Mean square deviation analysis of LMS and NLMS algorithms with white reference inputs

This paper investigates the mean square performance of the least mean square (LMS) and normalized LMS (NLMS) algorithms with white reference inputs. Their closed-form mean square deviation (MSD) expressions for the transient and steady-state regimes are derived. Additionally, bounds on the step-size which guarantee mean square stability are given. It is found that the step-size bound and transient behavior of the LMS and the steady-state MSD of the NLMS depend on the kurtosis of the input signal. Convergence rates and steady-state MSDs of the two algorithms are then compared, which shows that the normalized variant with a large step-size would offer faster convergence rate than the LMS scheme. However, when small step-sizes are employed, the LMS achieves lower steady-state MSD than the NLMS at the same convergence rate.