Synchronous deautoconvolution algorithm for discrete-time positive signals via I-divergence approximation

We pose the problem of the optimal approximation of a given nonnegative signal y_t with the scalar autoconvolution (x∗x)_t of a nonnegative signal x_t, where x_t and y_t are signals of equal length. The I-divergence has been adopted as optimality criterion, being well suited to incorporate nonnegativity constraints. To find a minimizer we derive an iterative descent algorithm of the alternating minimization type. The algorithm is based on the lifting of the original problem to a larger space, a relaxation technique developed by Csiszár and Tusnády in [Statistics & Decisions (S1) (1984), 205–237] which, in the present context, requires the solution of a hard partial minimization problem. We study the asymptotic behavior of the algorithm exploiting the optimality properties of the partial minimization problems and prove, among other results, that its limit points are Kuhn–Tucker points of the original minimization problem. Numerical experiments illustrate the results.