Stochastic Approximation with Correlated Data

Steady-state MSE convergence of LMS adaptive filters with deterministic reference inputs with applications to biomedical signals

“…Equivalently, the steady-state weight vector is an unbiased estimate of . The weight error vector at any occurrence time instant of the th occurrence can be written using 3as (11) In the complete expansion case, is due to the noise because the truncation error is null, and . The minimum error can be calculated from (1) as .…”

“…In addition, the term is also null if zero-mean white noise is assumed. Hence 14The weight error correlation matrix at the end of the th occurrence can be calculated from (11) as 15The second and third terms are null if zero-mean white noise is assumed.Therefore, in that case (16)…”

“…In summary, when incomplete expansions are used and the transition matrix products in (4) have a more complex description. The steady-state weight error vector can be calculated from (11) by taking the limit as . The first term will converge to the null matrix for small values of because (23) The difference is that two different driving terms must be considered now because has two components (1).…”

“…The minimum error is now time-variant (26) The excess MSE can be calculated using (12) and (13). The weight error correlation matrix can be calculated by multiplying (11) and applying the expected value (27) To calculate the steady-state value, we take the limit as . All the first three terms are transient (null at steady-state) if the step-size is selected to accomplish (23).…”

“…The discrepancies between theoretical results based on this assumption and the true algorithm behavior was investigated in [10] and found to be relatively small. A more realistic assumption (less strong) has also been used by several authors [11], [12], where statistically dependent reference inputs are considered.…”