DocumentCode :
1007866
Title :
Weak convergence and local stability properties of fixed step size recursive algorithms
Author :
Bucklew, James A. ; Kurtz, Thomas G. ; Sethares, William A.
Author_Institution :
Wisconsin Univ., Madison, WI, USA
Volume :
39
Issue :
3
fYear :
1993
fDate :
5/1/1993 12:00:00 AM
Firstpage :
966
Lastpage :
978
Abstract :
A recursive equation that 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 behavior of an unforced deterministic ordinary differential equation (ODE). The ODEs describing the motion of several common adaptive filters are examined in some simple settings, including the least mean square (LMS) algorithm and all three of its signed variants (the signed regressor, the signed error, and the sign-sign algorithms). Stability and instability results are presented in terms of the eigenvalues of a correlation-like matrix. This generalizes known results for LMS, signed regressor LMS, and signed error LMS, and gives new stability criteria for the sign-sign algorithm. The ability 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 an Ornstein-Uhlenbeck process, the properties of which can be described in a straightforward manner
Keywords :
adaptive filters; convergence of numerical methods; correlation theory; differential equations; eigenvalues and eigenfunctions; filtering and prediction theory; least squares approximations; parameter estimation; recursive functions; stability; LMS algorithm; Ornstein-Uhlenbeck process; adaptive filtering algorithms; correlation-like matrix; eigenvalues; fixed step size recursive algorithms; instability results; least mean square; local stability properties; parameter estimate errors; sign-sign algorithms; signed error; signed regressor; stochastic inputs; time varying system; unforced deterministic ordinary differential equation; weak convergence; Adaptive filters; Algorithm design and analysis; Convergence; Differential equations; Filtering algorithms; Least squares approximation; Motion analysis; Motion estimation; Stability; Stochastic processes;
fLanguage :
English
Journal_Title :
Information Theory, IEEE Transactions on
Publisher :
ieee
ISSN :
0018-9448
Type :
jour
DOI :
10.1109/18.256503
Filename :
256503
Link To Document :
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