Design of robust envelope-constrained filter with orthonormal bases

In the continuous-time envelope-constrained (EC) filtering problem using an orthonormal filter structure, the aim is to synthesize an orthonormal filter such that the noise enhancement is minimized while the noiseless output response of the filter with respect to a specified input signal stays within the upper and lower bounds of the envelope. The noiseless output response of the optimum filter to the prescribed input signal touches the output boundaries at some points. Consequently, any disturbance in the prescribed input signal or error in the implementation of the optimal filter will result in the output constraints being violated. In this paper, we review a semi-infinite envelope-constrained filtering problem in which the constraint robustness margin of the filter is maximized, subject to a specified allowable increase in the optimal noisy power gain. Using a smoothing technique, it is shown that the solution of the optimization problem can be obtained by solving a sequence of strictly convex optimization problems with integral cost. An efficient optimization algorithm is developed based on a combination of the golden section search method and the quasi-Newton method.