Optimization of adaptive identification for time-varying filters

Adaptive identification in a time-varying context is studied when controlled by the LMS algorithm with constant gain μ, under the assumption of correlated successive input vectors, it is well known by experience that the tracking mean square error (MSE)
\epsilon(\mu)
results from the tradeoff between the gradient part which is μ-increasing and the lag contribution which is μ-decreasing. In this note we clarify the relative roles of the gradient and lag errors by proving their decoupled character. This property relies upon independence between the additive noise at the output of the plant to be identified and the information vector at the plant input. Convergence of the MSE is established rather than assumed. Quantitative evaluations of upper and lower bounds allow an approximate optimization of the gain. In two important cases the optimum is exact. One of these cases is "slow-variations." It is defined in a quantitative manner thanks to the ratio of the "variation" noise to the output additive noise.