A Note on Closed Loop Adaptive

Given the mean limit ODE for the stochastic approximation deening the adaptive algorithm for a closed loop ANC, we characterize the limit points. Under appropriate conditions, it is shown that as the dimension of the weight vector increases, the sequence of corresponding limit points converges in the sense of l2 to the innnite dimensional optimal weight vector. Also, the limit point of the algorithm is nearly optimal if the dimension of the weight vector is large enough. The gradient of the mean square error with respect to the weight vector, evaluated at the limit, goes to zero in l1 and l2 as the dimension increases, as does the gradient with respect to the coeecients in the transfer function connecting the reference noise signal with the error output. Thus the algorithm is \nearly" a gradient descent algorithm and is indeed error reducing for large enough dimension. Under broad conditions, iterate averaging can be used to get a nearly optimal rate of convergence. Adaptive noise cancellation (ANC) methods have been successful in many applications in control and communications. A brief outline is in 5, 18]. In the basic model, one is given a process p n which consists of a signal and noise, and an auxiliary process z n which is correlated with the noise, but is independent of the signal. Given an integer m > 0 and a parameter vector = (0