On the least squares tap adjustment algorithm in adaptive digital echo cancellers

Fast recursive algorithms for updating coefficients in digital echo cancellers can be derived from the well-known method of least squares. Unless zero initial conditions are assumed, the exact initialization of these algorithms is yet unknown. For random data of more than twice the order of the filter, the existence of a unique least squares solution is proven. A constructive recursive procedure in time and order for computing the pseudoinverse solution for the initial steps is derived. Since the data matrix is composed of integers, this technique facilitates the implementation of stable tap-update algorithms