (pdf). This is demonstrated through an approximation of the optimum nonlinearities by expanding the noise pdf in a Gram-Charlier series. Thus a link is established between intuitively proposed and theoretically justi ed variants of the LMS algorithm. The approximation has also a practical advantage in that it provides a trade-o between simplicity and more accurate realization of the optimum nonlinearities.