Envelope-constrained filters: adaptive algorithms

In this paper, we consider an envelope-constrained (EC) optimal filter design problem involving a quadratic cost function and a number of linear inequality constraints. Using the duality theory and the space transformation function, the optimal solution of the dual problem can be computed by finding the limiting point of an ordinary differential equation given in terms of the gradient flow. An iterative algorithm is developed via discretizing the differential equation. From the primal-dual relationship, the corresponding sequence of approximate solutions to the original EC filtering problem is obtained. Based on these results, an adaptive algorithm is constructed for solving the stochastic EC filtering problem in which the input signal is corrupted by an additive random noise. For illustration, a practical example is solved for both noise-free and noisy cases.