Necessary and sufficient conditions and robustness for convergence of adaptive filtering algorithms

Abstract

The necessary and sufficient conditions for the almost sure convergence of adaptive filtering algorithms based on (a/n) stepsize are established. Assuming that the covariance of the output signal (Y/sub n/) and the cross-covariance of (Y/sub n/) and the reference signal ( psi /sub n/) are constant, if (Y/sub n/ Y/sub n/') satisfies a law of large numbers (where ' denotes transpose), then the necessary and sufficient condition for almost sure convergence of adaptive filtering algorithms is that (Y/sub n/ psi /sub n/') also satisfies the law of large numbers. Moreover when there exists a small deviation from the law of large numbers for (Y/sub n/ Y/sub n/') and (Y/sub n/ psi /sub n/'), there is also a bounded deviation dependent on the former from the convergence of the algorithms, and the latter tends to zero as the former goes to zero. >