When is the least-mean fourth algorithm mean-square stable?

We show that the least-mean fourth (LMF) and the least-mean mixed-norm (LMMN) algorithms are not mean-square stable when the input regressor is Gaussian-distributed. For the LMF algorithm, we propose an upper bound for the algorithm's probability of divergence, given the input and noise statistics, the stepsize and the filter length. We show that the upper bound can also be used for the LMMN algorithm, which is a combination of LMS and LMF.

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