Short-time instabilities in the LMS algorithm

The least mean square (LMS) algorithm is known to converge in the mean and in the mean square. This does not imply that the algorithm converges strictly at every time step k. In short-time periods the algorithm´s convergence can burst up and cause severe disturbances in speech applications. As long as Gaussian processes are used to drive the filter input and the order of the filter is relatively large, the occurrence of these instabilities is very rare. However, for other statistics this does not need to be true. The paper closes this gap in the literature by discussing potential short-time unstable behavior of the LMS algorithm. For spherically invariant random processes (SIRP), like Gaussian, Laplacian, and K₀, the probabilities for the occurrence of instability at a single time instant k are investigated