Transient Analysis of Adaptive Filters Algorithm

This paper develops a framework for the mean-square analysis of adaptive filters with general data and error nonlinearities. The approach relies on energy conservation arguments and is carried out without restrictions on the probability distribution of the input sequence. In particular, for adaptive filters with diagonal matrix nonlinearities, we provide closed form expressions for the steady-state performance and necessary and sufficient conditions for stability. We carry out a similar study for long adaptive filters that employ error nonlinearities relying on a weaker form of the independence assumption. We provide expressions for the steady-state error and bounds on the step-size for stability by exploiting the Cramer-Rao bound of the underlying estimation process. 1. ADAPTIVE FILTERING MODEL Consider noisy measurements d ( i ) = uiw"+v(i) , where WO denotes an unknown column vector that we wish to estimate, uj is a row regression vector, and v(i) is measurement noise. Adaptive schemes for estimating W O rely on recursive updates of the general form I = wi + pH(ui)uTf(e( i ) ) , i 2 o I (1) where w, is the estimate of wo at time i , p is the stepsize, and I e(i) = d ( i ) uiwi I (2) is the estimation error. The correction term in (1) is usually expressed in a separable form, H(ui )uTf(e( i ) ) , This work was partially supported by the National Science Foundation under awards ECS-9820765 and CCR-9732376. The work of T. Y . AI-Naffouri was also partially supported by a fellowship from King Fahd University of Petroleum and Minerals, Saudi Arabia. 1 Sign error I sign(e(i)l' ti I 1 ALGORITHM I H(U;) I1