On closed-loop adaptive noise cancellation

Given the mean limit ordinary differential equation for the stochastic approximation defining the adaptive algorithm for a closed-loop adaptive noise cancellation, 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 infinite-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 coefficients in the transfer function connecting the reference noise signal with the error output. Thus the algorithm is “nearly” a gradient descent algorithm and is error-reducing for large enough dimension. Under broad conditions, iterative averaging can be used to get a nearly optimal rate of convergence