Assessment of the efficiency of the LMS algorithm based on spectral information

The LMS algorithm is used to find the optimal minimum mean-squared error (MMSE) solutions for a wide variety of problems. Unfortunately, its convergence speed depends heavily on its initial conditions when the autocorrelation matrix R of its input vector has a high eigenvalue spread. In many applications such as system identification or channel equalization, R is Toeplitz. In this paper we exploit the Toeplitz structure of R to show that when the weight vector is initialized to zero, the convergence speed of LMS is related to the similarity between the input PSD and the power spectrum of the optimum solution.