WEAK-CONVERGENCE AND LOCAL STABILITY PROPERTIES OF FIXED STEP-SIZE RECURSIVE ALGORITHMS

A recursive equation which subsumes several common adaptive filtering algorithms is analyzed for general stochastic inputs and disturbances by relating the motion of the parameter estimate errors to the behavio r of an unforced deterministic ordinary differential equation (ODE). L ocal stability of the ODE implies long term stability of the algorithm while instability of the differential equation implies nonconvergence of the parameter estimates. The analysis does not require continuity of the update equation, and the asymptotic distribution of the paramet er trajectories for all stable cases (under some mild conditions) is s hown to be an Ornstein-Uhlenbeck process. The ODE's describing the mot ion of several common adaptive filters are examined in some simple set tings, including the least mean square (LMS) algorithm and all three o f its signed variants (the signed regressor, the signed error, and the sign-sign algorithms). Stability and instability results are presente d in terms of the eigenvalues of a correlation-like matrix. This gener alizes known results for LMS, signed regressor and signed error LMS, a nd gives new stability criteria for the sign-sign algorithm. The abili ty of the algorithms to track moving parameterizations can be analyzed in a similar manner, by relating the time varying system to a forced ODE. The asymptotic distribution about the forced ODE is again (under similar conditions) an Ornstein-Uhlenbeck process, whose properties ca n be described in a straightforward manner.