Iterative algorithms for envelope-constrained filter design.

Abstract

The design of envelope-constrained (EC) filters is considered for the time-domain synthesis of filters for signal processing problems. The objective is to achieve minimal noise enhancement where the shape of the filter output to a specified input signal is constrained to lie within prescribed upper and lower bounds. Traditionally, problems of this type were treated by using the least-square (LS) approach. However, in many practical signal processing problems, this "soft" least-square approach is unsatisfactory because large narrow excursions from the desired shape occur so that the norm of the filter can be large and the choice of an appropriate weighting function is not obvious. Moreover, the solution can be sensitive to the detailed structure of the desired pulse, and it is usually not obvious how the shape of the desired pulse should be altered in order to improve on the solution. The "hard" EC filter formulation is more relevant than the "soft" LS approach in a variety of signal processing fields such as robust antenna and filter design, communication channel equalization, and pulse compression in radar and sonar. The distinctive feature is the set of inequality constraints on the output waveform: rather than attempting to match a specific desired pulse, we deal with a whole set of allowable outputs and seek an optimal point of that set.The EC optimal filter design problems involve a convex quadratic cost function and a number of linear inequality constraints. These EC filtering problems are classified into: discrete-time EC filtering problem, continuous-time EC filtering problem, and adaptive discrete-time EC filtering problem.The discrete-time EC filtering problem is handled using the discrete Lagrangian duality theory in combination with 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. Two iterative algorithms utilizing the simple structure of the gradient flow are developed via discretizing the differential equations. Their convergence properties are derived for a deterministic environment. From the primal-dual relationship, the corresponding sequence of approximate solutions to the original discrete-time EC filtering problem is obtained.The continuous-time EC filtering problem (semi-infinite convex programming problem) is handled using the continuous Lagrangian duality theory and Caratheodory's dimensionality theory. Several important properties are derived and discussed in relation to practical engineering requirements. These include the observation that the continuous-time optimal filter via orthonormal filters has the structure of a matched filter in cascade with another filter. Furthermore, the semi-infinite convex programming problem is converted into an equivalent finite dual optimization problem, which can be solved by the optimization methods developed. Another issue, which relates to the continuous-time optimal filter design problem, is the design of robust optimal EC filters. The robustness issue arises because the solution of the EC filtering problem lies on the boundary of the feasible region. Thus, any disturbance in the prescribed input signal or errors in the implementation of the optimal filter are likely to result in the output constraints being violated. A detailed formulation and a corresponding design method for improving the robustness of optimal EC filters are given.Finally, an adaptive algorithm suitable for a stochastic environment is presented. The convergence properties of the algorithm in a stochastic environment are established.