NLMS is More Robust to Input-Correlation Than LMS: A Proof

In this work, we comparatively analyze the least mean squares (LMS) algorithm and the normalized least mean squares (NLMS) algorithm. We use the input moment matrices for comparison as the mean-square behavior of both algorithms is determined by the input moment matrices. First, we derive the closed-form expressions of the input moment matrices of the NLMS. Second, we do a numerical and theoretical comparison of the input moment matrices of the LMS and the NLMS. The analysis shows why the performance of the NLMS is less sensitive to the changes in eigenvalue-spread (of the input-correlation matrix) than the LMS.