An easy demonstration of the optimum value of the adaptation constant in the LMS algorithm [FIR filter theory]

The least mean squares (LMS) is the most widely used algorithm among those proposed to adapt the coefficients of an FIR filter in order to minimize the mean-square error (MSE) between its output and the desired signal. Since the introduction of the LMS algorithm, many variants have been proposed to improve its performance. Doubtless, the most popular is the normalized LMS algorithm, which uses a value for the adaptation constant that assures the fastest convergence. This correspondence shows a new demonstration of the algorithm based on a mathematical approach easier than that usually proposed.