H2 design of adaptation laws with constant gains

We present a method for optimizing adaptation laws that are generalizations of the LMS algorithm. The proposed technique has been applied successfully for designing estimators of rapidly time-varying mobile radio channels. The estimators apply time-invariant filtering on the instantaneous gradient. Time-varying parameters of linear regression models are estimated in situations where the regressors are stationary or have slowly time-varying properties. The structure and gains of these adaptation laws are optimized in MSE for time-variations modeled as correlated stochastic processes. The aim is to systematically use such prior information to provide filtering, prediction or fixed lag smoothing estimates for arbitrary lags. Our design method is based on a novel transformation that recasts the adaptation problem into a Wiener filter design problem. The filter works in open loop for slow parameter variations while a time-varying closed loop is important for fast variations. In closed loop, the filter design is performed iteratively. The solution at one iteration can be obtained by a bilateral Diophantine polynomial matrix equation and a spectral factorization. For white noise, the Diophantine equation has a closed-form solution. When one filter is known, a set of predictors and smoothers, up to a predefined prediction horizon or smoothing lag, is obtained by analytical expressions.