An Exact Expectation Model for the LMS Tracking Abilities

Nonstationary environments are ubiquitous in communications and acoustic systems. The ability to track their dynamics is one of the most desirable features of adaptive processing algorithms. The designers of these algorithms employ guidelines derived from stochastic analyses to adjust user-defined parameters to maximize performance or avoid stability issues. It is therefore important that analyzes of adaptive processing algorithms take into account the built-in sophisticated learning capabilities. This work presents a comprehensive model of the performance of the least mean square algorithm, operating under Markovian time-varying channels. Our advanced analysis considers both transient and steady-state regimes. Furthermore, in our analysis, the popular independence assumption is not adopted, resulting in a stochastic model which is accurate even when: (i) the step size is not infinitesimally small; or (ii) when the unknown system presents a high nonstationary degree. In addition, our evaluation is also able to provide a deterministic theoretical step-size sequence that optimizes algorithmic performance, as well as an accurate step size upper bound that guarantees algorithm stability. Computer simulations performed are in accordance with our theoretical predictions.