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It is shown that the MSN decrease is almost proportional to the mean square error (MSE). This allows obtaining simple expressions for the steady-state MSE. Also, it allows limiting the amount of time that the MSE takes large values and a curve that limits the MSE of LMS at any given time, independent of the input and background noise signals\u2019 properties. Finally, it is also shown that many complications in the analysis of the LMS and NLMS algorithms can come from variations in the input vector square norm. 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