Performance analysis of the deficient length LMS adaptive algorithm

In almost all analyzes of the least mean-square (LMS) finite impulse response (FIR) adaptive algorithm, it is assumed that the length of the adaptive filter is equal to that of the unknown system impulse response. However, in many practical situations, a deficient length adaptive filter, whose length is less than that of the unknown system, is employed, and analysis results for the sufficient length LMS algorithm are not necessarily applicable to the deficient length case. Therefore, there is an essential need to accurately quantify the behavior of the LMS algorithm for realistic situations where the length of the adaptive filter is deficient. In this paper, we present a performance analysis for the deficient length LMS adaptive algorithm for correlated Gaussian input data and using the common independence assumption. Exact expressions that completely characterize the transient and steady-state mean-square performances of the algorithm are developed, which lead to new insights into the statistical behavior of the deficient length LMS algorithm. Simulation experiments illustrate the accuracy of the theoretical results in predicting the convergence behavior of the algorithm.