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GAL and LSL Revisited: New Convergence Results

Two popular adaptive lattice algorithms-the gradient adaptive lattice (GAL) and the least squares lattice (LSL)-are revisited. Using the well-known ordinary differential equation (ODE) approach, two new results are presented in this paper. First, a global convergence proof of both algorithms is developed, which is, surprisingly, still missing in literature despite the popularity of the two algorithms. Second, in contrast to a general belief, it is shown that neither algorithm is completely immune to data condition or eigenvalue spread of the input correlation matrix, although significant improvement is achieved over the LMS algorithm.