Convergence analysis of two-dimensional LMS FIR filters

In this paper we consider the steady state mean square error (MSE) analysis for a 2-D LMS algorithm in which the filter´s weights are updated in both vertical and horizontal directions using the Fornasini and Marchesini (F-M) (1980) state space model. The MSE analysis is conducted using the well-known independence assumption. First we shown that computation of the weight-error correlation matrix (WECM) for the F-M model-based 2-D LMS algorithm requires an approximation for the WECMs at large spatial lags. Then, we propose a method to solve this problem. Further discussion is carried out for the special case when the input signal is white Gaussian. It is shown that a more strict condition on the upper bounds of the used step size values is required to ensure the convergence of the 2-D LMS in the MSE sense. Simulation experiments are presented to support the obtained analytical results.