Steady State Analysis of Two-Dimensional LMS Adaptive Filters Using the Independence Assumption

In this paper, we consider the steady state Mean Square Er-ror (MSE) analysis for 2-D LMS algorithm in which the filter's weights are updated in both vertical and horizontal directions using Fornasini and Marchesini (F-M) state space model. The MSE analysis is conducted using the well-known independence assumption. First we show that com-putation of the Weight-Error Correlation Matrix (WECM) for F-M model-based 2-D LMS algorithm requires an ap-proximation for the WECMs at large spatial lags. Then we propose a method to solve this problem. Further dis-cussion is carried out for the special case when the input signal is white Gaussian. It is shown that a more strict con-dition 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 sup-port the obtained analytical results