Exact tracking analysis of the NLMS algorithm for correlated Gaussian inputs

This work presents an exact tracking analysis of the Normalized Least Mean Square (NLMS) algorithm for circular complex correlated Gaussian inputs. Unlike the existing works, the analysis presented neither uses separation principle nor small step-size assumption. The approach is based on the derivation of a closed form expression for the cumulative distribution function (CDF) of random variables of the form (∥u∥
_D1
²
)(∥u∥
_D2
²
)
^-1
where u is a white Gaussian vector and D
₁
and D
₂
are diagonal matrices and using that to derive the first and second moments of such variables. These moments are then used to evaluate the tracking behavior of the NLMS algorithm in closed form. Thus, both the steady-state mean-square-error (MSE) and mean-square-deviation (MSD )tracking behaviors of the NLMS algorithm are evaluated. The analysis is also used to derive the optimum step-size that minimizes the excess MSE (EMSE). Simulations presented for the steady-state tracking behavior support the theoretical findings for a wide range of step-size and input correlation.