By L. Morales
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Jimaa, “An efficient and effective variable step size NLMS algorithm”, 42nd Asilomar conference on signals, systems and computers, CA, USA, October 26-29, 2008.  Ling Quin and M. G. Bellanger,” Convergence analysis of a variable step-size normalized adaptive filter algorithm”, Proc. 4, 1996.  Y. Wang, C. Zhang, and Z. Wang,“ A new variable step-size LMS algorithm with application to active noise control“, Proc. IEEE ICASSP, Vol. 5, pp. 573-575, 2003. Widrow, The Least Mean Fourth (LMF) Adaptive Algorithm and its Family" IEEE Transactions on Information Theory, 30(2), 275-283, 1984.
1 is not always satisfied for all kinds of adaptive filters. 1 can not be a strict mean square stability condition for an adaptive filter with error nonlinearity. 26 Adaptive Filtering where O ea denotes third and higher-power terms of ea . 2, we get where TEMSE is defined by (5). 2, we get 1 2 Eq e , e v TEMSE 2 Eq e Eq v (28) Substituting (27) and (28) into (11), and using Lemma 2, we can obtain (24), where parameters A, B, C are defined by (18b). 1 is satisfied, we can obtain (16) for real-valued cases.
61) Example 6: Consider the cases with g u 1 . g. 1) in . Example 7: In Gaussian noise environments, based on (46) and (52), we can obtain EMSE 2 u 2 v2 k3 v4 k4 2 v6 k0 v2 u 2 u k1 v2 k2 2 v4 . (62) where k0 3, k1 12, k2 45, k3 6, k4 15 for real-valued cases, k0 2, k1 8, k2 18, k3 4, k 4 6 for complex-valued cases. 3 Normalized type algorithms Being similar with LMF algorithm -, there are the stability and convergence problems in the LMP algorithm with p 2 , LMMN algorithm, and other adaptive filters with error nonlinearities.