Is negative step size lms algorithm stable operation possible?

The Least Mean Squares (LMS) algorithm and its variants are the most popular choice in many systems that require gradient-based adaptation. Examples of such applications include system identification, line enhancement, line equalization, adaptive echo cancellation and active noise cancellation. The only drawback of the LMS-family algorithms is the need for careful step size choice. Too small step size, although giving good excess mean squared error (MSE), results in slow convergence speed. Too large step size results in large excess MSE and may lead to the loss of convergence and instability. Therefore, there are many theoretical studies of the algorithm behavior with the aim to provide useful bounds on the step size. Regardless of the analytic method applied, the common result of many investigations seems to be the lower bound for LMS-like algorithms convergence given as zero (e.g. μ > 0). In this paper we show that at least one of the LMS-family algorithms- The Leaky LMS algorithm, is capable of stable operation even if the step size has (small) negative value. Theoretical derivations of the necessary stability condition has been validated by a number of simulations.